Instructions

  1. Update the author line at the top to have your name in it.
  2. You must knit this document to an html file and publish it to RPubs. Once you have published your project to the web, you must copy the web url link into the appropriate Course Project assignment in MyOpenMath before 11:59pm on the due date.
  3. Answer all the following questions completely. Some may ask for written responses.
  4. Use R chunks for code to be evaluated where needed and always comment all of your code so the reader can understand what your code aims to accomplish.
  5. Proofread your knitted document before publishing it to ensure it looks the way you want it to. Tip: Use double spaces at the end of a line to create a line break and make sure text does not have a header label that isn’t supposed to.
  6. DELETE ALL THESE INSTRUCTIONS BEFORE PUBLISHING YOUR DOCUMENT.

Purpose

In this project, students will demonstrate their understanding of probability and the normal and binomial distributions.

Question 1

Assume IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. If a person is randomly selected, find each of the requested probabilities. Here, x, denotes the IQ of the randomly selected person.

IQ score: x~N (mean=100,sd=15)

  1. P(x > 65)
#This code finds the probability of a random person having an IQ score greater than 65
1-pnorm(q=65,mean=100, sd=15)
## [1] 0.9901847

The probability of a random person having in IQ score greater than 65 is 0.9902.

  1. P(x < 150)
#This code finds the probability of a random person having an IQ score less than 150
pnorm(q=150,mean=100,sd=15)
## [1] 0.9995709

The probability of a random person having in IQ score less than 150 is 0.9996.

Question 2

Assume the same mean and standard deviation of IQ scores that was described in question 1.

  1. A high school offers a special program for gifted students. In order to qualify, students must have IQ scores in the top 5%. What is the minimum qualifying IQ?
#These codes will find the IQ score that denotes the 95th percentile (top 5%) and the equivalent way to calculate it
qnorm(0.95,mean = 100,sd=15)
## [1] 124.6728
qnorm(0.05, mean = 100, sd=15, lower.tail = FALSE)
## [1] 124.6728

The minimum qualifying IQ to be in the top 5%is 124.67.

  1. If one person is randomly selected, what is the probability that their IQ score is greater than 110?
#This code will find the probabilty of a randomly selected person having an IQ score of more than 110
1-pnorm(q=110, mean=100, sd=15)
## [1] 0.2524925

The probability of a randomly selected person having an IQ score of more than 110 is 0.2525.

Question 3

  1. Still using the mean and standard deviation from question 1, what is the z-score for an IQ of 140?
#This code finds the Z-Score for an IQ of 140 using a mean of 100 and sd of 15. 
(140 -100)/15
## [1] 2.666667

The Z-Score for an IQ of 140 is 2.67.

  1. We mentioned in week 6 that a data value is considered “unusual” if it lies more than two standard deviations from the mean. Is an IQ of 140 considered unusual?

An IQ of 140 is considered to be “unusal” because it is more than two standard deviations from the mean (mean=100 and sd=15).

  1. What is the probability of getting an IQ greater than 140?
#Code for the probability of getting IQ greater than 140 and an alternative code of finding the same informtion.
1-pnorm(q=140, mean=100, sd=15)
## [1] 0.003830381
pnorm(q=140, mean = 100, sd=15, lower.tail=FALSE)
## [1] 0.003830381

The probability of getting an IQ greater than 140 is 0.0038.

Question 4

You are taking a 15-question multiple choice quiz and each question has 5 options (a,b,c,d,e) and you randomly guess every question.

  1. How many questions do you expect to answer correctly on average?
#This will give us the probability of getting one question correct
(1/5)
## [1] 0.2
#How many questions you can expect to answer correctly on average (n*p)
15*0.2
## [1] 3

On average, you would expect to answer 3 questions correctly.

  1. What is the probability that you get every question correct?
#Probability of getting exactly 15 questions correct out of 15 questions with p=0.2
dbinom(x=15, size=15, prob=0.2)
## [1] 3.2768e-11

The probability is very low, less than 0% chance of getting every question correct.

  1. What is the probability that you get every question incorrect?
#Probability of getting exactly 0 questions incorrect out of 15 questions with p=0.2
dbinom(x=0, size=15, prob=0.2)
## [1] 0.03518437

The probability of getting every question incorrect is 0.0352.

