Discussion 4
Andrew Gregory
2024-02-07
Week 4 Discussion - Central Limit Theorem
- Please Google and describe Law of Large Numbers. in your own words.
Law of Large Numbers (LLN) is the theory that states that your sample statistics, such as sample mean, become more and more reliable as you collect more and more random samples. In theory, if you collect an infinite number of samples, LLN states that the sample mean will more closely resemble the mean of the population.
- Please explain CLT in your own words.
Simply put, as the sample size increases, CLT states that a sample distribution variable approximates a normal distribution. In practice, one use of CLT is to analyze the returns of a stock. The most common uses of the central limit theorem include, estimating population parameters using sample statistics, confidence intervals and margin of error, and hypothesis testing. Central Limit Theorem is also typically used in situations of quality control, election polling, and medical research. CLT is useful in these situations because CLT allows one to aggregate the mean of the sampling distribution and assume the sampling distribution normally-distributed.
Sources:
https://www.investopedia.com/terms/c/central_limit_theorem.asp
https://medium.com/analytics-vidhya/law-of-large-numbers-vs-central-limit-theorem-7819f32c67b2
- What are the similarities and differences between LLN and CLT? Write a few lines.
LLN and CLT are the basic framework to conduct hypothesis testing. Since CLT states that the sampling distribution is approximately normally distributed if the sample size is large enough, the LLN provides the parameters that a sample mean of a specific size will tend to be close to the population mean as an infinite number of samples are collected. Therefore, the difference between Central Limit Theorem and Law of Large Numbers is that LLN is about the mean of a curve, while CLT is the theory related to the sampling distribution of a curve.
- Pick up any distribution apart from normal, uniform or poisson. You can Wikipedia about the distribution and/or read how to implement the distribution in R (what parameters are required to generate the distribution). Please describe this distribution first in 5 lines.
Binomial Distribution is a distribution that summarizes the probability that a value will take one of two independent values. Binomial Distributions have a finite number of independent trials, and each trial has only one of the two possible results either success or failure. Binomial Distributions are often used to depict the probability of obtaining a certain number of successes in a fixed number of trials. One use of binomial distribution includes conducting analysis of the probability of a certain number of successful investments in a portfolio. In R, the arguments required to generate a Binomial distribution are, number of observations (n), number of trials (size), and probability of success on each trial (prob).
# example of binomial paremeters
n <- 10000
p <- 0.5
set.seed(13579) # for reproducibility
binomial_values <- rbinom(100, n, p)
binomial_values## [1] 4921 4918 5023 4905 5012 4996 4945 5037 5058 5001 4960 4988 4935 4993 5031
## [16] 4989 5087 4973 5016 4969 4963 4909 4967 5001 4876 5082 4999 4964 5030 5027
## [31] 4943 4984 4978 5063 4927 5051 5050 5021 5094 5105 4929 4986 4927 4962 5015
## [46] 4985 5062 5029 5003 5068 4934 4999 4921 5093 5044 5008 5009 4963 4995 4983
## [61] 4978 5065 5053 4921 5001 5036 5090 5004 5037 5094 4974 5014 5056 5076 5017
## [76] 4915 4989 5025 5007 5037 4974 5008 5090 4980 5046 4998 4984 4987 5010 4895
## [91] 5075 5000 5047 4923 5111 4984 4945 4977 4987 49585A. Then, apply the CLT on the sample mean of this chosen distribution in R (adapt our class R code, or you can find an alternative code on the web too).
# Step 1 Create the distribution
set.seed(13579) # Set seed for reproducibility
N <- 10000 # Specify sample size
mybinom <- rbinom(N, size = 100, prob = 0.5)
mybinom # check if created correctly## [1] 45 44 55 43 35 47 56 52 49 51 47 50 51 54 53 48 57 55 51 52 48 44 46 49
## [25] 50 59 55 46 54 51 61 45 49 45 49 50 40 52 44 60 45 50 55 51 43 49 58 43
## [49] 44 51 56 42 53 50 53 47 44 49 42 56 60 51 53 49 57 51 51 51 51 54 53 56
## [73] 45 48 49 40 49 50 51 50 50 55 41 52 55 47 47 52 49 51 47 49 46 53 51 56
## [97] 50 47 49 51 41 49 50 52 56 57 48 54 51 41 46 54 54 43 52 56 49 56 47 52
## [121] 48 51 50 53 40 43 48 44 49 46 46 57 42 49 47 53 49 54 37 54 52 49 50 42
## [145] 47 48 55 61 54 53 33 48 52 58 46 53 48 54 48 56 50 43 48 46 50 57 38 37
## [169] 50 49 50 48 56 52 51 50 52 58 53 49 52 44 45 49 45 51 56 52 41 50 51 52
## [193] 53 46 51 43 49 55 46 44 51 37 56 48 55 50 54 44 59 48 47 44 53 54 41 51
## [217] 51 46 48 51 47 43 58 51 51 47 49 43 55 53 50 49 56 56 54 49 55 48 53 63
## [241] 49 53 50 48 48 59 41 40 49 48 44 42 64 53 45 50 50 46 52 52 52 58 52 52
## [265] 49 40 57 59 52 50 52 44 50 52 48 41 58 43 53 48 59 50 57 48 45 45 56 42
## [289] 47 50 52 51 44 55 52 47 47 50 42 48 49 56 55 55 46 60 47 43 44 52 56 43
## [313] 54 43 54 47 49 51 40 61 37 41 47 48 46 52 49 50 54 43 49 56 50 55 52 53
## [337] 54 49 49 38 48 51 50 60 60 46 49 56 53 44 42 43 45 43 46 56 48 41 51 55
## [361] 52 51 50 60 46 60 53 52 51 56 39 49 53 60 44 46 43 42 51 56 61 49 48 55
## [385] 44 53 50 53 58 47 42 44 51 55 49 57 53 45 48 35 52 57 57 49 51 53 38 47
## [409] 52 52 59 57 54 47 49 59 58 47 54 55 55 49 57 47 54 52 51 45 52 43 44 50
## [433] 61 53 55 45 47 44 41 41 47 53 49 48 44 50 60 57 53 49 46 45 47 50 46 47
## [457] 56 50 53 53 48 57 55 49 49 50 46 54 45 52 54 45 44 51 52 50 54 51 51 42
## [481] 47 48 52 49 47 58 42 58 51 58 38 50 44 56 47 50 50 50 49 45 53 47 53 46
## [505] 49 47 48 46 51 50 53 47 50 51 58 41 45 49 49 46 47 40 34 55 51 48 50 52
## [529] 52 52 48 53 51 52 56 48 45 52 49 47 59 49 55 54 53 43 56 48 46 38 54 49
## [553] 46 47 58 47 47 54 53 59 44 45 48 53 49 55 42 50 53 49 53 52 48 46 58 53
## [577] 38 47 57 53 45 58 50 43 50 47 45 52 47 59 42 57 44 51 55 51 58 46 49 62
## [601] 53 50 58 50 43 48 47 54 43 47 49 56 54 56 52 53 44 39 58 41 52 58 47 52
## [625] 51 37 56 49 52 56 47 61 50 49 51 55 59 38 49 53 42 51 51 59 53 55 48 52
## [649] 55 50 47 47 46 50 44 51 42 53 53 49 40 58 47 48 53 50 55 46 48 47 49 49
## [673] 48 55 45 48 44 52 45 45 53 56 51 48 45 44 53 53 41 47 46 55 58 54 43 53
## [697] 59 60 49 53 46 46 50 49 49 52 44 56 58 53 49 38 53 53 44 52 53 58 52 60
## [721] 46 46 57 52 50 50 51 53 50 56 49 52 56 52 42 49 54 51 46 45 40 55 42 52
## [745] 47 56 54 43 45 36 47 57 51 45 57 51 49 46 55 51 49 55 54 45 53 53 50 49
## [769] 52 48 48 53 40 45 48 47 52 45 55 56 59 38 53 53 47 49 48 52 49 49 46 51
## [793] 56 51 48 48 47 49 51 50 50 45 49 54 58 53 49 47 47 45 43 55 54 51 53 58
## [817] 48 41 56 53 53 61 48 44 48 49 58 47 48 60 51 52 53 40 48 48 49 54 42 61
## [841] 43 48 47 54 50 53 51 47 54 47 51 54 61 53 48 48 55 43 64 49 52 49 49 46
## [865] 68 40 48 57 42 53 50 52 41 45 52 54 44 45 47 51 49 56 51 56 51 56 39 52
## [889] 51 47 52 47 53 49 49 50 47 51 46 50 46 55 50 42 47 52 50 49 57 47 60 48
## [913] 55 52 55 42 50 45 56 56 42 42 47 54 54 56 62 57 53 52 47 45 49 49 51 42
## [937] 50 44 47 45 54 47 45 51 54 61 48 43 44 52 57 45 55 51 50 52 50 53 41 49
## [961] 57 46 36 43 47 55 40 47 44 58 55 60 43 59 50 55 52 53 46 47 49 54 57 43
## [985] 44 53 57 50 45 58 49 40 51 46 51 43 47 48 44 46 49 54 50 43 54 52 41 52
## [1009] 54 54 50 57 50 45 51 52 44 51 48 50 49 50 48 53 53 51 47 48 50 52 46 57
## [1033] 46 45 45 51 50 49 52 51 49 46 55 53 46 42 52 47 41 54 58 58 46 50 52 52
## [1057] 55 53 45 54 45 48 49 57 50 44 47 45 54 46 49 52 45 49 54 51 46 46 46 51
## [1081] 54 48 51 47 45 43 55 43 45 51 50 49 46 51 58 46 49 47 47 44 55 52 55 53
## [1105] 55 43 47 48 46 54 58 52 56 48 49 52 46 52 59 54 52 51 54 45 58 47 57 47
## [1129] 54 46 55 49 47 60 48 49 53 59 60 57 59 51 49 45 43 45 50 51 43 40 54 53
## [1153] 51 49 45 42 50 50 49 49 42 48 54 55 55 46 49 63 50 53 50 56 55 48 53 54
## [1177] 48 51 58 52 39 50 55 50 53 54 42 53 53 43 45 54 54 49 44 50 46 41 61 50
## [1201] 54 56 47 48 49 51 47 54 45 49 52 59 48 57 46 51 48 59 51 55 48 48 49 61
## [1225] 50 43 50 60 58 45 45 49 52 54 54 48 53 50 45 43 48 49 55 54 47 55 52 37
## [1249] 47 50 53 43 50 54 56 54 50 52 46 59 58 49 51 44 43 41 50 40 43 48 44 41
## [1273] 54 49 54 42 40 52 51 53 51 52 50 52 54 52 49 52 47 53 47 50 50 55 61 47
## [1297] 51 51 54 49 49 51 54 45 55 56 48 55 59 51 53 46 53 45 45 50 49 44 43 46
## [1321] 51 57 49 50 50 47 49 47 47 47 53 48 60 50 53 49 39 49 60 47 51 50 56 51
## [1345] 48 51 45 53 42 55 48 47 48 43 48 52 51 57 45 53 58 59 57 60 59 45 62 52
## [1369] 55 54 52 56 53 51 49 37 56 51 58 46 48 50 44 56 55 55 60 50 51 56 57 50
## [1393] 45 44 46 52 49 49 55 44 44 56 46 56 43 52 58 48 47 50 58 51 45 50 57 48
## [1417] 41 47 47 58 50 56 43 49 47 38 53 54 51 52 54 50 50 58 41 45 49 45 47 41
## [1441] 43 56 47 54 50 49 45 58 52 50 48 47 51 54 51 51 51 50 49 50 52 51 45 51
## [1465] 50 47 54 61 43 51 40 49 52 47 47 46 46 44 55 41 51 50 45 36 52 46 51 49
## [1489] 47 46 46 51 40 53 52 48 50 46 54 44 53 49 45 63 49 50 49 45 52 56 57 57
## [1513] 44 51 45 39 49 44 57 52 48 49 55 55 56 62 46 44 60 55 47 50 42 43 52 48
## [1537] 48 48 49 54 52 55 51 52 46 56 48 53 46 51 47 49 50 48 49 58 51 47 52 54
## [1561] 56 48 53 53 58 49 52 53 53 46 38 52 57 53 49 45 51 55 57 56 50 39 51 51
## [1585] 44 43 51 46 53 55 51 54 57 48 56 57 52 44 50 56 45 44 50 48 46 42 51 53
## [1609] 48 54 47 50 47 44 51 42 46 52 54 44 55 50 45 52 52 55 53 41 56 58 48 54
## [1633] 54 56 50 48 46 53 53 47 45 40 46 49 43 51 52 55 47 46 61 48 49 52 46 54
## [1657] 52 54 56 45 50 49 51 47 41 48 50 55 53 57 45 44 49 50 43 44 46 54 59 51
## [1681] 47 48 47 51 56 47 48 46 40 52 47 53 47 48 44 57 52 45 52 47 49 52 52 54
## [1705] 59 40 47 47 54 59 52 50 42 47 41 54 49 49 49 59 47 52 56 49 50 48 51 48
## [1729] 56 48 51 50 47 49 52 51 57 46 55 49 47 49 52 47 47 54 61 56 48 56 46 49
## [1753] 48 48 44 52 60 58 58 60 47 49 47 55 48 54 57 64 44 52 58 50 59 42 50 54
## [1777] 51 47 58 57 48 53 49 49 52 58 52 51 54 52 49 45 50 53 52 49 45 48 42 57
## [1801] 50 59 45 50 47 52 44 51 60 47 40 54 49 48 57 51 47 46 53 45 47 53 53 50
## [1825] 42 46 55 47 39 49 51 49 46 46 49 46 38 46 48 43 52 47 57 42 45 55 52 59
## [1849] 52 57 52 49 46 53 43 41 52 47 55 50 39 50 44 54 51 49 57 47 45 57 42 55
## [1873] 54 52 51 55 61 46 49 49 45 46 44 40 58 51 48 50 43 48 48 46 50 47 46 50
## [1897] 51 43 49 49 52 60 58 57 50 55 47 47 49 40 57 45 51 54 46 53 50 47 45 51
## [1921] 54 50 52 61 50 54 47 53 49 45 48 47 41 42 47 54 52 46 58 51 51 43 48 48
## [1945] 58 53 55 44 48 59 45 54 51 42 42 45 46 55 49 49 55 46 42 50 49 55 50 56
## [1969] 42 58 46 54 54 53 56 53 51 51 46 45 52 51 57 54 44 62 60 55 53 44 43 52
## [1993] 44 51 46 50 50 47 50 52 55 61 51 45 53 45 49 50 53 42 54 52 53 55 47 56
## [2017] 42 52 50 44 47 48 49 57 54 50 48 47 57 43 55 43 48 43 42 44 50 52 51 53
## [2041] 51 61 45 45 55 49 58 50 38 50 45 48 55 60 51 54 52 48 56 50 51 51 51 47
## [2065] 50 44 45 53 51 51 46 52 55 53 50 42 53 53 46 51 51 45 50 45 44 54 42 49
## [2089] 51 44 54 41 45 49 54 41 37 42 52 53 48 48 55 47 53 47 51 51 52 41 49 52
## [2113] 49 53 40 55 54 48 53 51 55 52 54 55 56 56 50 51 48 47 34 48 53 49 49 49
## [2137] 48 50 48 54 37 49 54 46 50 49 57 57 49 49 51 54 43 42 58 56 53 45 53 53
## [2161] 52 54 44 54 53 54 47 50 61 52 54 40 49 40 45 46 47 44 49 56 60 58 46 49
## [2185] 47 54 48 49 46 61 42 48 43 48 43 52 42 50 49 51 49 47 42 53 51 54 51 53
## [2209] 55 48 56 48 51 49 51 49 46 50 52 45 57 47 41 54 48 49 46 54 50 39 49 55
## [2233] 44 48 39 55 51 51 50 57 43 46 45 51 46 50 54 47 54 55 51 45 50 47 45 47
## [2257] 47 45 53 48 51 40 53 50 48 46 45 50 62 53 56 59 51 55 49 54 48 54 47 54
## [2281] 60 48 53 59 58 48 50 58 58 54 55 47 46 48 39 45 58 40 44 46 48 45 50 47
## [2305] 42 44 50 45 54 52 50 41 58 56 60 53 50 58 46 46 54 49 56 51 50 49 51 51
## [2329] 58 52 54 43 47 52 50 52 52 50 41 45 40 56 44 50 57 41 45 56 62 43 44 51
## [2353] 46 44 46 50 57 46 45 55 52 54 39 46 55 48 52 57 53 43 47 60 48 43 61 51
## [2377] 49 56 55 50 52 52 47 52 46 54 49 42 57 47 55 41 42 48 46 53 49 47 46 52
## [2401] 51 44 49 52 48 47 51 57 41 48 59 48 49 45 49 58 49 48 61 44 48 51 53 48
## [2425] 46 53 51 49 49 49 44 45 48 45 55 60 49 54 53 50 53 41 47 52 51 50 43 61
## [2449] 42 55 56 49 49 37 50 47 63 47 52 47 50 51 54 58 53 58 50 45 48 52 42 56
## [2473] 46 60 56 50 61 44 43 53 46 48 52 53 52 55 52 49 52 56 47 51 50 57 48 57
## [2497] 59 53 57 55 52 54 49 49 51 47 55 49 57 48 48 50 53 43 43 46 51 56 47 43
## [2521] 49 53 59 54 45 51 51 51 50 55 41 44 47 49 52 54 51 50 46 49 43 49 48 49
## [2545] 45 54 46 61 51 51 50 51 55 47 37 53 47 52 37 52 58 48 47 48 52 46 45 50
## [2569] 54 54 48 53 50 58 39 53 42 51 51 51 51 43 48 44 57 50 49 50 54 45 51 47
## [2593] 46 48 54 52 48 48 43 57 55 49 52 56 54 56 48 58 52 49 50 54 46 50 58 49
## [2617] 53 49 54 50 34 49 49 49 60 54 43 41 58 56 55 45 47 49 46 46 51 47 53 57
## [2641] 49 46 47 49 61 47 48 50 44 49 47 51 61 52 56 51 52 50 48 49 53 48 51 44
## [2665] 52 45 49 55 51 54 53 67 51 45 51 45 45 45 48 43 52 47 55 50 44 52 43 59
## [2689] 49 54 44 46 52 46 53 48 59 55 40 59 44 51 53 44 52 47 49 47 47 55 47 52
## [2713] 50 56 51 57 42 51 47 52 47 47 66 40 38 50 54 47 54 60 46 52 56 55 49 49
## [2737] 46 48 52 51 50 44 56 55 55 52 45 55 49 48 44 56 52 49 43 48 51 50 45 51
## [2761] 51 49 43 49 56 52 56 46 55 49 44 50 47 43 49 50 54 39 46 52 43 42 44 47
## [2785] 49 50 54 50 43 52 49 50 50 49 48 48 48 50 46 48 41 55 50 46 49 46 46 47
## [2809] 52 47 54 47 44 55 53 51 43 46 51 57 55 42 46 47 49 53 39 53 54 45 46 55
## [2833] 36 54 51 56 53 44 49 58 61 51 48 55 57 50 51 43 35 51 44 45 43 46 57 45
## [2857] 53 46 46 45 56 55 59 49 50 49 47 55 58 48 51 47 49 48 51 46 46 49 49 46
## [2881] 49 54 51 51 55 41 52 44 56 51 41 54 44 44 49 44 47 51 51 56 48 56 47 40
## [2905] 51 45 52 39 51 53 49 48 50 51 50 50 57 54 49 45 55 48 58 52 41 49 47 51
## [2929] 48 50 55 55 47 53 45 45 57 57 55 54 44 52 50 50 50 51 50 50 54 47 52 51
## [2953] 51 53 38 59 49 52 57 53 59 51 44 52 49 48 61 52 57 45 46 54 49 36 56 49
## [2977] 45 50 52 53 54 46 47 43 51 42 50 58 49 52 36 50 50 50 55 48 47 58 37 42
## [3001] 62 52 45 45 43 47 48 51 55 53 52 60 47 49 42 46 56 45 45 54 47 48 50 50
## [3025] 54 53 50 56 59 49 45 50 50 51 54 55 47 58 50 50 59 45 50 48 45 51 52 52
## [3049] 48 52 48 54 51 45 54 52 58 45 55 44 55 43 43 47 55 52 50 45 45 43 53 47
## [3073] 56 58 47 50 51 52 52 51 45 55 55 47 46 52 48 39 55 48 55 50 51 53 56 51
## [3097] 48 52 46 51 39 49 41 56 48 51 48 49 51 61 44 50 61 54 52 52 51 44 58 53
## [3121] 47 47 48 61 52 53 52 39 48 51 49 45 42 54 49 54 57 49 57 46 50 47 54 52
## [3145] 47 48 45 43 56 48 50 55 51 49 42 48 54 55 47 60 54 53 50 59 49 49 52 50
## [3169] 53 55 54 51 44 46 40 52 57 50 48 52 52 49 49 45 45 49 49 52 49 52 49 40
## [3193] 46 57 42 51 55 46 43 61 49 49 48 49 45 48 51 39 45 33 46 51 58 51 54 51
## [3217] 48 54 58 52 48 59 46 45 45 47 43 55 52 51 42 51 51 62 50 55 44 53 44 54
## [3241] 44 51 50 44 50 59 49 49 57 52 51 60 40 53 55 52 47 56 57 49 54 48 46 53
## [3265] 45 55 45 45 57 51 54 55 53 53 47 49 42 52 47 51 41 51 64 40 53 55 58 41
## [3289] 51 57 47 42 48 51 46 48 58 47 59 41 55 48 44 52 54 50 58 45 46 50 48 44
## [3313] 45 56 53 53 53 45 58 51 44 59 51 57 48 52 50 49 59 54 46 49 53 54 47 48
## [3337] 40 47 50 57 54 52 42 56 51 51 52 58 42 45 56 49 49 50 44 46 47 48 43 42
## [3361] 50 49 49 55 42 56 46 54 60 55 46 48 47 49 53 56 52 48 47 43 49 53 54 54
## [3385] 50 49 53 43 51 50 42 49 56 51 55 50 49 52 40 51 52 54 45 50 42 54 46 56
## [3409] 49 45 50 62 48 50 51 48 49 58 49 50 45 56 47 51 50 56 54 54 55 47 54 50
## [3433] 52 46 40 55 48 48 54 59 60 56 59 51 49 50 49 57 49 53 47 54 53 51 54 44
## [3457] 47 53 52 60 44 55 54 57 44 54 49 45 51 61 58 47 53 48 46 54 48 55 46 49
## [3481] 54 51 50 51 52 53 44 57 54 54 45 54 56 49 57 48 43 51 43 46 45 45 55 51
## [3505] 55 48 59 45 51 56 54 43 47 56 55 47 58 48 52 44 55 39 48 52 54 56 47 59
## [3529] 44 53 48 49 43 51 47 59 46 49 53 43 46 47 53 55 46 50 49 53 47 49 42 52
## [3553] 46 53 44 50 49 57 41 52 44 48 41 46 55 54 43 53 55 54 53 45 48 53 55 51
## [3577] 46 52 51 47 61 53 55 50 58 55 51 46 59 44 51 52 51 57 52 51 62 54 56 45
## [3601] 52 45 46 50 50 52 50 47 49 58 53 53 49 45 61 47 54 54 45 42 48 52 50 49
## [3625] 46 47 52 56 67 45 55 49 54 53 50 56 49 48 43 52 49 45 47 52 50 50 57 62
## [3649] 46 51 53 52 49 61 51 57 50 56 41 48 54 53 46 50 51 51 51 58 53 50 40 59
## [3673] 56 47 42 49 54 52 44 54 53 58 55 58 52 46 56 53 52 44 39 50 61 47 58 46
## [3697] 52 57 50 48 55 40 61 46 53 45 59 48 46 58 39 50 50 45 44 47 53 55 51 59
## [3721] 54 49 52 55 48 49 43 48 47 51 43 58 55 54 50 56 50 44 46 50 59 54 56 47
## [3745] 50 45 53 50 52 53 51 43 47 43 41 37 56 54 48 51 51 50 46 46 48 45 54 52
## [3769] 53 49 50 57 51 55 44 42 48 56 48 47 55 55 46 62 51 58 56 54 53 48 35 49
## [3793] 52 51 45 52 50 56 53 56 46 38 45 55 54 63 48 44 50 43 53 50 49 49 49 52
## [3817] 51 47 61 51 54 55 48 48 56 55 50 48 47 49 52 41 55 59 48 42 58 53 47 45
## [3841] 47 50 57 43 47 40 46 51 47 57 52 56 43 45 44 45 55 49 52 45 49 60 54 56
## [3865] 46 49 56 42 58 49 54 51 61 42 54 47 52 43 51 51 55 51 41 58 53 50 49 48
## [3889] 49 52 53 40 49 50 44 51 46 44 45 49 62 44 51 55 39 61 51 48 59 51 48 46
## [3913] 47 51 55 47 52 53 51 48 47 50 56 57 47 56 44 55 44 40 52 49 51 53 46 37
## [3937] 52 54 49 47 56 58 37 49 56 50 46 57 54 48 48 42 54 42 51 41 55 45 48 58
## [3961] 48 50 51 45 56 48 40 48 51 45 43 42 55 42 59 57 52 55 49 48 51 50 49 43
## [3985] 51 54 49 45 49 50 45 51 51 52 58 47 48 56 49 55 41 48 45 47 46 46 47 40
## [4009] 53 54 51 58 44 50 40 49 55 48 50 47 48 47 52 60 52 52 53 47 51 54 44 48
## [4033] 49 53 44 47 36 50 62 42 50 45 54 46 43 46 57 42 49 54 53 50 47 51 39 49
## [4057] 48 48 48 53 45 53 52 46 60 43 50 43 51 55 50 55 45 54 50 58 47 59 47 55
## [4081] 47 56 44 42 52 46 49 45 42 51 53 41 53 41 52 44 48 49 48 58 55 45 49 56
## [4105] 47 52 40 50 54 48 47 58 50 51 48 47 51 56 57 52 47 52 52 48 51 58 56 46
## [4129] 52 53 48 50 43 37 46 49 44 50 64 48 48 50 56 57 47 54 56 54 48 50 48 47
## [4153] 44 48 44 48 40 44 50 52 61 55 56 53 58 60 49 45 52 49 44 43 52 40 47 53
## [4177] 52 42 46 42 49 55 58 54 53 44 49 57 46 53 45 46 50 52 47 52 39 58 57 55
## [4201] 41 51 47 42 48 48 46 57 57 52 47 43 42 47 50 53 54 46 42 54 48 53 48 60
## [4225] 47 56 48 47 43 45 53 49 47 46 42 47 50 55 50 49 56 52 56 54 48 51 52 52
## [4249] 55 48 51 43 62 46 46 50 54 49 51 57 50 54 55 47 50 57 54 44 50 51 43 45
## [4273] 55 49 48 52 45 59 50 50 54 52 50 52 51 53 53 58 47 51 50 48 57 50 46 47
## [4297] 51 53 52 49 46 59 45 54 48 53 55 45 50 55 50 50 43 50 49 51 40 57 49 50
## [4321] 42 52 62 49 48 51 52 49 36 52 54 54 53 43 54 45 45 45 46 46 48 51 39 45
## [4345] 50 45 50 63 53 49 54 51 56 58 46 49 52 48 54 62 48 48 50 52 52 50 55 39
## [4369] 46 50 57 46 49 45 51 51 46 46 51 54 47 42 54 44 59 53 36 49 50 48 44 43
## [4393] 52 53 41 44 46 54 44 51 36 57 49 51 51 44 49 45 52 40 60 53 44 52 49 51
## [4417] 52 47 54 52 57 48 49 51 47 51 42 53 49 47 56 54 43 50 57 53 48 45 53 44
## [4441] 51 55 50 58 37 50 42 42 55 46 46 51 51 56 53 42 55 50 55 52 51 56 53 52
## [4465] 53 57 58 50 48 54 46 48 45 47 56 52 54 49 60 57 56 58 39 56 46 47 51 45
## [4489] 52 58 52 51 48 47 58 51 51 56 48 53 55 54 49 46 41 50 49 55 51 46 54 45
## [4513] 51 54 51 46 54 56 47 49 47 60 49 50 49 48 52 53 44 48 54 55 48 52 53 57
## [4537] 56 54 46 48 52 47 58 55 50 50 50 47 48 56 51 46 55 62 46 49 51 52 49 49
## [4561] 55 52 51 47 52 50 47 50 53 48 47 49 56 48 42 57 50 47 57 50 52 49 