Question 5

Consider still the 15-question multiple choice quiz that each question has 5 options (a,b,c,d,e) and you randomly guess every question.

  1. How many questions does one need to answer correctly in order score exactly a 60%?
#How many questions out of 15 need to be answered correctly to score a 60% (n*p)
15*.60
## [1] 9

One would need to answer 9 questions out of 15 correctly to get exactly a 60%.

  1. If a grade of 60% or lower is considered failing, then what is the probability of you failing?
#Probability of answering 9 or fewer correct 
pbinom(9,size=15,prob=0.2)
## [1] 0.9998868

The probability of failing by randomly guessing each question is 0.9998, which is a very high probability.

  1. If you need a grade of 80% or higher on this quiz to maintain a passing grade, what is the probability of you maintaining that passing grade?
#How many questions need to be answered correctly to get an 80%
.80*15
## [1] 12
#Probability of maintaining passing grade 
1-pbinom(11, size=15, prob=0.2)
## [1] 1.011253e-06

The probability of maintaining a passing grade of 80% or higher on this quiz is 1.011253e-06 which is very low and below 0%.

Question 6

Suppose you own a catering company. You hire local college students as servers. Not being the most reliable employees, there is an 80% chance that any one server will actually show up for a scheduled event. For a wedding scheduled on Saturday, you need at least 5 servers.
a. Suppose you schedule 5 employees, what is the probability that all 5 come to work?

#Probability that all 5 employees come to work 
dbinom(x=5, size=5, prob=.80)
## [1] 0.32768

The probability of all 5 scheduled employees coming to work is 0.3277.

  1. Suppose you schedule 7 employees, what is the probability that at least 5 come to work?
#Probability that 5 out of 7 scheduled employees come to work 
1-pbinom(4, size=7, prob=.80)
## [1] 0.851968

The probability of 5 out of 7 scheduled employees coming to work is 0.8520.

  1. It is really important that you have at least 5 servers show up! How many employees should you schedule in order to be 99% confident that at least 5 show up? Hint: there is no single formula for the answer here, so maybe use some kind of trial and error method.
#Show data frame with range of probabilities to determine how many employees need to be scheduled to achieve 99% confidence 5 will show up
data.frame(employees_scheduled = c(5:10), prob = pbinom(q =4, size = c(5:10), prob = 0.80, lower.tail = FALSE))
##   employees_scheduled      prob
## 1                   5 0.3276800
## 2                   6 0.6553600
## 3                   7 0.8519680
## 4                   8 0.9437184
## 5                   9 0.9804186
## 6                  10 0.9936306
#This code will confirm whether the suspected number of employees needed to be scheduled in order for 5 to show up from the step above will give us 99% 
1-pbinom(4, size=10, prob=.80)
## [1] 0.9936306

You would need to schedule at least 10 employees to be 99% confident that at least 5 show up.