51 49
## [4585] 43 52 47 45 48 46 59 46 55 50 54 48 56 49 45 50 51 51 58 42 48 42 44 51
## [4609] 57 51 51 52 50 55 60 47 54 49 51 49 45 49 43 39 50 46 57 37 50 52 59 50
## [4633] 52 55 50 50 54 46 52 58 52 48 51 44 60 52 59 51 46 56 48 43 47 42 55 45
## [4657] 42 49 52 50 43 58 58 48 46 49 48 55 48 55 52 48 54 62 39 54 53 46 58 42
## [4681] 50 55 57 48 50 42 43 50 44 56 47 60 47 44 48 49 42 55 42 48 56 51 59 44
## [4705] 64 48 54 44 47 55 46 57 54 44 48 49 48 50 57 48 41 56 57 47 44 58 51 46
## [4729] 45 58 52 49 55 52 51 49 48 45 42 47 60 54 53 53 44 48 58 47 58 51 56 51
## [4753] 40 60 59 53 56 41 50 48 49 49 47 50 46 54 52 55 56 52 54 53 49 65 51 51
## [4777] 49 54 40 49 39 47 56 53 43 50 48 56 50 48 48 48 48 51 51 42 52 53 50 49
## [4801] 46 50 53 53 46 52 63 46 55 56 46 60 51 58 50 52 53 52 48 51 48 50 52 56
## [4825] 48 55 44 51 50 54 52 51 51 52 61 53 46 50 42 45 48 47 50 53 47 48 53 42
## [4849] 53 56 45 57 57 53 44 56 43 57 57 48 56 52 53 43 54 39 47 52 51 49 54 55
## [4873] 49 46 48 54 50 45 47 46 49 44 54 59 51 58 51 40 45 50 49 43 52 50 54 47
## [4897] 46 49 46 42 44 53 47 51 45 53 49 60 37 62 53 46 55 48 50 53 56 53 52 50
## [4921] 53 41 52 49 40 54 55 45 59 41 45 56 50 46 45 40 51 49 52 48 43 47 45 48
## [4945] 48 48 49 54 51 48 44 58 49 52 54 51 48 58 54 51 53 55 44 47 56 58 53 59
## [4969] 49 59 56 52 50 51 57 42 51 45 47 52 43 57 49 44 49 46 54 49 47 48 50 44
## [4993] 56 42 58 56 55 50 53 51 51 49 54 48 47 46 44 50 52 49 52 54 42 63 43 49
## [5017] 59 57 50 46 48 55 57 48 48 45 42 48 49 48 52 44 54 61 53 45 48 51 56 50
## [5041] 51 50 58 50 53 58 46 53 47 49 59 52 58 38 44 43 53 56 49 51 50 58 52 44
## [5065] 44 58 47 58 49 47 47 43 39 38 46 42 51 44 39 56 45 52 48 47 56 51 51 49
## [5089] 56 48 50 53 53 52 58 58 50 47 51 48 47 45 47 39 54 40 44 56 47 54 51 47
## [5113] 44 56 59 51 52 55 54 55 43 43 49 53 56 58 51 44 56 54 48 46 56 48 55 52
## [5137] 42 52 52 55 48 59 59 46 59 60 58 53 48 52 53 47 47 52 57 49 58 54 58 50
## [5161] 49 56 54 53 49 52 50 43 54 48 45 44 51 41 50 51 51 51 47 51 56 54 48 58
## [5185] 48 52 52 59 54 56 52 55 46 47 57 55 52 55 58 50 45 45 43 47 52 47 51 49
## [5209] 48 48 47 49 54 52 49 42 52 52 52 40 55 44 56 41 49 55 44 50 54 45 47 40
## [5233] 51 52 49 52 51 48 46 54 54 42 53 52 48 58 45 49 46 50 54 46 44 54 55 58
## [5257] 53 49 57 58 52 52 55 49 55 51 53 52 50 48 55 52 42 46 49 54 49 51 53 46
## [5281] 55 48 42 58 55 43 54 40 52 55 50 51 54 47 48 53 50 55 53 52 49 45 51 50
## [5305] 45 52 59 53 54 48 38 52 59 48 39 56 49 52 53 51 58 47 39 54 42 51 55 50
## [5329] 49 58 49 51 58 53 42 50 61 58 37 47 47 56 46 54 48 50 52 49 50 51 47 42
## [5353] 57 52 48 45 50 55 52 41 46 44 46 47 58 52 49 55 56 52 46 60 49 49 47 45
## [5377] 56 47 45 50 53 53 49 47 53 57 55 52 55 48 58 53 47 47 51 44 50 48 53 58
## [5401] 41 52 52 47 44 54 59 44 43 53 39 41 43 56 44 48 42 50 55 41 47 44 41 57
## [5425] 43 54 42 39 51 47 47 43 49 55 46 47 57 51 43 53 48 49 48 53 49 52 60 45
## [5449] 53 61 47 50 47 48 58 42 45 54 57 50 44 56 57 49 45 50 53 54 53 56 50 48
## [5473] 60 47 53 50 56 56 48 48 45 60 44 43 56 52 50 48 49 38 53 49 49 48 46 38
## [5497] 54 50 55 49 48 55 43 50 52 50 53 54 51 63 44 54 44 51 51 50 47 53 54 50
## [5521] 43 57 55 48 39 47 52 53 54 56 55 54 47 53 52 48 47 49 47 44 49 50 56 43
## [5545] 56 44 35 49 51 52 45 47 44 52 48 51 44 55 54 64 55 60 51 52 52 52 49 51
## [5569] 48 55 52 48 58 59 43 58 52 53 55 44 49 43 51 49 46 50 45 58 46 47 49 53
## [5593] 45 47 50 40 53 62 45 52 49 51 37 47 49 58 47 50 55 49 43 57 50 49 57 56
## [5617] 53 51 44 46 51 58 49 38 43 54 40 45 45 48 50 51 50 55 56 51 53 44 51 48
## [5641] 51 45 52 51 52 51 53 48 41 58 51 54 43 54 47 44 56 54 61 52 57 48 55 52
## [5665] 48 44 50 61 51 50 59 52 53 49 52 50 53 51 62 50 49 51 58 63 54 57 48 43
## [5689] 53 53 44 47 51 41 48 53 55 46 53 53 50 42 41 55 43 52 49 53 55 46 44 53
## [5713] 58 43 48 52 53 44 53 51 51 43 60 58 51 58 51 52 52 43 54 65 53 54 50 51
## [5737] 49 49 46 54 52 54 38 43 44 47 38 53 48 53 56 50 47 50 48 50 51 52 55 49
## [5761] 51 49 49 53 50 49 51 46 49 41 44 47 51 62 48 44 47 56 47 52 50 47 50 51
## [5785] 53 52 49 41 45 54 45 58 49 55 49 48 52 53 61 54 40 57 56 53 53 51 42 55
## [5809] 50 62 47 43 46 53 50 44 52 46 45 53 54 39 44 53 53 47 55 48 47 52 45 49
## [5833] 52 54 56 44 56 49 53 45 49 54 44 44 48 57 50 50 53 46 47 38 47 58 43 47
## [5857] 62 44 49 61 52 50 44 53 49 43 42 50 49 51 47 52 57 45 54 44 55 43 55 41
## [5881] 57 48 52 50 50 49 57 56 37 44 44 53 56 50 44 35 60 50 52 46 56 48 52 46
## [5905] 55 50 47 50 52 54 52 52 49 53 50 50 50 52 38 45 48 46 50 47 56 58 49 55
## [5929] 54 45 54 58 47 50 43 48 55 54 49 51 39 50 43 51 42 47 54 50 55 60 37 43
## [5953] 48 43 54 52 48 49 52 52 57 53 49 49 45 45 50 46 43 53 52 50 51 51 53 50
## [5977] 57 48 47 47 51 49 56 46 61 53 57 52 52 51 57 45 51 47 59 51 45 50 47 47
## [6001] 50 34 55 47 53 56 46 47 56 46 43 51 53 51 47 49 49 60 45 47 57 50 49 52
## [6025] 49 39 52 53 53 57 46 48 49 40 55 48 56 47 54 45 52 51 60 50 48 57 50 58
## [6049] 52 55 45 44 43 57 40 58 47 49 40 69 54 46 45 44 55 48 52 55 38 60 51 52
## [6073] 47 43 45 59 49 52 54 54 45 50 49 49 53 47 49 48 50 53 49 49 52 45 54 54
## [6097] 43 54 50 50 52 49 48 42 52 47 49 51 56 51 52 55 57 63 55 44 51 50 51 48
## [6121] 49 46 46 56 44 55 47 47 58 53 51 39 52 50 50 50 49 58 44 59 58 50 53 54
## [6145] 56 59 62 50 54 55 49 59 40 50 52 55 48 52 51 62 57 46 47 57 49 51 48 50
## [6169] 52 53 46 45 55 50 51 48 53 48 57 43 51 43 36 57 54 50 43 45 49 48 49 52
## [6193] 50 50 52 49 51 54 51 46 59 51 57 51 43 45 53 58 60 56 51 48 47 51 38 53
## [6217] 40 44 46 50 49 44 52 50 57 61 48 53 44 52 47 48 54 50 49 47 46 48 50 56
## [6241] 53 52 54 48 50 47 44 47 45 44 54 50 52 60 50 45 51 46 54 58 62 53 46 51
## [6265] 51 48 48 47 50 53 48 53 46 46 51 56 44 47 52 44 57 54 53 46 43 50 50 45
## [6289] 51 45 48 40 55 48 54 49 49 51 51 52 52 41 42 45 50 52 55 50 49 59 55 46
## [6313] 54 50 53 48 44 43 46 49 54 51 53 56 57 51 51 51 51 59 43 45 60 43 51 52
## [6337] 53 56 57 48 51 48 44 46 56 48 41 46 56 43 54 48 55 55 55 44 44 55 56 57
## [6361] 44 43 49 64 49 56 43 47 58 50 53 45 50 54 46 45 49 48 53 50 52 42 55 51
## [6385] 48 46 46 50 49 57 60 49 45 49 56 42 54 51 58 56 55 48 53 51 46 47 49 51
## [6409] 52 52 48 54 49 44 50 41 59 42 60 42 49 54 51 51 58 51 49 48 49 41 52 54
## [6433] 52 57 60 50 55 56 58 54 48 52 46 50 57 55 53 52 51 42 46 43 49 50 54 55
## [6457] 49 45 58 52 53 51 45 45 51 48 52 49 51 52 42 47 51 52 45 57 50 48 45 57
## [6481] 64 53 59 54 56 45 46 46 51 56 49 53 50 60 54 47 47 42 46 50 53 51 60 55
## [6505] 51 58 52 54 59 45 44 56 42 50 51 50 54 45 51 53 42 48 53 52 46 52 47 47
## [6529] 48 58 38 50 49 51 55 50 56 54 59 47 51 54 59 48 62 56 43 50 60 52 50 54
## [6553] 45 52 57 47 48 46 52 44 43 50 56 48 47 49 49 45 45 55 47 51 56 40 52 59
## [6577] 51 56 60 46 48 46 48 62 57 61 46 49 56 48 46 58 55 43 53 56 44 54 47 55
## [6601] 49 48 60 44 50 48 50 41 59 51 56 49 46 53 49 48 52 50 45 56 47 46 52 56
## [6625] 45 48 52 50 51 52 45 50 57 43 46 53 48 46 52 55 47 42 50 47 49 54 51 55
## [6649] 51 44 52 46 44 47 51 49 56 43 57 60 52 52 57 46 45 40 49 42 52 48 61 62
## [6673] 53 63 49 55 53 47 51 54 50 44 49 49 47 56 55 49 50 55 50 56 45 51 49 49
## [6697] 50 49 50 51 55 47 47 58 51 49 42 45 53 56 51 44 45 42 44 53 62 50 42 43
## [6721] 43 52 44 44 45 51 52 50 52 57 38 46 50 54 42 48 46 47 44 51 47 44 51 44
## [6745] 44 53 48 59 56 56 57 42 51 51 58 53 50 51 50 58 47 51 47 46 44 51 39 50
## [6769] 48 51 52 53 53 51 49 49 53 60 50 54 47 55 63 54 50 55 46 48 42 46 61 50
## [6793] 43 51 41 51 55 48 53 51 44 43 55 51 48 49 51 57 42 67 47 43 44 53 46 56
## [6817] 53 49 50 50 56 38 47 57 53 50 58 51 50 42 63 51 52 55 51 50 52 42 41 44
## [6841] 43 51 52 46 47 43 59 51 46 46 54 52 49 54 47 54 46 48 56 48 52 49 47 52
## [6865] 55 47 49 47 48 41 54 49 48 51 42 53 58 57 52 52 59 58 48 49 45 57 50 45
## [6889] 56 45 41 51 49 43 40 57 52 46 47 56 40 42 54 47 50 52 45 49 54 47 49 57
## [6913] 51 58 59 48 53 55 43 56 43 47 37 49 46 52 50 46 53 44 49 59 51 51 55 51
## [6937] 57 53 50 50 55 59 49 53 49 48 53 53 50 53 44 47 59 48 52 43 49 50 49 49
## [6961] 45 49 42 49 49 50 54 48 52 55 50 46 46 43 54 50 56 47 44 53 50 46 53 49
## [6985] 47 51 57 57 54 52 60 44 51 45 53 42 42 52 49 53 44 53 48 47 44 52 47 62
## [7009] 47 47 47 54 45 46 56 48 52 39 44 50 50 51 58 52 50 52 50 52 46 41 57 48
## [7033] 52 44 54 60 55 48 52 51 49 51 52 48 44 45 42 49 50 43 47 55 54 42 55 51
## [7057] 53 57 55 50 56 44 47 44 44 41 46 47 50 49 48 47 59 53 46 53 51 60 53 48
## [7081] 56 38 38 50 51 47 54 47 47 39 57 45 50 50 51 50 44 44 45 44 45 51 50 51
## [7105] 56 51 47 43 58 51 47 49 45 52 44 51 48 46 53 52 50 61 47 47 55 45 51 57
## [7129] 48 50 58 47 42 52 55 50 51 49 63 54 50 56 46 47 58 53 51 57 52 38 53 51
## [7153] 50 57 43 45 49 60 45 49 43 56 53 49 53 47 54 49 54 44 40 49 42 42 53 58
## [7177] 51 43 44 49 49 45 50 43 48 61 54 56 50 49 47 42 51 61 56 41 47 50 55 48
## [7201] 50 53 36 46 52 53 46 53 44 46 56 51 58 42 57 40 55 54 47 45 51 56 44 45
## [7225] 49 48 44 48 40 47 46 50 46 45 52 49 46 51 55 38 51 57 49 58 53 50 46 55
## [7249] 51 55 45 49 45 44 41 49 46 53 56 46 48 52 56 55 44 57 52 54 54 39 52 44
## [7273] 51 46 47 45 47 43 41 40 63 47 49 58 52 56 47 50 56 56 55 49 53 52 47 47
## [7297] 51 44 56 53 55 52 43 44 50 49 44 47 61 56 51 47 52 46 56 48 55 49 46 59
## [7321] 54 49 56 46 39 55 54 52 47 49 50 42 51 55 50 50 52 53 50 53 45 50 54 55
## [7345] 49 44 60 47 56 49 53 54 45 49 54 46 48 50 55 51 50 48 51 41 41 57 48 52
## [7369] 50 47 58 53 44 62 50 52 54 48 47 46 52 52 54 57 50 51 48 48 54 53 53 49
## [7393] 52 42 58 48 47 54 58 45 54 63 44 59 47 46 51 60 54 46 51 55 53 54 48 42
## [7417] 49 44 43 50 52 54 53 54 49 41 48 53 50 50 56 54 51 50 53 47 51 55 50 55
## [7441] 52 49 47 50 53 53 52 47 42 46 55 39 