Question 7

  1. Generate a random sample of 10,000 numbers from a normal distribution with mean of 51 and standard deviation of 7. Store that data in object called rand_nums.
#set seed function to get reproducible results
set.seed(3)
#create a vector of 10,000 randomly sampled numbers (rounded to whole number) and store in an object named rand_nums
rand_nums<- round(rnorm(n=10000, mean=51, sd=7), 0)
rand_nums
##     [1] 44 49 53 43 52 51 52 59 42 60 46 43 46 53 52 49 44 46 60 52 47 44
##    [23] 50 39 48 46 59 58 50 43 57 57 56 56 49 56 60 51 44 57 57 49 63 45
##    [45] 53 35 50 59 48 45 56 45 53 39 41 48 44 61 57 46 55 57 53 53 59 48
##    [67] 48 58 42 52 51 54 58 53 53 56 60 54 44 50 63 49 56 60 57 51 53 45
##    [89] 54 45 45 44 46 58 48 51 48 55 58 50 55 48 58 55 58 55 52 38 46 54
##   [111] 48 49 48 62 46 48 51 53 52 51 41 54 55 48 56 46 54 46 39 59 56 52
##   [133] 61 40 52 34 61 45 42 45 43 37 44 50 57 54 42 47 65 54 42 69 54 57
##   [155] 59 53 51 44 42 68 46 57 58 62 64 56 60 43 56 60 53 48 53 47 47 59
##   [177] 66 39 40 44 53 45 54 53 48 47 42 56 66 48 41 46 56 56 70 41 51 50
##   [199] 42 46 42 53 59 40 48 61 39 62 41 43 49 45 57 48 56 47 53 64 61 54
##   [221] 52 63 44 56 47 68 49 50 58 55 43 49 54 39 46 49 44 45 54 49 49 60
##   [243] 53 60 58 56 49 54 54 69 47 61 52 46 64 55 38 47 51 36 63 56 48 48
##   [265] 53 44 47 51 52 59 54 50 48 60 46 55 53 53 57 45 51 50 35 54 62 43
##   [287] 63 52 61 45 56 47 46 54 60 53 47 60 49 57 42 56 45 48 41 76 43 52
##   [309] 52 45 53 56 56 35 45 50 51 47 55 51 45 57 49 52 66 52 44 49 52 43
##   [331] 37 48 55 45 56 62 52 63 53 47 50 68 58 53 53 51 53 62 55 52 34 51
##   [353] 53 54 52 52 37 62 53 54 56 54 37 43 58 66 47 54 65 50 61 47 46 50
##   [375] 48 57 45 44 54 66 48 43 47 48 52 50 48 37 46 40 49 46 50 49 57 43
##   [397] 46 45 50 53 66 57 55 52 49 56 66 55 62 63 48 48 47 54 49 46 62 46
##   [419] 49 47 55 51 53 49 51 42 43 53 63 44 50 46 58 64 44 55 51 49 53 49
##   [441] 52 55 59 42 57 45 41 32 57 68 51 40 54 44 59 43 51 53 73 52 55 55
##   [463] 48 46 66 55 48 56 63 59 56 51 43 51 65 43 59 47 36 42 51 62 57 57
##   [485] 58 48 62 47 41 49 63 33 51 65 48 57 41 43 44 51 36 42 52 52 52 53
##   [507] 46 53 48 43 52 44 50 44 51 36 58 54 56 64 48 52 47 54 47 59 55 50
##   [529] 52 47 51 50 41 53 45 50 49 51 49 48 44 59 57 52 51 59 54 43 52 62
##   [551] 49 53 50 44 44 57 57 54 42 43 38 54 46 63 49 36 47 47 55 47 52 41
##   [573] 54 59 53 44 52 58 59 53 48 60 49 43 53 55 41 40 49 54 45 40 47 47
##   [595] 46 49 54 45 47 59 48 52 40 61 41 43 37 46 55 56 52 48 58 57 39 46
##   [617] 49 58 44 53 46 61 47 49 54 57 55 62 50 48 34 47 51 50 38 50 49 52
##   [639] 55 50 63 53 55 41 48 48 58 43 45 58 41 55 44 52 50 47 52 44 54 62
##   [661] 49 57 42 38 45 56 56 38 52 50 41 54 48 50 35 58 55 53 48 48 53 47
##   [683] 55 57 61 46 51 