43 40 49 56 46 47 52 44 50 50 56 53
## [7465] 53 56 50 46 51 51 54 53 44 51 45 48 43 46 51 46 57 44 48 50 53 54 51 48
## [7489] 53 47 56 52 48 48 45 42 59 44 48 54 47 46 49 52 53 55 49 56 48 47 45 51
## [7513] 54 44 55 53 54 52 56 58 60 45 44 45 37 41 47 47 51 53 52 54 53 48 50 38
## [7537] 54 48 44 43 42 49 49 39 39 49 44 45 51 49 60 53 51 54 59 50 52 50 49 47
## [7561] 44 50 53 52 56 56 49 53 52 44 47 51 46 48 59 48 54 59 48 46 50 49 58 48
## [7585] 55 43 55 55 51 56 47 45 49 47 42 58 42 47 60 61 50 40 60 47 54 54 46 48
## [7609] 44 52 51 53 47 57 44 53 47 47 54 58 49 51 46 51 47 49 54 44 47 55 56 45
## [7633] 50 53 42 42 59 53 52 56 58 53 61 49 52 51 51 46 44 46 48 58 49 50 49 52
## [7657] 44 53 61 56 51 61 58 54 47 47 47 47 53 52 44 54 50 56 52 57 56 57 52 54
## [7681] 50 48 54 42 46 39 52 53 46 45 52 45 52 54 55 49 51 56 46 50 56 48 57 49
## [7705] 51 47 49 50 60 43 50 41 55 47 63 54 55 52 52 53 53 45 57 48 47 50 54 48
## [7729] 46 46 44 51 47 55 52 49 46 44 55 55 55 51 51 41 58 46 40 45 47 55 53 51
## [7753] 54 55 56 51 49 52 51 56 55 60 49 50 54 52 57 43 60 45 46 52 52 56 52 43
## [7777] 56 50 48 48 39 48 54 44 60 50 52 41 41 54 54 48 49 52 45 53 50 49 45 58
## [7801] 53 48 45 55 49 50 53 59 48 46 46 56 55 45 43 50 48 54 49 49 44 48 49 48
## [7825] 36 45 44 49 55 48 56 49 51 47 50 51 50 43 50 56 49 41 47 48 43 42 52 59
## [7849] 58 50 46 51 52 52 55 48 51 51 50 36 46 58 51 38 51 52 44 54 44 55 45 48
## [7873] 54 41 48 45 46 53 51 43 44 50 52 55 56 44 50 48 43 48 49 52 37 44 50 49
## [7897] 50 52 49 51 49 57 47 48 59 53 48 50 42 52 52 47 46 54 54 51 48 52 48 51
## [7921] 51 55 55 57 51 41 43 61 42 57 45 43 47 41 46 53 50 50 56 49 52 51 46 45
## [7945] 49 45 48 50 51 47 43 54 51 48 57 52 48 43 56 50 53 45 55 51 53 53 47 44
## [7969] 51 52 48 50 39 44 46 44 56 46 59 54 59 47 53 44 50 59 39 61 45 54 57 42
## [7993] 43 54 54 52 51 45 55 53 63 41 50 51 49 53 58 45 50 48 49 55 57 57 43 53
## [8017] 49 45 57 53 54 50 54 52 54 55 49 55 57 45 46 49 52 48 51 56 48 51 47 44
## [8041] 42 48 57 45 54 47 48 51 59 43 45 43 58 48 50 54 42 61 42 55 51 41 52 44
## [8065] 40 49 50 52 47 54 50 46 58 54 51 44 53 47 41 52 55 45 52 47 53 59 57 57
## [8089] 39 50 56 46 55 49 48 48 45 50 63 52 53 50 53 48 51 45 50 51 55 53 48 44
## [8113] 53 58 47 42 54 57 55 48 43 50 51 54 46 49 44 52 56 49 53 47 44 52 40 53
## [8137] 51 53 54 49 49 47 52 51 53 46 46 50 51 60 42 54 50 42 48 51 50 46 53 45
## [8161] 52 51 53 45 48 43 53 56 49 48 56 52 53 48 54 49 48 50 51 48 48 54 47 51
## [8185] 44 43 53 45 51 51 39 39 63 57 44 48 57 48 53 53 44 42 50 44 51 46 53 55
## [8209] 44 58 47 49 54 48 52 48 50 40 50 48 42 44 51 47 48 43 45 47 49 50 47 48
## [8233] 52 47 38 48 53 56 45 44 55 51 47 48 54 52 50 44 53 48 47 52 44 45 51 50
## [8257] 44 48 48 46 56 54 50 47 48 55 50 55 47 49 49 45 44 62 41 45 48 50 56 52
## [8281] 52 54 49 49 45 48 41 42 51 46 48 49 46 46 55 49 45 43 46 53 65 47 55 47
## [8305] 60 53 43 50 47 52 44 49 50 44 52 36 54 45 56 51 49 58 49 56 52 51 48 45
## [8329] 48 55 45 58 51 55 53 55 46 44 50 52 55 55 46 51 55 53 40 46 53 54 52 51
## [8353] 47 51 50 46 54 49 52 51 54 49 51 51 46 41 46 43 50 44 45 43 52 54 50 50
## [8377] 53 58 51 50 47 46 43 45 51 43 56 48 48 54 49 55 55 48 40 53 48 43 46 48
## [8401] 51 51 48 59 57 52 56 56 51 50 55 54 47 45 45 50 46 56 50 44 42 55 54 60
## [8425] 57 42 46 49 51 46 47 47 53 44 49 50 45 60 40 51 41 48 53 46 49 47 50 50
## [8449] 47 54 51 53 51 44 55 56 41 51 61 57 51 50 55 52 48 43 57 52 44 47 55 48
## [8473] 50 49 47 51 48 43 58 44 50 43 39 45 45 43 53 44 48 50 51 52 50 45 58 47
## [8497] 51 49 47 57 56 49 40 51 61 54 46 60 52 54 45 57 48 57 56 53 54 59 55 48
## [8521] 51 48 48 45 45 51 47 56 45 50 53 49 47 55 51 55 51 54 53 58 55 44 59 51
## [8545] 52 59 44 55 53 46 48 50 38 54 50 58 43 58 52 53 51 51 53 56 47 53 46 56
## [8569] 47 49 47 42 49 52 50 48 53 49 57 48 55 58 42 45 46 50 48 40 51 52 47 53
## [8593] 38 49 50 53 57 41 46 47 52 54 49 59 52 50 46 42 48 54 42 56 50 48 49 51
## [8617] 55 46 50 42 46 60 37 56 50 42 45 53 53 54 52 48 52 49 55 52 48 39 47 55
## [8641] 54 44 45 53 49 56 51 45 45 46 47 42 53 50 57 49 58 47 46 49 44 49 51 54
## [8665] 45 58 46 41 55 47 50 42 55 44 55 57 54 47 50 54 42 49 60 49 61 43 48 51
## [8689] 47 45 47 44 52 48 44 57 55 49 53 53 52 47 52 55 50 52 49 40 50 44 51 56
## [8713] 47 46 53 46 48 49 56 50 61 49 54 45 50 47 51 51 50 48 47 51 49 53 53 50
## [8737] 47 50 55 48 45 45 48 48 54 47 52 46 49 54 51 47 48 62 55 55 50 57 52 51
## [8761] 49 50 50 61 37 52 49 49 48 53 62 51 45 62 48 44 56 52 60 52 54 51 56 51
## [8785] 49 54 39 51 50 48 49 50 45 53 49 43 53 55 49 40 52 57 45 47 52 46 47 51
## [8809] 43 52 56 49 53 49 47 55 53 62 49 52 42 60 47 51 53 46 46 36 55 52 44 56
## [8833] 47 50 52 50 50 50 48 47 54 56 50 49 50 54 48 46 51 46 50 46 42 45 53 49
## [8857] 41 45 51 50 46 63 50 50 48 55 54 53 49 55 50 46 47 43 52 49 44 54 48 52
## [8881] 51 52 54 56 48 48 51 57 40 55 48 46 42 55 54 50 52 50 50 38 56 52 60 49
## [8905] 39 49 53 46 44 54 56 46 50 53 54 52 54 52 50 57 41 51 56 53 53 41 48 48
## [8929] 52 47 51 48 52 41 53 47 41 45 44 52 49 52 48 49 50 59 45 48 50 55 61 49
## [8953] 58 55 58 55 52 54 43 50 43 57 54 49 48 57 56 46 51 47 52 48 42 52 46 45
## [8977] 45 51 49 50 50 48 51 52 52 41 40 50 44 56 50 55 50 49 44 52 46 47 53 47
## [9001] 54 36 51 57 51 51 52 51 54 56 55 50 60 45 51 54 46 52 53 53 51 51 44 47
## [9025] 55 53 50 45 50 44 57 45 48 45 47 50 51 50 51 51 54 46 50 51 53 49 52 56
## [9049] 54 47 44 44 53 50 46 39 48 55 55 56 41 49 56 61 52 46 43 41 46 54 50 48
## [9073] 57 50 49 45 41 48 52 56 58 46 45 58 44 48 44 50 33 44 56 41 50 55 39 47
## [9097] 45 46 55 51 58 44 62 59 53 49 51 52 49 50 48 50 46 47 48 54 49 40 38 47
## [9121] 57 49 55 46 46 47 53 50 53 49 56 42 45 48 56 44 43 53 47 58 60 50 52 40
## [9145] 46 59 48 50 58 53 48 38 54 48 51 60 49 58 49 42 46 50 52 45 47 50 51 54
## [9169] 42 51 53 51 50 53 50 44 42 51 56 60 50 43 56 52 48 48 57 50 36 53 56 54
## [9193] 51 44 52 52 54 43 55 60 50 48 46 50 42 48 53 58 45 53 51 43 52 50 45 51
## [9217] 51 56 59 52 59 52 48 42 52 50 42 53 48 46 49 57 50 45 44 51 47 46 42 44
## [9241] 56 45 49 64 54 56 54 47 52 47 63 58 50 53 50 50 43 59 52 45 50 55 51 58
## [9265] 51 49 50 59 45 45 60 47 45 51 49 49 47 44 48 48 41 53 47 48 53 52 63 53
## [9289] 51 48 54 47 52 59 53 52 53 46 47 57 51 46 53 52 52 46 54 53 60 54 50 54
## [9313] 46 44 52 47 46 54 52 53 53 50 46 40 48 51 48 60 42 51 51 45 45 50 50 44
## [9337] 59 43 57 56 50 51 50 49 50 41 53 50 41 49 56 50 55 40 50 53 44 53 56 51
## [9361] 42 51 49 52 47 42 53 43 47 57 57 51 57 53 47 55 56 52 61 46 55 49 55 55
## [9385] 38 58 49 40 49 47 50 55 47 48 49 51 42 54 51 44 48 53 46 49 53 48 47 58
## [9409] 58 48 45 48 47 53 50 51 53 53 51 51 44 45 47 50 49 47 47 53 49 49 48 46
## [9433] 47 57 42 43 52 54 59 48 49 48 52 48 60 48 51 43 41 45 59 47 55 53 45 44
## [9457] 46 53 54 43 48 47 46 50 48 51 40 47 50 49 54 43 54 50 46 50 40 53 45 43
## [9481] 43 52 48 51 54 52 58 58 46 54 48 51 42 49 42 51 51 57 57 44 45 52 45 48
## [9505] 42 50 47 46 46 45 59 43 56 60 42 57 60 39 45 44 46 42 51 50 49 55 47 37
## [9529] 48 43 58 64 53 52 48 56 50 53 45 57 50 46 46 46 51 49 51 51 40 47 43 46
## [9553] 47 48 48 55 37 55 56 58 45 47 47 54 44 58 52 53 49 48 46 47 46 46 49 46
## [9577] 53 50 46 57 46 55 52 54 49 43 56 42 47 46 35 50 50 56 42 55 47 49 48 49
## [9601] 46 53 51 49 45 45 47 53 55 46 44 44 51 52 45 59 51 58 43 55 57 49 51 56
## [9625] 51 45 52 56 51 51 54 47 51 45 47 47 52 38 46 41 58 42 52 55 52 45 50 60
## [9649] 55 56 45 47 51 46 48 46 50 45 55 58 50 57 44 46 54 40 55 52 49 54 45 48
## [9673] 56 51 43 54 47 55 45 46 48 49 45 48 55 55 55 48 47 44 65 52 47 47 53 53
## [9697] 51 49 49 46 49 51 47 47 55 51 51 46 58 54 49 59 49 56 56 43 52 45 54 47
## [9721] 50 47 49 57 51 53 52 52 50 56 48 53 51 45 52 41 47 58 54 46 43 47 45 49
## [9745] 56 51 51 52 51 50 48 45 45 52 45 43 47 50 49 59 58 42 53 58 53 60 45 43
## [9769] 49 51 54 47 51 46 51 53 52 55 52 54 51 48 48 56 41 54 56 59 54 55 43 45
## [9793] 53 52 43 47 59 56 54 45 35 40 53 43 52 43 54 46 46 52 47 48 47 46 45 43
## [9817] 52 52 42 56 49 44 51 55 45 51 48 54 44 52 38 50 45 44 42 51 52 52 50 51
## [9841] 45 47 47 55 59 55 53 53 54 46 47 54 51 36 57 48 50 53 35 50 54 52 53 55
## [9865] 49 60 49 50 45 55 53 50 54 50 43 43 51 49 45 49 46 54 50 47 50 49 44 42
## [9889] 53 53 51 50 57 45 57 49 48 41 53 47 56 49 58 48 56 55 39 48 50 49 49 44
## [9913] 57 51 48 53 48 64 52 51 46 42 49 52 41 63 52 47 41 58 51 49 44 59 55 41
## [9937] 44 49 50 47 55 51 52 58 55 54 53 57 47 42 48 51 44 44 52 45 54 48 46 56
## [9961] 54 57 33 60 47 50 52 46 48 50 49 51 50 50 38 50 54 52 46 50 48 47 43 49
## [9985] 38 50 52 48 58 47 54 43 52 47 49 52 53 52 51 46## determine mean and sd
mu <- mean(mybinom) # check mean of actual distribution
mu## [1] 50.0209sigma <- sd(mybinom) # check sd of actual distribution
sigma## [1] 4.995133## run summary commands to cross reference mean and sd
library("psych")
describe(mybinom) # summary statistics## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 10000 50.02 5 50 50.03 4.45 33 69 36 -0.03 -0.07 0.05### Plot out the distribution
hist(mybinom,
breaks = 100,
xlab = "Mybinom",
ylab = "Frequency",
main = "Acutal Binomial Distribution")
# Step 2: Create an empty matrix
z <- matrix(data = rep(x = 0,
times = 10000
),
nrow = 10000,
ncol = 1)
z[1:51]## [1] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## [39] 0 0 0 0 0 0 0 0 0 0 0 0 0describe(z)## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 10000 0 0 0 0 0 0 0 0 NaN NaN 0# Step 3: Take random samples
for (i in 1:10000){
z[i,] <- mean(sample( x = mybinom, # a vector of elements from which to choose, or a positive integer.
size = 100, # a non-negative integer giving the number of items to choose.
replace = TRUE # should sampling be with replacement?