52 45 55 53 45 45 54 55 61 51 47 53 59 50 47 55 59
##   [705] 47 54 43 47 57 51 50 54 41 41 52 59 49 49 38 40 49 38 49 63 50 57
##   [727] 44 37 64 60 59 44 55 46 55 52 40 46 30 57 52 57 47 59 63 51 56 51
##   [749] 60 46 46 42 54 52 49 40 53 48 59 53 47 44 56 45 54 50 48 52 53 47
##   [771] 55 53 60 41 43 51 52 58 48 47 62 48 46 44 44 47 61 51 51 55 53 51
##   [793] 66 53 46 65 46 55 35 41 58 53 59 62 49 43 57 53 51 50 37 51 48 54
##   [815] 56 52 51 55 41 45 55 55 59 54 50 43 39 59 55 47 45 58 45 60 53 60
##   [837] 66 64 56 45 45 42 62 54 62 55 44 39 44 49 60 47 58 50 48 34 63 46
##   [859] 56 61 56 54 49 46 55 46 57 58 45 61 51 43 53 55 41 58 53 47 50 41
##   [881] 55 57 54 42 56 51 51 56 52 47 51 47 63 52 58 62 56 45 54 48 65 44
##   [903] 50 52 55 46 51 51 49 40 49 54 48 41 47 53 42 60 66 43 56 57 48 50
##   [925] 58 49 43 55 45 46 56 58 61 57 58 41 43 54 55 52 45 58 59 49 49 60
##   [947] 58 67 41 44 52 42 52 59 46 53 54 55 43 51 58 60 50 57 49 53 42 54
##   [969] 45 56 56 53 43 55 47 41 45 55 69 49 56 50 53 46 51 51 48 36 53 66
##   [991] 60 50 41 48 41 59 42 30 61 56 54 43 44 36 55 46 43 60 42 45 57 55
##  [1013] 60 45 60 39 51 54 52 39 59 58 57 49 49 47 37 56 47 48 65 54 46 61
##  [1035] 48 59 57 55 42 53 53 52 50 65 63 51 55 61 66 52 48 63 57 43 50 61
##  [1057] 51 39 47 55 41 38 48 47 49 38 50 40 56 45 55 48 51 55 58 61 59 46
##  [1079] 28 56 66 58 35 55 61 38 40 42 39 57 46 50 42 60 45 51 59 53 67 49
##  [1101] 49 49 51 46 69 53 51 51 38 54 53 46 48 48 55 45 45 54 45 49 52 54
##  [1123] 44 60 52 42 45 50 51 50 37 53 46 56 56 58 39 45 53 50 50 41 55 54
##  [1145] 46 46 44 52 42 42 53 53 48 38 47 44 48 54 51 53 47 56 53 62 53 48
##  [1167] 54 51 41 60 45 51 63 48 57 43 58 55 47 43 35 62 43 54 48 42 44 54
##  [1189] 47 44 50 50 45 54 49 52 37 41 56 55 47 57 49 55 53 51 44 51 50 57
##  [1211] 44 55 47 50 61 53 51 55 58 55 50 53 42 58 54 55 57 42 49 58 50 51
##  [1233] 49 53 61 46 51 51 50 46 44 43 49 51 63 55 47 37 55 47 51 56 59 49
##  [1255] 47 57 52 43 54 66 51 46 44 57 51 49 45 49 51 51 52 44 54 44 51 41
##  [1277] 37 45 46 38 37 62 55 45 59 49 55 54 55 53 43 45 43 62 49 51 51 51
##  [1299] 47 51 55 52 63 60 52 47 49 43 59 54 46 54 49 53 46 55 59 52 42 39
##  [1321] 60 50 47 60 50 54 56 55 50 48 37 48 54 44 61 53 57 48 46 47 53 46
##  [1343] 35 53 46 52 50 59 58 62 50 55 44 52 48 48 57 39 53 65 54 49 42 44
##  [1365] 47 56 60 53 59 60 56 59 62 38 51 36 53 47 50 44 47 52 61 54 36 49
##  [1387] 41 60 61 62 51 52 58 50 66 51 59 55 49 50 39 51 52 46 45 51 51 62
##  [1409] 54 47 46 55 62 66 57 40 59 51 47 41 52 57 39 53 56 42 59 53 50 47
##  [1431] 62 47 50 63 58 48 58 36 45 46 51 47 49 56 50 46 41 50 49 54 52 61
##  [1453] 52 60 46 47 52 50 56 56 54 63 65 61 59 40 48 49 40 53 56 63 47 52
##  [1475] 44 52 45 56 61 48 59 59 73 51 45 50 56 39 55 54 31 61 41 41 43 48
##  [1497] 38 52 53 51 54 52 43 55 58 56 40 48 51 65 52 48 52 49 56 55 59 41
##  [1519] 53 49 47 44 59 41 46 58 63 56 57 40 61 44 55 57 43 53 52 56 55 47
##  [1541] 43 62 41 56 55 47 53 57 45 43 40 42 53 68 66 47 54 58 52 51 50 55
##  [1563] 59 55 48 40 52 45 46 53 60 52 45 51 61 57 57 41 48 61 56 38 45 49
##  [1585] 43 56 46 58 51 47 46 48 50 56 38 59 56 54 40 45 43 56 64 41 49 58
##  [1607] 55 44 61 41 46 58 56 45 51 60 52 49 54 54 34 45 54 63 54 53 55 50
##  [1629] 49 58 53 46 52 55 56 49 62 49 41 54 59 37 50 36 44 52 42 45 42 50
##  [1651] 35 56 43 49 54 43 62 50 64 46 49 46 57 49 43 51 54 45 54 44 41 55
##  [1673] 51 50 39 54 58 46 48 47 45 52 55 57 38 48 49 48 42 44 52 66 48 51
##  [1695] 54 61 52 50 44 51 47 53 52 41 51 48 56 58 58 46 42 54 51 41 53 44
##  [1717] 54 62 61 62 45 49 35 39 65 56 53 44 47 59 62 50 48 52 40 58 56 58
##  [1739] 49 41 58 53 52 48 50 60 53 57 58 44 57 45 48 56 47 48 48 47 42 50
##  [1761] 58 50 41 38 44 50 45 61 54 54 48 55 43 51 52 51 59 62 57 40 34 51
##  [1783] 48 63 62 52 31 35 56 36 63 36 53 52 52 58 49 60 53 60 51 53 42 68
##  [1805] 47 52 51 49 43 45 58 51 57 50 45 60 53 43 51 52 46 55 63 59 52 59
##  [1827] 55 55 52 35 49 56 49 63 36 46 44 49 56 65 65 53 45 58 56 46 54 61
##  [1849] 41 60 51 43 49 44 54 50 49 46 61 46 47 49 56 46 48 36 54 62 60 46
##  [1871] 54 51 48 49 60 43 47 52 56 67 41 50 58 46 52 49 47 56 49 52 59 52
##  [1893] 50 60 43 51 53 42 61 43 54 43 51 45 34 58 57 58 44 46 48 39 51 42
##  [1915] 55 60 47 48 47 51 46 59 54 53 50 45 48 50 48 52 56 52 40 41 51 46
##  [1937] 48 49 43 53 61 54 64 61 45 43 45 64 57 42 65 55 65 48 68 48 55 51
##  [1959] 54 55 56 36 52 57 49 54 53 45 46 49 50 44 50 51 53 50 51 57 44 56
##  [1981] 52 53 46 48 48 43 58 55 45 52 56 50 47 45 65 72 47 54 60 56 44 52
##  [2003] 55 45 53 48 46 52 47 42 55 61 56 51 41 59 44 42 45 66 47 65 48 62
##  [2025] 51 36 61 50 67 57 42 53 52 45 43 57 61 39 54 52 43 40 55 48 53 43
##  [2047] 39 50 53 48 49 46 53 49 60 56 60 75 50 54 41 39 47 50 43 52 53 54
##  [2069] 66 42 58 54 49 45 53 60 62 54 46 42 35 52 50 47 60 54 54 55 56 49
##  [2091] 47 60 50 52 58 55 47 61 49 48 45 57 53 54 38 54 41 50 54 55 48 50
##  [2113] 57 42 42 58 38 48 41 38 45 47 60 59 53 45 60 42 48 49 42 57 40 41
##  [2135] 58 54 52 49 51 62 46 48 52 61 60 48 44 48 54 52 53 51 36 52 43 50
##  [2157] 50 44 55 51 54 52 39 55 49 52 51 43 51 55 63 57 45 51 49 42 51 51
##  [2179] 55 52 61 62 42 49 58 55 59 48 44 43 45 49 55 47 39 63 48 57 56 50
##  [2201] 69 54 47 56 44 54 50 58 54 51 51 49 55 48 51 45 52 54 43 47 49 47
##  [2223] 45 57 41 48 51 63 51 45 58 51 51 45 49 40 39 57 50 53 64 34 43 50
##  [2245] 50 48 64 58 64 47 56 54 59 51 46 52 57 45 36 43 61 55 65 52 50 58
##  [2267] 46 47 57 51 49 48 44 50 46 55 47 44 54 57 47 53 46 64 53 47 63 52
##  [2289] 43 51 55 49 54 44 63 55 52 59 38 49 55 39 57 46 61 