)
)
}
z[1:51]## [1] 50.86 49.42 50.79 49.43 50.08 50.32 50.80 49.31 49.93 50.11 50.63 49.11
## [13] 49.25 49.82 50.49 49.70 50.08 48.50 50.01 50.75 50.07 49.73 49.90 49.83
## [25] 50.05 51.42 50.63 49.81 48.31 48.87 49.87 49.50 49.80 50.54 49.45 49.94
## [37] 50.68 50.70 49.82 49.91 50.13 50.29 49.99 48.76 49.98 50.29 49.91 49.65
## [49] 50.12 50.07 49.62describe(z)## vars n mean sd median trimmed mad min max range skew kurtosis
## X1 1 10000 50.02 0.5 50.02 50.02 0.5 48.25 51.91 3.66 -0.03 0.01
## se
## X1 0.01hist(z, xlab = "", main = "Histogram of Sample Mean (n = 100)")
## Step 4: EXPANDING COLUMNS OF NULL MATRIX CODE and replace null matrix
z <- matrix(data = rep(x = 0,
times = 40000
),
nrow = 10000,
ncol = 4)
n <- c(2, 6, 30, 1000) # let n be the sample size
#larger sample size should have closer sample mean to true population mean
# i indexes rows and j indexes columns
# As column j increases from 1 to 4 (1-2-3-4), sample size also increases from 2 to 1000 (2-6-30-1000) respectively.
# Sample (of size 2-6-30-1000) pulled out from the original distribution i=10000 times.
for (j in 1:4){ # indexing columns of matrix, where each column will represent different sample size
for (i in 1:10000){ # indexes the rows of matrix
z[i,j] <- mean(sample( x = mybinom, # compute mean and assign
size = n[j],
replace = TRUE
)
)
}
}
#Step 5: Confrim CLT Intuition
colnames(z) <- c("Sample size=2", "Sample size=6", "Sample size=30", "Sample size=1000")
summary(z) ## Sample size=2 Sample size=6 Sample size=30 Sample size=1000
## Min. :37.0 Min. :42.50 Min. :46.57 Min. :49.40
## 1st Qu.:47.5 1st Qu.:48.67 1st Qu.:49.43 1st Qu.:49.91
## Median :50.0 Median :50.00 Median :50.03 Median :50.02
## Mean :50.0 Mean :50.05 Mean :50.02 Mean :50.02
## 3rd Qu.:52.5 3rd Qu.:51.33 3rd Qu.:50.63 3rd Qu.:50.13
## Max. :62.5 Max. :58.33 Max. :53.83 Max. :50.64par(mfrow = c(3,2))
hist(x = mybinom, main = "Binomial Distribution"
)
for (k in 1:4){
hist(x = z[,k], # matrix column 1-4 which contain sample means of randomly chosen samples from uniform dist
main = "Binomial Distribution of Sample Mean",
xlim = c(30,60), # plot the domain of X axis from x=10 to x=20 only
xlab = paste0("Sample Size ", n[k], " (Column ", k, " from matrix)")
)
}
colnames(z) <- c("Sample size=2", "Sample size=6", "Sample size=30", "Sample size=1000")
summary(z)## Sample size=2 Sample size=6 Sample size=30 Sample size=1000
## Min. :37.0 Min. :42.50 Min. :46.57 Min. :49.40
## 1st Qu.:47.5 1st Qu.:48.67 1st Qu.:49.43 1st Qu.:49.91
## Median :50.0 Median :50.00 Median :50.03 Median :50.02
## Mean :50.0 Mean :50.05 Mean :50.02 Mean :50.02
## 3rd Qu.:52.5 3rd Qu.:51.33 3rd Qu.:50.63 3rd Qu.:50.13
## Max. :62.5 Max. :58.33 Max. :53.83 Max. :50.64# Step 6:Backtrace mean and sd
mu # original binomial distribution mean## [1] 50.0209colMeans(z) ## Sample size=2 Sample size=6 Sample size=30 Sample size=1000
## 49.99595 50.05242 50.01983 50.02121apply(X = z,
MARGIN = 2, # columns
FUN = sd) * c(sqrt(2),
sqrt(6),
sqrt(30),
sqrt(1000)
) ## Sample size=2 Sample size=6 Sample size=30 Sample size=1000
## 4.979036 4.957149 4.961859 4.989873
5B. Alternatively, apply the CLT on any other sample statistic like say the sample median, sample 25th percentile or even the sample 80th percentile. This may be marginally harder than the last part, but you can try to submit both.
# Step 1 Create the distribution
set.seed(13579) # Set seed for reproducibility
N <- 10000 # Specify sample size
mybinom2 <- rbinom(N, size = 100, prob = 0.5)
mybinom2 # check if created correctly## [1] 45 44 55 43 35 47 56 52 49 51 47 50 51 54 53 48 57 55 51 52 48 44 46 49
## [25] 50 59 55 46 54 51 61 45 49 45 49 50 40 52 44 60 45 50 55 51 43 49 58 43
## [49] 44 51 56 42 53 50 53 47 44 49 42 56 60 51 53 49 57 51 51 51 51 54 53 56
## [73] 45 48 49 40 49 50 51 50 50 55 41 52 55 47 47 52 49 51 47 49 46 53 51 56
## [97] 50 47 49 51 41 49 50 52 56 57 48 54 51 41 46 54 54 43 52 56 49 56 47 52
## [121] 48 51 50 53 40 43 48 44 49 46 46 57 42 49 47 53 49 54 37 54 52 49 50 42
## [145] 47 48 55 61 54 53 33 48 52 58 46 53 48 54 48 56 50 43 48 46 50 57 38 37
## [169] 50 49 50 48 56 52 51 50 52 58 53 49 52 44 45 49 45 51 56 52 41 50 51 52
## [193] 53 46 51 43 49 55 46 44 51 37 56 48 55 50 54 44 59 48 47 44 53 54 41 51
## [217] 51 46 48 51 47 43 58 51 51 47 49 43 55 53 50 49 56 56 54 49 55 48 53 63
## [241] 49 53 50 48 48 59 41 40 49 48 44 42 64 53 45 50 50 46 52 52 52 58 52 52
## [265] 49 40 57 59 52 50 52 44 50 52 48 41 58 43 53 48 59 50 57 48 45 45 56 42
## [289] 47 50 52 51 44 55 52 47 47 50 42 48 49 56 55 55 46 60 47 43 44 52 56 43
## [313] 54 43 54 47 49 51 40 61 37 41 47 48 46 52 49 50 54 43 49 56 50 55 52 53
## [337] 54 49 49 38 48 51 50 60 60 46 49 56 53 44 42 43 45 43 46 56 48 41 51 55
## [361] 52 51 50 60 46 60 53 52 51 56 39 49 53 60 44 46 43 42 51 56 61 49 48 55
## [385] 44 53 50 53 58 47 42 44 51 55 49 57 53 45 48 35 52 57 57 49 51 53 38 47
## [409] 52 52 59 57 54 47 49 59 58 47 54 55 55 49 57 47 54 52 51 45 52 43 44 50
## [433] 61 53 55 45 47 44 41 41 47 53 49 48 44 50 60 57 53 49 46 45 47 50 46 47
## [457] 56 50 53 53 48 57 55 49 49 50 46 54 45 52 54 45 44 51 52 50 54 51 51 42
## [481] 47 48 52 49 47 58 42 58 51 58 38 50 44 56 47 50 50 50 49 45 53 47 53 46
## [505] 49 47 48 46 51 50 53 47 50 51 58 41 45 49 49 46 47 40 34 55 51 48 50 52
## [529] 52 52 48 53 51 52 56 48 45 52 49 47 59 49 55 54 53 43 56 48 46 38 54 49
## [553] 46 47 58 47 47 54 53 59 44 45 48 53 49 55 42 50 53 49 53 52 48 46 58 53
## [577] 38 47 57 53 45 58 50 43 50 47 45 52 47 59 42 57 44 51 55 51 58 46 49 62
## [601] 53 50 58 50 43 48 47 54 43 47 49 56 54 56 52 53 44 39 58 41 52 58 47 52
## [625] 51 37 56 49 52 56 47 61 50 49 51 55 59 38 49 53 42 51 51 59 53 55 48 52
## [649] 55 50 47 47 46 50 44 51 42 53 53 49 40 58 47 48 53 50 55 46 48 47 49 49
## [673] 48 55 45 48 44 52 45 45 53 56 51 48 45 44 53 53 41 47 46 55 58 54 43 53
## [697] 59 60 49 53 46 46 50 49 49 52 44 56 58 53 49 38 53 53 44 52 53 58 52 60
## [721] 46 46 57 52 50 50 51 53 50 56 49 52 56 52 42 49 54 51 46 45 40 55 42 52
## [745] 47 56 54 43 45 36 47 57 51 45 57 51 49 46 55 51 49 55 54 45 53 53 50 49
## [769] 52 48 48 53 40 45 48 47 52 45 55 56 59 38 53 53 47 49 48 52 49 49 46 51
## [793] 56 51 48 48 47 49 51 50 50 45 49 54 58 53 49 47 47 45 43 55 54 51 53 58
## [817] 48 41 56 53 53 61 48 44 48 49 58 47 48 60 51 52 53 40 48 48 49 54 42 61
## [841] 43 48 47 54 50 53 51 47 54 47 51 54 61 53 48 48 55 43 64 49 52 49 49 46
## [865] 68 40 48 57 42 53 50 52 41 45 52 54 44 45 47 51 49 56 51 56 51 56 39 52
## [889] 51 47 52 47 53 49 49 50 47 51 46 50 46 55 50 42 47 52 50 49 57 47 60 48
## [913] 55 52 55 42 50 45 56 56 42 42 47 54 54 56 62 57 53 52 47 45 49 49 51 42
## [937] 50 44 47 45 54 47 45 51 54 61 48 43 44 52 57 45 55 51 50 52 50 53 41 49
## [961] 57 46 36 43 47 55 40 47 44 58 55 60 43 59 50 55 52 53 46 47 49 54 57 43
## [985] 44 53 57 50 45 58 49 40 51 46 51 43 47 48 44 46 49 54 50 43 54 52 41 52
## [1009] 54 54 50 57 50 45 51 52 44 51 48 50 49 50 48 53 53 51 47 48 50 52 46 57
## [1033] 46 45 45 51 50 49 52 51 49 46 55 53 46 42 52 47 41 54 58 58 46 50 52 52
## [1057] 55 53 45 54 45 48 49 57 50 44 47 45 54 46 49 52 45 49 54 51 46 46 46 51
## [1081] 54 48 51 47 45 43 55 43 45 51 50 49 46 51 58 46 49 47 47 44 55 52 55 53
## [1105] 55 43 47 48 46 54 58 52 56 48 49 52 46 52 59 54 52 51 54 45 58 47 57 47
## [1129] 54 46 55 49 47 60 48 49 53 59 60 57 59 51 49 45 43 45 50 51 43 40 54 53
## [1153] 51 49 45 42 50 50 49 49 42 48 54 55 55 46 49 63 50 53 50 56 55 48 53 54
## [1177] 48 51 58 52 39 50 55 50 53 54 42 53 53 43 45 54 54 49 44 50 46 41 61 50
## [1201] 54 56 47 48 49 51 47 54 45 49 52 59 48 57 46 51 48 59 51 55 48 48 49 61
## [1225] 50 43 50 60 58 45 45 49 52 54 54 48 53 50 45 43 48 49 55 54 47 55 52 37
## [1249] 47 50 53 43 50 54 56 54 50 52 46 59 58 49 51 44 43 41 50 40 43 48 44 41
## [1273] 54 49 54 42 40 52 51 53 51 52 50 52 54 52 49 52 47 53 47 50 50 55 61 47
## [1297] 51 51 54 49 49 51 54 45 55 56 48 55 59 51 53 46 53 45 45 50 49 44 43 46
## [1321] 51 57 49 50 50 47 49 47 47 47 53 48 60 50 53 49 39 49 60 47 51 50 56 51
## [1345] 48 51 45 53 42 55 48 47 48 43 48 52 51 57 45 53 58 59 57 60 59 45 62 52
## [1369] 55 54 52 56 53 51 49 37 56 51 58 46 48 50 44 56 55 55 60 50 51 56 57 50
## [1393] 45 44 46 52 49 49 55 44 44 56 46 56 43 52 58 48 47 50 58 51 45 50 57 48
## [1417] 41 47 47 58 50 56 43 49 47 38 53 54 51 52 54 50 50 58 41 45 49 45 47 41
## [1441] 43 56 47 54 50 49 45 58 52 50 48 47 51 54 51 51 51 50 49 50 52 51 45 51
## [1465] 50 47 54 61 43 51 40 49 52 47 47 46 46 44 55 41 51 50 45 36 52 46 51 49
## [1489] 47 46 46 51 40 53 52 48 50 46 54 44 53 49 45 63 49 50 49 45 52 56 57 57
## [1513] 44 51 45 39 49 44 57 52 48 49 55 55 56 62 46 44 60 55 47 50 42 43 52 48
## [1537] 48 48 49 54 52 55 51 52 46 56 48 53 46 51 47 49 50 48 49 58 51 47 52 54
## [1561] 56 48 53 53 58 49 52 53 53 46 38 52 57 53 49 45 51 55 57 56 50 39 51 51
## [1585] 44 43 51 46 53 55 51 54 57 48 56 57 52 44 50 56 45 44 50 48 46 42 51 53
## [1609] 48 54 47 50 47 44 51 42 46 52 54 44 55 50 45 52 52 55 53 41 56 58 48 54
## [1633] 54 56 50 48 46 53 53 47 45 40 46 49 43 51 52 55 47 46 61 48 49 52 46 54
## [1657] 52 54 56 45 50 49 51 47 41 48 50 55 53 57 45 44 49 50 43 44 46 54 59 51
## [1681] 47 48 47 51 56 47 48 46 40 52 47 53 47 48 44 57 52 45 52 47 49 52 52 54
## [1705] 59 40 47 47 54 59 52 50 42 47 41 54 49 49 49 59 47 52 56 49 50 48 51 48
## [1729] 56 48 51 50 47 49 52 51 57 46 55 49 47 49 52 47 47 54 61 56 48 56 46 49
## [1753] 48 48 44 52 60 58 58 60 47 49 47 55 48 54 57 64 44 52 58 50 59 42 50 54
## [1777] 51 47 58 57 48 53 49 49 52 58 52 51 54 52 49 45 50 53 52 49 45 48 42 57
## [1801] 50 59 45 50 47 52 44 51 60 47 40 54 49 48 57 51 47 46 53 45 47 53 53 50
## [1825] 42 46 55 47 39 49 51 49 46 46 49 46 38 46 48 43 52 47 57 42 45 55 52 59
## [1849] 52 57 52 49 46 53 43 41 52 47 55 50 39 50 44 54 51 49 57 47 45 57 42 55
## [1873] 54 52 51 55 61 46 49 49 45 46 44 40 58 51 48 50 43 48 48 46 50 47 46 50
## [1897] 51 43 49 49 52 60 58 57 50 55 47 47 49 40 57 45 51 54 46 53 50 47 45 51
## [1921] 54 50 52 61 50 54 47 53 49 45 48 47 41 42 47 54 52 46 58 51 51 43 48 48
## [1945] 58 53 55 44 48 59 45 54 51 42 42 45 46 55 49 49 55 46 42 50 49 55 50 56
## [1969] 42 58 46 54 54 53 56 53 51 51 46 45 52 51 57 54 44 62 60 55 53 44 43 52
## [1993] 44 51 46 50 50 47 50 52 55 61 51 45 53 45 49 50 53 42 54 52 53 55 47 56
## [2017] 42 52 50 44 47 48 49 57 54 50 48 47 57 43 55 43 48 43 42 44 50 52 51 53
## [2041] 51 61 45 45 55 49 58 50 38 50 45 48 55 60 51 54 52 48 56 50 51 51 51 47
## [2065] 50 44 45 53 51 51 46 52 55 53 50 42 53 53 46 51 51 45 50 45 44 54 42 49
## [2089] 51 44 54 41 45 49 54 41 37 42 52 53 48 48 55 47 53 47 51 51 52 41 49 52
## [2113] 49 53 40 55 54 48 53 51 55 52 54 55 56 56 50 51 48 47 34 48 53 49 49 49
## [2137] 48 50 48 54 37 49 54 46 50 49 57 57 49 49 51 54 43 42 58 56 53 45 53 53
## [2161] 52 54 44 54 53 54 47 50 61 52 54 40 49 40 45 46 47 44 49 56 60 58 46 49
## [2185] 47 54 48 49 46 61 42 48 43 48 43 52 42 50 49 51 49 47 42 53 51 54 51 53
## [2209] 55 48 56 48 51 49 51 49 46 50 52 45 57 47 41 54 48 49 46 54 50 39 49 55
## [2233] 44 48 39 55 51 51 50 57 43 46 45 51 46 50 54 47 54 55 51 45 50 47 45 47
## [2257] 47 45 53 48 51 40 53 50 48 46 45 50 62 53 56 59 51 55 49 54 48 54 47 54
## [2281] 60 48 53 59 58 48 50 58 58 54 55 47 46 48 39 45 58 40 44 46 48 45 50 47
## [2305] 42 44 50 45 54 52 50 41 58 56 60 53 50 58 46 46 54 49 56 51 50 49 51 51
## [2329] 58 52 54 43 47 52 50 52 52 50 41 45 40 56 44 50 57 41 45 56 62 43 44 51
## [2353] 46 44 46 50 57 46 45 55 52 54 39 46 55 48 52 57 53 43 47 60 48 43 61 51
## [2377] 49 56 55 50 52 52 47 52 46 54 49 42 57 47 55 41 42 48 46 53 49 47 46 52
## [2401] 51 44 49 52 48 47 51 57 41 48 59 48 49 45 49 58 49 48 61 44 48 51 53 48
## [2425] 46 53 51 49 49 49 44 45 48 45 55 60 49 54 53 50 53 41 47 52 51 50 43 61
## [2449] 42 55 56 49 49 37 50 47 63 47 52 47 50 51 54 58 53 58 50 45 48 52 42 56
## [2473] 46 60 56 50 61 44 43 53 46 48 52 53 52 55 52 49 52 56 47 51 50 57 48 57
## [2497] 59 53 57 55 52 54 49 49 51 47 55 49 57 48 48 50 53 43 43 46 51 56 47 43
## [2521] 49 53 59 54 45 51 51 51 50 55 41 44 47 49 52 54 51 50 46 49 43 49 48 49
## [2545] 45 54 46 61 51 51 50 51 55 47 37 53 47 52 37 52 58 48 47 48 52 46 45 50
## [2569] 54 54 48 53 50 58 39 53 42 51 51 51 51 43 48 44 57 50 49 50 54 45 51 47
## [2593] 46 48 54 52 48 48 43 57 55 49 52 56 54 56 48 58 52 49 50 54 46 50 58 49
## [2617] 53 49 54 50 34 49 49 49 60 54 43 41 58 56 55 45 47 49 46 46 51 47 53 57
## [2641] 49 46 47 49 61 47 48 50 44 49 47 51 61 52 56 51 52 50 48 49 53 48 51 44
## [2665] 52 45 49 55 51 54 53 67 51 45 51 45 45 45 48 43 52 47 55 50 44 52 43 59
## [2689] 49 54 44 46 52 46 53 48 59 55 40 59 44 51 53 44 52 47 49 47 47 55 47 52
## [2713] 50 56 51 57 42 51 47 52 47 47 66 40 38 50 54 47 54 60 46 52 56 55 49 49
## [2737] 46 48 52 51 50 44 56 55 55 52 45 55 49 48 44 56 52 49 43 48 51 50 45 51
## [2761] 51 49 43 49 56 52 56 46 55 49 44 50 47 43 49 50 54 39 46 52 43 42 44 47
## [2785] 49 50 54 50 43 52 49 50 50 49 48 48 48 50 46 48 41 55 50 46 49 46 46 47
## [2809] 52 47 54 47 44 55 53 51 43 46 51 57 55 42 46 47 49 53 39 53 54 45 46 55
## [2833] 36 54 51 56 53 44 49 58 61 51 48 55 57 50 51 43 35 51 44 45 43 46 57 45
## [2857] 53 46 46 45 56 55 59 49 50 49 47 55 58 48 51 47 49 48 51 46 46 49 49 46
## [2881] 49 54 51 51 55 41 52 44 56 51 41 54 44 44 49 44 47 51 51 56 48 56 47 40
## [2905] 51 45 52 39 51 53 49 48 50 51 50 50 57 54 49 45 55 48 58 52 41 49 47 51
## [2929] 48 50 55 55 47 53 45 45 57 57 55 54 44 52 50 50 50 51 50 50 54 47 52 51
## [2953] 51 53 38 59 49 52 57 53 59 51 44 52 49 48 61 52 57 45 46 54 49 36 56 49
## [2977] 45 50 52 53 54 46 47 43 51 42 50 58 49 52 36 50 50 50 55 48 47 58 37 42
## [3001] 62 52 45 45 43 47 48 51 55 53 52 60 47 49 42 46 56 45 45 54 47 48 50 50
## [3025] 54 53 50 56 59 49 45 50 50 51 54 55 47 58 50 50 59 45 50 48 45 51 52 52
## [3049] 48 52 48 54 51 45 54 52 58 45 55 44 55 43 43 47 55 52 50 45 45 43 53 47
## [3073] 56 58 47 50 51 52 52 51 45 55 55 47 46 52 48 39 55 48 55 50 51 53 56 51
## [3097] 48 52 46 51 39 49 41 56 48 51 48 49 51 61 44 50 61 54 52 52 51 44 58 53
## [3121] 47 47 48 61 52 53 52 39 48 51 49 45 42 54 49 54 57 49 57 46 50 47 54 52
## [3145] 47 48 45 43 56 48 50 55 51 49 42 48 54 55 47 60 54 53 50 59 49 49 52 50
## [3169] 53 55 54 51 44 46 40 52 57 50 48 52 52 49 49 45 45 49 49 52 49 52 49 40
## [3193] 46 57 42 51 55 46 43 61 49 49 48 49 45 48 51 39 45 33 46 51 58 51 54 51
## [3217] 48 54 58 52 48 59 46 45 45 47 43 55 52 51 42 51 51 62 50 55 44 53 44 54
## [3241] 44 51 50 44 50 59 49 49 57 52 51 60 40 53 55 52 47 56 57 49 54 48 46 53
## [3265] 45 55 45 45 57 51 54 55 53 53 47 49 42 52 47 51 41 51 64 40 53 55 58 41
## [3289] 51 57 47 42 48 51 46 48 58 47 59 41 55 48 44 52 54 50 58 45 46 50 48 44
## [3313] 45 56 53 53 53 45 58 51 44 59 51 57 48 52 50 49 59 54 46 49 53 54 47 48
## [3337] 40 47 50 57 54 52 42 56 51 51 52 58 42 45 56 49 49 50 44 46 47 48 43 42
## [3361] 50 49 49 55 42 56 46 54 60 55 46 48 47 49 53 56 52 48 47 43 49 53 54 54
## [3385] 50 49 53 43 51 50 42 49 56 51 55 50 49 52 40 51 52 54 45 50 42 54 46 56
## [3409] 49 45 50 62 48 50 51 48 49 58 49 50 45 56 47 51 50 56 54 54 55 47 54 50
## [3433] 52 46 40 55 48 48 54 59 60 56 59 51 49 50 49 57 49 53 47 54 53 51 54 44
## [3457] 47 53 52 60 44 55 54 57 44 54 49 45 51 61 58 47 53 48 46 54 48 55 46 49
## [3481] 54 51 50 51 52 53 44 57 54 54 45 54 56 49 57 48 43 51 43 46 45 45 55 51
## [3505] 55 48 59 45 51 56 54 43 47 56 55 47 58 48 52 44 55 39 48 52 54 56 47 59
## [3529] 44 53 48 49 43 51 47 59 46 49 53 43 46 47 53 55 46 50 49 53 47 49 42 52
## [3553] 46 53 44 50 49 57 41 52 44 48 41 46 55 54 43 53 55 54 53 45 48 53 55 51
## [3577] 46 52 51 47 61 53 55 50 58 55 51 46 59 44 51 52 51 57 52 51 62 54 56 45
## [3601] 52 45 46 50 50 52 50 47 49 58 53 53 49 45 61 47 54 54 45 42 48 52 50 49
## [3625] 46 47 52 56 67 45 55 49 54 53 50 56 49 48 43 52 49 45 47 52 50 50 57 62
## [3649] 46 51 53 52 49 61 51 57 50 56 41 48 54 53 46 50 51 51 51 58 53 50 40 59
## [3673] 56 47 42 49 54 52 44 54 53 58 55 58 52 46 56 53 52 44 39 50 61 47 58 46
## [3697] 52 57 50 48 55 40 61 46 53 45 59 48 46 58 39 50 50 45 44 47 53 55 51 59
## [3721] 54 49 52 55 48 49 43 48 47 51 43 58 55 54 50 56 50 44 46 50 59 54 56 47
## [3745] 50 45 53 50 52 53 51 43 47 43 41 37 56 54 48 51 51 50 46 46 48 45 54 52
## [3769] 53 49 50 57 51 55 44 42 48 56 48 47 55 55 46 62 51 58 56 54 53 48 35 49
## [3793] 52 51 45 52 50 56 53 56 46 38 45 55 54 63 48 44 50 43 53 50 49 49 49 52
## [3817] 51 47 61 51 54 55 48 48 56 55 50 48 47 49 52 41 55 59 48 42 58 53 47 45
## [3841] 47 50 57 43 47 40 46 51 47 57 52 56 43 45 44 45 55 49 52 45 49 60 54 56
## [3865] 46 49 56 42 58 49 54 51 61 42 54 47 52 43 51 51 55 51 41 58 53 50 49 48
## [3889] 49 52 53 40 49 50 44 51 46 44 45 49 62 44 51 55 39 61 51 48 59 51 48 46
## [3913] 47 51 55 47 52 53 51 48 47 50 56 57 47 56 44 55 44 40 52 49 51 53 46 37
## [3937] 52 54 49 47 56 58 37 49 56 50 46 57 54 48 48 42 54 42 51 41 55 45 48 58
## [3961] 48 50 51 45 56 48 40 48 51 45 43 42 55 42 59 57 52 55 49 48 51 50 49 43
## [3985] 51 54 49 45 49 50 45 51 51 52 58 47 48 56 49 55 41 48 45 47 46 46 47 40
## [4009] 53 54 51 58 44 50 40 49 55 48 50 47 48 47 52 60 52 52 53 47 51 54 44 48
## [4033] 49 53 44 47 36 50 62 42 50 45 54 46 43 46 57 42 49 54 53 50 47 51 39 49
## [4057] 48 48 48 53 45 53 52 46 60 43 50 43 51 55 50 55 45 54 50 58 47 59 47 55
## [4081] 47 56 44 42 52 46 49 45 42 51 53 41 53 41 52 44 48 49 48 58 55 45 49 56
## [4105] 47 52 40 50 54 48 47 58 50 51 48 47 51 56 57 52 47 52 52 48 51 58 56 46
## [4129] 52 53 48 50 43 37 46 49 44 50 64 48 48 50 56 57 47 54 56 54 48 50 48 47
## [4153] 44 48 44 48 40 44 50 52 61 55 56 53 58 60 49 45 52 49 44 43 52 40 47 53
## [4177] 52 42 46 42 49 55 58 54 53 44 49 57 46 53 45 46 50 52 47 52 39 58 57 55
## [4201] 41 51 47 42 48 48 46 57 57 52 47 43 42 47 50 53 54 46 42 54 48 53 48 60
## [4225] 47 56 48 47 43 45 53 49 47 46 42 47 50 55 50 49 56 52 56 54 48 51 52 52
## [4249] 55 48 51 43 62 46 46 50 54 49 51 57 50 54 55 47 50 57 54 44 50 51 43 45
## [4273] 55 49 48 52 45 59 50 50 54 52 50 52 51 53 53 58 47 51 50 48 57 50 46 47
## [4297] 51 53 52 49 46 59 45 54 48 53 55 45 50 55 50 50 43 50 49 51 40 57 49 50
## [4321] 42 52 62 49 48 51 52 49 36 52 54 54 53 43 54 45 45 45 46 46 48 51 39 45
## [4345] 50 45 50 63 53 49 54 51 56 58 46 49 52 48 54 62 48 48 50 52 52 50 55 39
## [4369] 46 50 57 46 49 45 51 51 46 46 51 54 47 42 54 44 59 53 36 49 50 48 44 43
## [4393] 52 53 41 44 46 54 44 51 36 57 49 51 51 44 49 45 52 40 60 53 44 52 49 51
## [4417] 52 47 54 52 57 48 49 51 47 51 42 53 49 47 56 54 43 50 57 53 48 45 53 44
## [4441] 51 55 50 58 37 50 42 42 55 46 46 51 51 56 53 42 55 50 55 52 51 56 53 52
## [4465] 53 57 58 50 48 54 46 48 45 47 56 52 54 49 60 57 56 58 39 56 46 47 51 45
## [4489] 52 58 52 51 48 47 58 51 51 56 48 53 55 54 49 46 41 50 49 55 51 46 54 45
## [4513] 51 54 51 46 54 56 47 49 47 60 49 50 49 48 52 53 44 48 54 55 48 52 53 57
## [4537] 56 54 46 48 52 47 58 55 50 50 50 47 48 56 51 46 55 62 46 49 51 52 49 49
## [4561] 55 52 51 47 52 50 47 50 53 48 47 49 56 48 42 57 50 47 57 50 52 49 51 49
## [4585] 43 52 47 45 48 46 59 46 55 50 54 48 56 49 45 50 51 51 58 42 48 42 44 51
## [4609] 57 51 51 52 50 55 60 47 54 49 51 49 45 49 43 39 50 46 57 37 50 52 59 50
## [4633] 52 55 50 50 54 46 52 58 52 48 51 44 60 52 59 51 46 56 48 43 47 42 55 45
## [4657] 42 49 52 50 43 58 58 48 46 49 48 55 48 55 52 48 54 62 39 54 53 46 58 42
## [4681] 50 55 57 48 50 42 43 50 44 56 47 60 47 44 48 49 42 55 42 48 56 51 59 44
## [4705] 64 48 54 44 47 55 46 57 54 44 48 49 48 50 57 48 41 56 57 47 44 58 51 46
## [4729] 45 58 52 49 55 52 51 49 48 45 42 47 60 54 53 53 44 48 58 47 58 51 56 51
## [4753] 40 60 59 53 56 41 50 48 49 49 47 50 46 54 52 55 56 52 54 53 49 65 51 51
## [4777] 49 54 40 49 39 47 56 53 43 50 48 56 50 48 48 48 48 51 51 42 52 53 50 49
## [4801] 46 50 53 53 46 52 63 46 55 56 46 60 51 58 50 52 53 52 48 51 48 50 52 56
## [4825] 48 55 44 51 50 54 52 51 51 52 61 53 46 50 42 45 48 47 50 53 47 48 53 42
## [4849] 53 56 45 57 57 53 44 56 43 57 57 48 56 52 53 43 54 39 47 52 51 49 54 55
## [4873] 49 46 48 54 50 45 47 46 49 44 54 59 51 58 51 40 45 50 49 43 52 50 54 47
## [4897] 46 49 46 42 44 53 47 51 45 53 49 60 37 62 53 46 55 48 50 53 56 53 52 50
## [4921] 53 41 52 49 40 54 55 45 59 41 45 56 50 46 45 40 51 49 52 48 43 47 45 48
## [4945] 48 48 49 54 51 48 44 58 49 52 54 51 48 58 54 51 53 55 44 47 56 58 53 59
## [4969] 49 59 56 52 50 51 57 42 51 45 47 52 43 57 49 44 49 46 54 49 47 48 50 44
## [4993] 56 42 58 56 55 50 53 51 51 49 54 48 47 46 44 50 52 49 52 54 42 63 43 49
## [5017] 59 57 50 46 48 55 57 48 48 45 42 48 49 48 52 44 54 61 53 45 48 51 56 50
## [5041] 51 50 58 50 53 58 46 53 47 49 59 52 58 38 44 43 53 56 49 51 50 58 52 44
## [5065] 44 58 47 58 49 47 47 43 39 38 46 42 51 44 39 56 45 52 48 47 56 51 51 49
## [5089] 56 48 50 53 53 52 58 58 50 47 51 48 47 45 47 39 54 40 44 56 47 54 51 47
## [5113] 44 56 59 51 52 55 54 55 43 43 49 53 56 58 51 44 56 54 48 46 56 48 55 52
## [5137] 42 52 52 55 48 59 59 46 59 60 58 53 48 52 53 47 47 52 57 49 58 54 58 50
## [5161] 49 56 54 53 49 52 50 43 54 48 45 44 51 41 50 51 51 51 47 51 56 54 48 58
## [5185] 48 52 52 59 54 56 52 55 46 47 57 55 52 55 58 50 45 45 43 47 52 47 51 49
## [5209] 48 48 47 49 54 52 49 42 52 52 52 40 55 44 56 41 49 55 44 50 54 45 47 40
## [5233] 51 52 49 52 51 48 46 54 54 42 53 52 48 58 45 49 46 50 54 46 44 54 55 58
## [5257] 53 49 57 58 52 52 55 49 55 51 53 52 50 48 55 52 42 46 49 54 49 51 53 46
## [5281] 55 48 42 58 55 43 54 40 52 55 50 51 54 47 48 53 50 55 53 52 49 45 51 50
## [5305] 45 52 59 53 54 48 38 52 59 48 39 56 49 52 53 51 58 47 39 54 42 51 55 50
## [5329] 49 58 49 51 58 53 42 50 61 58 37 47 47 56 46 54 48 50 52 49 50 51 47 42
## [5353] 57 52 48 45 50 55 52 41 46 44 46 47 58 52 49 55 56 52 46 60 49 49 47 45
## [5377] 56 47 45 50 53 53 49 47 53 57 55 52 55 48 58 53 47 47 51 44 50 48 53 58
## [5401] 41 52 52 47 44 54 59 44 43 53 39 41 43 56 44 48 42 50 55 41 47 44 41 57
## [5425] 43 54 42 39 51 47 47 43 49 55 46 47 57 51 43 53 48 49 48 53 49 52 60 45
## [5449] 53 61 47 50 47 48 58 42 45 54 57 50 44 56 57 49 45 50 53 54 53 56 50 48
## [5473] 60 47 53 50 56 56 48 48 45 60 44 43 56 52 50 48 49 38 53 49 49 48 46 38
## [5497] 54 50 55 49 48 55 43 50 52 50 53 54 51 63 44 54 44 51 51 50 47 53 54 50
## [5521] 43 57 55 48 39 47 52 53 54 56 55 54 47 53 52 48 47 49 47 44 49 50 56 43
## [5545] 56 44 35 49 51 52 45 47 44 52 48 51 44 55 54 64 55 60 51 52 52 52 49 51
## [5569] 48 55 52 48 58 59 43 58 52 53 55 44 49 43 51 49 46 50 45 58 46 47 49 53
## [5593] 45 47 50 40 53 62 45 52 49 51 37 47 49 58 47 50 55 49 43 57 50 49 57 56
## [5617] 53 51 44 46 51 58 49 38 43 54 40 45 45 48 50 51 50 55 56 51 53 44 51 48
## [5641] 51 45 52 51 52 51 53 48 41 58 51 54 43 54 47 44 56 54 61 52 57 48 55 52
## [5665] 48 44 50 61 51 50 59 52 53 49 52 50 53 51 62 50 49 51 58 63 54 57 48 43
## [5689] 53 53 44 47 51 41 48 53 55 46 53 53 50 42 41 55 43 52 49 53 55 46 44 53
## [5713] 58 43 48 52 53 44 53 51 51 43 60 58 51 58 51 52 52 43 54 65 53 54 50 51
## [5737] 49 49 46 54 52 54 38 43 44 47 38 53 48 53 56 50 47 50 48 50 51 52 55 49
## [5761] 51 49 49 53 50 49 51 46 49 41 44 47 51 62 48 44 47 56 47 52 50 47 50 51
## [5785] 53 52 49 41 45 54 45 58 49 55 49 48 52 53 61 54 40 57 56 53 53 51 42 55
## [5809] 50 62 47 43 46 53 50 44 52 46 45 53 54 39 44 53 53 47 55 48 47 52 45 49
## [5833] 52 54 56 44 56 49 53 45 49 54 44 44 48 57 50 50 53 46 47 38 47 58 43 47
## [5857] 62 44 49 61 52 50 44 53 49 43 42 50 49 51 47 52 57 45 54 44 55 43 55 41
## [5881] 57 48 52 50 50 49 57 56 37 44 44 53 56 50 44 35 60 50 52 46 56 48 52 46
## [5905] 55 50 47 50 52 54 52 52 49 53 50 50 50 52 38 45 48 46 50 47 56 58 49 55
## [5929] 54 45 54 58 47 50 43 48 55 54 49 51 39 50 43 51 42 47 54 50 55 60 37 43
## [5953] 48 43 54 52 48 49 52 52 57 53 49 49 45 45 50 46 43 53 52 50 51 51 53 50
## [5977] 57 48 47 47 51 49 56 46 61 53 57 52 52 51 57 45 51 47 59 51 45 50 47 47
## [6001] 50 34 55 47 53 56 46 47 56 46 43 51 53 51 47 49 49 60 45 47 57 50 49 52
## [6025] 49 39 52 53 53 57 46 48 49 40 55 48 56 47 54 45 52 51 60 50 48 57 50 58
## [6049] 52 55 45 44 43 57 40 58 47 49 40 69 54 46 45 44 55 48 52 55 38 60 51 52
## [6073] 47 43 45 59 49 52 54 54 45 50 49 49 53 47 49 48 50 53 49 49 52 45 54 54
## [6097] 43 54 50 50 52 49 48 42 52 47 49 51 56 51 52 55 57 63 55 44 51 50 51 48
## [6121] 49 46 46 56 44 55 47 47 58 53 51 39 52 50 50 50 49 58 44 59 58 50 53 54
## [6145] 56 59 62 50 54 55 49 59 40 50 52 55 48 52 51 62 57 46 47 57 49 51 48 50
## [6169] 52 53 46 45 55 50 51 48 53 48 57 43 51 43 36 57 54 50 43 45 49 48 49 52
## [6193] 50 50 52 49 51 54 51 46 59 51 57 51 43 45 53 58 60 56 51 48 47 51 38 53
## [6217] 40 44 46 50 49 44 52 50 57 61 48 53 44 52 47 48 54 50 49 47 46 48 50 56
## [6241] 53 52 54 48 50 47 44 47 45 44 54 50 52 60 50 45 51 46 54 58 62 53 46 51
## [6265] 51 48 48 47 50 53 48 53 46 46 51 56 44 47 52 44 57 54 53 46 43 50 50 45
## [6289] 51 45 48 40 55 48 54 49 49 51 51 52 52 41 42 45 50 52 55 50 49 59 55 46
## [6313] 54 50 53 48 44 43 46 49 54 51 53 56 57 51 51 51 51 59 43 45 60 43 51 52
## [6337] 53 56 57 48 51 48 44 46 56 48 41 46 56 43 54 48 55 55 55 44 44 55 56 57
## [6361] 44 43 49 64 49 56 43 47 58 50 53 45 50 54 46 45 49 48 53 50 52 42 55 51
## [6385] 48 46 46 50 49 57 60 49 45 49 56 42 54 51 58 56 55 48 53 51 46 47 49 51
## [6409] 52 52 48 54 49 44 50 41 59 42 60 42 49 54 51 51 58 51 49 48 49 41 52 54
## [6433] 52 57 60 50 55 56 58 54 48 52 46 50 57 55 53 52 51 42 46 43 49 50 54 55
## [6457] 49 45 58 52 53 51 45 45 51 48 52 49 51 52 42 47 51 52 45 57 50 48 45 57
## [6481] 64 53 59 54 56 45 46 46 51 56 49 53 50 60 54 47 47 42 46 50 53 51 60 55
## [6505] 51 58 52 54 59 45 44 56 42 50 51 50 54 45 51 53 42 48 53 52 46 52 47 47
## [6529] 48 58 38 50 49 51 55 50 56 54 59 47 51 54 59 48 62 56 43 50 60 52 50 54
## [6553] 45 52 57 47 48 46 52 44 43 50 56 48 47 49 49 45 45 55 47 51 56 40 52 59
## [6577] 51 56 60 46 48 46 48 62 57 61 46 49 56 48 46 58 55 43 53 56 44 54 47 55
## [6601] 49 48 60 44 50 48 50 41 59 51 56 49 46 53 49 48 52 50 45 56 47 46 52 56
## [6625] 45 48 52 50 51 52 45 50 57 43 46 53 48 46 52 55 47 42 50 47 49 54 51 55
## [6649] 51 44 52 46 44 47 51 49 56 43 57 60 52 52 57 46 45 40 49 42 52 48 61 62
## [6673] 53 63 49 55 53 47 51 54 50 44 49 49 47 56 55 49 50 55 50 56 45 51 49 49
## [6697] 50 49 50 51 55 47 47 58 51 49 42 45 53 56 51 44 45 42 44 53 62 50 42 43
## [6721] 43 52 44 44 45 51 52 50 52 57 38 46 50 54 42 48 46 47 44 51 47 44 51 44
## [6745] 44 53 48 59 56 56 57 42 51 51 58 53 50 51 50 58 47 51 47 46 44 51 39 50
## [6769] 48 51 52 53 53 51 49 49 53 60 50 54 47 55 63 54 50 55 46 48 42 46 61 50
## [6793] 43 51 41 51 55 48 53 51 44 43 55 51 48 49 51 57 42 67 47 43 44 53 46 56
## [6817] 53 49 50 50 56 38 47 57 53 50 58 51 50 42 63 51 52 55 51 50 52 42 41 44
## [6841] 43 51 52 46 47 43 59 51 46 46 54 52 49 54 47 54 46 48 56 48 52 49 47 52
## [6865] 55 47 49 47 48 41 54 49 48 51 42 53 58 57 52 52 59 58 48 49 45 57 50 45
## [6889] 56 45 41 51 49 43 40 57 52 46 47 56 40 42 54 47 50 52 45 49 54 47 49 57
## [6913] 51 58 59 48 53 55 43 56 43 47 37 49 46 52 50 46 53 44 49 59 51 51 55 51
## [6937] 57 53 50 50 55 59 49 53 49 48 53 53 50 53 44 47 59 48 52 43 49 50 49 49
## [6961] 45 49 42 49 49 50 54 48 52 55 50 46 46 43 54 50 56 47 44 53 50 46 53 49
## [6985] 47 51 57 57 54 52 60 44 51 45 53 42 42 52 49 53 44 53 48 47 44 52 47 62
## [7009] 47 47 47 54 45 46 56 48 52 39 44 50 50 51 58 52 50 52 50 52 46 41 57 48
## [7033] 52 44 54 60 55 48 52 51 49 51 52 48 44 45 42 49 50 43 47 55 54 42 55 51
## [7057] 53 57 55 50 56 44 47 44 44 41 46 47 50 49 48 47 59 53 46 53 51 60 53 48
## [7081] 56 38 38 50 51 47 54 47 47 39 57 45 50 50 51 50 44 44 45 44 45 51 50 51
## [7105] 56 51 47 43 58 51 47 49 45 52 44 51 48 46 53 52 50 61 47 47 55 45 51 57
## [7129] 48 50 58 47 42 52 55 50 51 49 63 54 50 56 46 47 58 53 51 57 52 38 53 51
## [7153] 50 57 43 45 49 60 45 49 43 56 53 49 53 47 54 49 54 44 40 49 42 42 53 58
## [7177] 51 43 44 49 49 45 50 43 48 61 54 56 50 49 47 42 51 61 56 41 47 50 55 48
## [7201] 50 53 36 46 52 53 46 53 44 46 56 51 58 42 57 40 55 54 47 45 51 56 44 45
## [7225] 49 48 44 48 40 47 46 50 46 45 52 49 46 51 55 38 51 57 49 58 53 50 46 55
## [7249] 51 55 45 49 45 44 41 49 46 53 56 46 48 52 56 55 44 57 52 54 54 39 52 44
## [7273] 51 46 47 45 47 43 41 40 63 47 49 58 52 56 47 50 56 56 55 49 53 52 47 47
## [7297] 51 44 56 53 55 52 43 44 50 49 44 47 61 56 51 47 52 46 56 48 55 49 46 59
## [7321] 54 49 56 46 39 55 54 52 47 49 50 42 51 55 50 50 52 53 50 53 45 50 54 55
## [7345] 49 44 60 47 56 49 53 54 45 49 54 46 48 50 55 51 50 48 51 41 41 57 48 52
## [7369] 50 47 58 53 44 62 50 52 54 48 47 46 52 52 54 57 50 51 48 48 54 53 53 49
## [7393] 52 42 58 48 47 54 58 45 54 63 44 59 47 46 51 60 54 46 51 55 53 54 48 42
## [7417] 49 44 43 50 52 54 53 54 49 41 48 53 50 50 56 54 51 50 53 47 51 55 50 55
## [7441] 52 49 47 50 53 53 52 47 42 46 55 39 43 40 49 56 46 47 52 44 50 50 56 53
## [7465] 53 56 50 46 51 51 54 53 44 51 45 48 43 46 51 46 57 44 48 50 53 54 51 48
## [7489] 53 47 56 52 48 48 45 42 59 44 48 54 47 46 49 52 53 55 49 56 48 47 45 51
## [7513] 54 44 55 53 54 52 56 58 60 45 44 45 37 41 47 47 51 53 52 54 53 48 50 38
## [7537] 54 48 44 43 42 49 49 39 39 49 44 45 51 49 60 53 51 54 59 50 52 50 49 47
## [7561] 44 50 53 52 56 56 49 53 52 44 47 51 46 48 59 48 54 59 48 46 50 49 58 48
## [7585] 55 43 55 55 51 56 47 45 49 47 42 58 42 47 60 61 50 40 60 47 54 54 46 48
## [7609] 44 52 51 53 47 57 44 53 47 47 54 58 49 51 46 51 47 49 54 44 47 55 56 45
## [7633] 50 53 42 42 59 53 52 56 58 53 61 49 52 51 51 46 44 46 48 58 49 50 49 52
## [7657] 44 53 61 56 51 61 58 54 47 47 47 47 53 52 44 54 50 56 52 57 56 57 52 54
## [7681] 50 48 54 42 46 39 52 53 46 45 52 45 52 54 55 49 51 56 46 50 56 48 57 49
## [7705] 51 47 49 50 60 43 50 41 55 47 63 54 55 52 52 53 53 45 57 48 47 50 54 48
## [7729] 46 46 44 51 47 55 52 49 46 44 55 55 55 51 51 41 58 46 40 45 47 55 53 51
## [7753] 54 55 56 51 49 52 51 56 55 60 49 50 54 52 57 43 60 45 46 52 52 56 52 43
## [7777] 56 50 48 48 39 48 54 44 60 50 52 41 41 54 54 48 49 52 45 53 50 49 45 58
## [7801] 53 48 45 55 49 50 53 59 48 46 46 56 55 45 43 50 48 54 49 49 44 48 49 48
## [7825] 36 45 44 49 55 48 56 49 51 47 50 51 50 43 50 56 49 41 47 48 43 42 52 59
## [7849] 58 50 46 51 52 52 55 48 51 51 50 36 46 58 51 38 51 52 44 54 44 55 45 48
## [7873] 54 41 48 45 46 53 51 43 44 50 52 55 56 44 50 48 43 48 49 52 37 44 50 49
## [7897] 50 52 49 51 49 57 47 48 59 53 48 50 42 52 52 47 46 54 54 51 48 52 48 51
## [7921] 51 55 55 57 51 41 43 61 42 57 45 43 47 41 46 53 50 50 56 49 52 51 46 45
## [7945] 49 45 48 50 51 47 43 54 51 48 57 52 48 43 56 50 53 45 55 51 53 53 47 44
## [7969] 51 52 48 50 39 44 46 44 56 46 59 54 59 47 53 44 50 59 39 61 45 54 57 42
## [7993] 43 54 54 52 51 45 55 53 63 41 50 51 49 53 58 45 50 48 49 55 57 57 43 53
## [8017] 49 45 57 53 54 50 54 52 54 55 49 55 57 45 46 49 52 48 51 56 48 51 47 44
## [8041] 42 48 57 45 54 47 48 51 59 43 45 43 58 48 50 54 42 61 42 55 51 41 52 44
## [8065] 40 49 50 52 47 54 50 46 58 54 51 44 53 47 41 52 55 45 52 47 53 59 57 57
## [8089] 39 50 56 46 55 49 48 48 45 50 63 52 53 50 53 48 51 45 50 51 55 53 48 44
## [8113] 53 58 47 42 54 57 55 48 43 50 51 54 46 49 44 52 56 49 53 47 44 52 40 53
## [8137] 51 53 54 49 49 47 52 51 53 46 46 50 51 60 42 54 50 42 48 51 50 46 53 45
## [8161] 52 51 53 45 48 43 53 56 49 48 56 52 53 48 54 49 48 50 51 48 48 54 47 51
## [8185] 44 43 53 45 51 51 39 39 63 57 44 48 57 48 53 53 44 42 50 44 51 46 53 55
## [8209] 44 58 47 49 54 48 52 48 50 40 50 48 42 44 51 47 48 43 45 47 49 50 47 48
## [8233] 52 47 38 48 53 56 45 44 55 51 47 48 54 52 50 44 53 48 47 52 44 45 51 50
## [8257] 44 48 48 46 56 54 50 47 48 55 50 55 47 49 49 45 44 62 41 45 48 50 56 52
## [8281] 52 54 49 49 45 48 41 42 51 46 48 49 46 46 55 49 45 43 46 53 65 47 55 47
## [8305] 60 53 43 50 47 52 44 49 50 44 52 36 54 45 56 51 49 58 49 56 52 51 48 45
## [8329] 48 55 45 58 51 55 53 55 46 44 50 52 55 55 46 51 55 53 40 46 53 54 52 51
## [8353] 47 51 50 46 54 49 52 51 54 49 51 51 46 41 46 43 50 44 45 43 52 54 50 50
## [8377] 53 58 51 50 47 46 43 45 51 43 56 48 48 54 49 55 55 48 40 53 48 43 46 48
## [8401] 51 51 48 59 57 52 56 56 51 50 55 54 47 45 45 50 46 56 50 44 42 55 54 60
## [8425] 57 42 46 49 51 46 47 47 53 44 49 50 45 60 40 51 41 48 53 46 49 47 50 50
## [8449] 47 54 51 53 51 44 55 56 41 51 61 57 51 50 55 52 48 43 57 52 44 47 55 48
## [8473] 50 49 47 51 48 43 58 44 50 43 39 45 45 43 53 44 48 50 51 52 50 45 58 47
## [8497] 51 49 47 57 56 49 40 51 61 54 46 60 52 54 45 57 48 57 56 53 54 59 55 48
## [8521] 51 48 48 45 45 51 47 56 45 50 53 49 47 55 51 55 51 54 53 58 55 44 59 51
## [8545] 52 59 44 55 53 46 48 50 38 54 50 58 43 58 52 53 51 51 53 56 47 53 46 56
## [8569] 47 49 47 42 49 52 50 48 53 49 57 48 55 58 42 45 46 50 48 40 51 52 47 53
## [8593] 38 49 50 53 57 41 46 47 52 54 49 59 52 50 46 42 48 54 42 56 50 48 49 51
## [8617] 55 46 50 42 46 60 37 56 50 42 45 53 53 54 52 48 52 49 55 52 48 39 47 55
## [8641] 54 44 45 53 49 56 51 45 45 46 47 42 53 50 57 49 58 47 46 49 44 49 51 54
## [8665] 45 58 46 41 55 47 50 42 55 44 55 57 54 47 50 54 42 49 60 49 61 43 48 51
## [8689] 47 45 47 44 52 48 44 57 55 49 53 53 52 47 52 55 50 52 49 40 50 44 51 56
## [8713] 47 46 53 46 48 49 56 50 61 49 54 45 50 47 51 51 50 48 47 51 49 53 53 50
## [8737] 47 50 55 48 45 45 48 48 54 47 52 46 49 54 51 47 48 62 55 55 50 57 52 51
## [8761] 49 50 50 61 37 52 49 49 48 53 62 51 45 62 48 44 56 52 60 52 54 51 56 51
## [8785] 49 54 39 51 50 48 49 50 45 53 49 43 53 55 49 40 52 57 45 47 52 46 47 51
## [8809] 43 52 56 49 53 49 47 55 53 62 49 52 42 60 47 51 53 46 46 36 55 52 44 56
## [8833] 47 50 52 50 50 50 48 47 54 56 50 49 50 54 48 46 51 46 50 46 42 45 53 49
## [8857] 41 45 51 50 46 63 50 50 48 55 54 53 49 55 50 46 47 43 52 49 44 54 48 52
## [8881] 51 52 54 56 48 48 51 57 40 55 48 46 42 55 54 50 52 50 50 38 56 52 60 49
## [8905] 39 49 53 46 44 54 56 46 50 53 54 52 54 52 50 57 41 51 56 53 53 41 48 48
## [8929] 52 47 51 48 52 41 53 47 41 45 44 52 49 52 48 49 50 59 45 48 50 55 61 49
## [8953] 58 55 58 55 52 54 43 50 43 57 54 49 48 57 56 46 51 47 52 48 42 52 46 45
## [8977] 45 51 49 50 50 48 51 52 52 41 40 50 44 56 50 55 50 49 44 52 46 47 53 47
## [9001] 54 36 51 57 51 51 52 51 54 56 55 50 60 45 51 54 46 52 53 53 51 51 44 47
## [9025] 55 53 50 45 50 44 57 45 48 45 47 50 51 50 51 51 54 46 50 51 53 49 52 56
## [9049] 54 47 44 44 53 50 46 39 48 55 55 56 41 49 56 61 52 46 43 41 46 54 50 48
## [9073] 57 50 49 45 41 48 52 56 58 46 45 58 44 48 44 50 33 44 56 41 50 55 39 47
## [9097] 45 46 55 51 58 44 62 59 53 49 51 52 49 50 48 50 46 47 48 54 49 40 38 47
## [9121] 57 49 55 46 46 47 53 50 53 49 56 42 45 48 56 44 43 53 47 58 60 50 52 40
## [9145] 46 59 48 50 58 53 48 38 54 48 51 60 49 58 49 42 46 50 52 45 47 50 51 54
## [9169] 42 51 53 51 50 53 50 44 42 51 56 60 50 43 56 52 48 48 57 50 36 53 56 54
## [9193] 51 44 52 52 54 43 55 60 50 48 46 50 42 48 53 58 45 53 51 43 52 50 45 51
## [9217] 51 56 59 52 59 52 48 42 52 50 42 53 48 46 49 57 50 45 44 51 47 46 42 44
## [9241] 56 45 49 64 54 56 54 47 52 47 63 58 50 53 50 50 43 59 52 45 50 55 51 58
## [9265] 51 49 50 59 45 45 60 47 45 51 49 49 47 44 48 48 41 53 47 48 53 52 63 53
## [9289] 51 48 54 47 52 59 53 52 53 46 47 57 51 46 53 52 52 46 54 53 60 54 50 54
## [9313] 46 44 52 47 46 54 52 53 53 50 46 40 48 51 48 60 42 51 51 45 45 50 50 44
## [9337] 59 43 57 56 50 51 50 49 50 41 53 50 41 49 56 50 55 40 50 53 44 53 56 51
## [9361] 42 51 49 52 47 42 53 43 47 57 57 51 57 53 47 55 56 52 61 46 55 49 55 55
## [9385] 38 58 49 40 49 47 50 55 47 48 49 51 42 54 51 44 48 53 46 49 53 48 47 58
## [9409] 58 48 45 48 47 53 50 51 53 53 51 51 44 45 47 50 49 47 47 53 49 49 48 46
## [9433] 47 57 42 43 52 54 59 48 49 48 52 48 60 48 51 43 41 45 59 47 55 53 45 44
## [9457] 46 53 54 43 48 47 46 50 48 51 40 47 50 49 54 43 54 50 46 50 40 53 45 43
## [9481] 43 52 48 51 54 52 58 58 46 54 48 51 42 49 42 51 51 57 57 44 45 52 45 48
## [9505] 42 50 47 46 46 45 59 43 56 60 42 57 60 39 45 44 46 42 51 50 49 55 47 37
## [9529] 48 43 58 64 53 52 48 56 50 53 45 57 50 46 46 46 51 49 51 51 40 47 43 46
## [9553] 47 48 48 55 37 55 56 58 45 47 47 54 44 58 52 53 49 48 46 47 46 46 49 46
## [9577] 53 50 46 57 46 55 52 54 49 43 56 42 47 46 35 50 50 56 42 55 47 49 48 49
## [9601] 46 53 51 49 45 45 47 53 55 46 44 44 51 52 45 59 51 58 43 55 57 49 51 56
## [9625] 51 45 52 56 51 51 54 47 51 45 47 47 52 38 46 41 58 42 52 55 52 45 50 60
## [9649] 55 56 45 47 51 46 48 46 50 45 55 58 50 57 44 46 54 40 55 52 49 54 45 48
## [9673] 56 51 43 54 47 55 45 46 48 49 45 48 55 55 55 48 47 44 65 52 47 47 53 53
## [9697] 51 49 49 46 49 51 47 47 55 51 51 46 58 54 49 59 49 56 56 43 52 45 54 47
## [9721] 50 47 49 57 51 53 52 52 50 56 48 53 51 45 52 41 47 58 54 46 43 47 45 49
## [9745] 56 51 51 52 51 50 48 45 45 52 45 43 47 50 49 59 58 42 53 58 53 60 45 43
## [9769] 49 51 54 47 51 46 51 53 52 55 52 54 51 48 48 56 41 54 56 59 54 55 43 45
## [9793] 53 52 43 47 59 56 54 45 35 40 53 43 52 43 54 46 46 52 47 48 47 46 45 43
## [9817] 52 52 42 56 49 44 51 55 45 51 48 54 44 52 38 50 45 44 42 51 52 52 50 51
## [9841] 45 47 47 55 59 55 53 53 54 46 47 54 51 36 57 48 50 53 35 50 54 52 53 55
## [9865] 49 60 49 50 45 55 53 50 54 50 43 43 51 49 45 49 46 54 50 47 50 49 44 42
## [9889] 53 53 51 50 57 45 57 49 48 41 53 47 56 49 58 48 56 55 39 48 50 49 49 44
## [9913] 57 51 48 53 48 64 52 51 46 42 49 52 41 63 52 47 41 58 51 49 44 59 55 41
## [9937] 44 49 50 47 55 51 52 58 55 54 53 57 47 42 48 51 44 44 52 45 54 48 46 56
## [9961] 54 57 33 60 47 50 52 46 48 50 49 51 50 50 38 50 54 52 46 50 48 47 43 49
## [9985] 38 50 52 48 58 47 54 43 52 47 49 52 53 52 51 46## determine 25th percentile
percentile_25th <- quantile(mybinom2) # check 25th percenitle of actual distribution
percentile_25th## 0% 25% 50% 75% 100%
## 33 47 50 53 69## run summary commands to check
summary(mybinom2) # summary statistics## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 33.00 47.00 50.00 50.02 53.00 69.00# generate sample
z <- matrix(data = rep(x = 0,
times = 40000
),
nrow = 10000,
ncol = 4)
n <- c(2, 6, 30, 1000) # let n be the sample size
#larger sample size should have closer sample mean to true population mean
for (j in 1:4){ # indexing columns of matrix, where each column will represent different sample size
for (i in 1:10000){ # indexes the rows of matrix
z[i,j] <- mean(sample( x = mybinom2, # compute mean and assign
size = n[j],
replace = TRUE
)
)
}
}
colnames(z) <- c("Sample size=2", "Sample size=6", "Sample size=30", "Sample size=1000")
summary(z) ## Sample size=2 Sample size=6 Sample size=30 Sample size=1000
## Min. :37.50 Min. :43.17 Min. :46.47 Min. :49.40
## 1st Qu.:47.50 1st Qu.:48.67 1st Qu.:49.40 1st Qu.:49.91
## Median :50.00 Median :50.00 Median :50.03 Median :50.02
## Mean :49.97 Mean :50.03 Mean :50.02 Mean :50.02
## 3rd Qu.:52.50 3rd Qu.:51.33 3rd Qu.:50.67 3rd Qu.:50.13
## Max. :63.00 Max. :57.33 Max. :53.33 Max. :50.64par(mfrow = c(3,2))
hist(x = mybinom2, main = "Binomial Distribution of Sample 25th Percentiles")
for (k in 1:4){
hist(x = z[,k], # matrix column 1-4 which contain sample means of randomly chosen samples from uniform dist
main = "",
xlim = c(30,70), # plot the domain of X axis from x=10 to x=20 only
xlab = paste0("Sample Size ", n[k], " (Column ", k, " from matrix)")
)
} 
Yes, my findings align witht the Central Limit Theorum, becuase as the sample statistics converge, sample mean, and sample 25th percentile, converge toward their population values as the same size increases. As seen above, as sample size increases, the peak of the sample narrows converging toward the population, respbling a near but not perfect normal distrubtion. These histograms support the overall meaning behind the central limit theorem, which states that the distrubuion if samples, tends to become almost normal as samle sizes increase infinitealy.