64 49 40 58 49
##  [2311] 56 56 60 42 45 42 51 67 52 62 64 63 55 40 44 45 58 55 46 46 44 53
##  [2333] 53 45 42 48 46 50 54 42 52 55 50 37 55 41 69 50 52 44 44 62 53 51
##  [2355] 55 53 49 45 48 63 50 51 47 47 52 54 61 57 46 43 32 47 50 62 53 53
##  [2377] 41 47 51 51 53 58 68 46 52 45 53 48 64 58 59 47 47 53 42 50 61 61
##  [2399] 64 45 63 42 52 53 44 51 58 52 60 61 50 53 53 50 38 48 47 62 55 68
##  [2421] 56 52 49 45 54 55 37 37 60 50 54 54 38 55 42 43 52 54 44 46 46 51
##  [2443] 50 51 56 51 45 49 47 69 51 63 63 49 49 52 43 50 56 44 56 55 54 46
##  [2465] 53 61 49 33 50 50 35 54 48 58 41 55 50 46 54 54 54 62 57 47 53 50
##  [2487] 32 59 48 51 46 64 51 47 44 51 62 47 51 48 52 38 54 40 45 50 56 50
##  [2509] 45 38 38 43 65 52 53 48 65 53 43 52 34 39 51 58 56 52 48 51 51 52
##  [2531] 45 47 49 55 40 43 58 38 59 34 49 62 52 53 60 37 55 47 61 48 57 46
##  [2553] 44 59 56 55 46 47 47 50 40 39 46 53 46 58 61 46 46 57 46 57 65 55
##  [2575] 54 49 59 52 53 57 57 59 65 52 57 53 58 53 45 45 58 41 55 51 57 50
##  [2597] 48 49 37 51 63 52 43 65 49 61 52 47 51 53 50 43 52 51 41 43 54 62
##  [2619] 63 52 46 54 43 49 45 56 44 43 51 44 42 56 63 45 57 52 45 55 53 62
##  [2641] 46 54 55 61 46 49 64 53 45 55 50 61 47 60 71 43 62 57 54 58 64 50
##  [2663] 49 48 45 49 47 47 53 44 41 56 47 47 53 44 64 56 46 62 51 58 58 45
##  [2685] 43 39 50 56 53 54 45 36 51 54 54 53 50 43 61 54 55 53 55 40 49 48
##  [2707] 47 54 56 58 58 56 50 49 54 43 52 37 66 46 54 50 59 52 48 57 45 64
##  [2729] 50 61 58 43 58 50 51 60 51 51 37 56 39 44 53 53 45 49 67 47 52 51
##  [2751] 54 45 58 60 50 59 57 57 50 51 58 51 58 50 66 57 62 43 48 42 54 60
##  [2773] 52 49 56 50 52 63 37 51 50 55 44 41 51 56 62 46 54 44 46 48 52 50
##  [2795] 51 46 53 64 64 55 56 54 48 49 45 66 43 45 41 46 51 50 62 52 41 55
##  [2817] 34 46 45 57 60 43 40 48 46 48 58 46 58 52 40 56 46 62 61 36 49 57
##  [2839] 51 50 49 37 39 56 42 61 54 50 48 58 51 51 49 47 38 50 53 44 46 50
##  [2861] 61 39 39 65 57 52 51 43 53 51 58 49 48 57 58 53 60 54 52 55 56 53
##  [2883] 35 41 44 49 56 44 51 54 56 44 50 57 53 49 47 59 51 38 57 46 44 42
##  [2905] 52 51 51 45 59 55 54 53 56 59 53 33 50 46 45 46 50 55 45 59 58 48
##  [2927] 43 56 53 38 50 26 39 44 54 45 40 54 45 47 41 49 51 46 56 39 52 43
##  [2949] 45 49 47 65 43 52 41 61 55 38 60 38 56 63 52 61 50 52 42 59 51 45
##  [2971] 39 39 48 56 46 62 56 59 54 42 54 54 47 42 48 43 55 37 60 54 48 49
##  [2993] 61 43 54 50 46 64 54 53 61 42 45 55 53 46 47 54 42 44 43 54 56 59
##  [3015] 51 52 38 58 59 49 51 54 46 41 51 54 47 54 48 46 51 50 45 61 39 57
##  [3037] 63 50 54 44 67 53 41 42 50 62 48 45 52 51 36 52 51 51 50 43 53 57
##  [3059] 34 62 47 41 45 51 40 52 55 59 46 44 59 40 52 50 65 54 43 43 37 52
##  [3081] 58 54 48 43 44 52 48 52 51 44 54 48 58 50 45 48 42 59 48 56 51 48
##  [3103] 54 51 59 53 54 55 55 50 43 55 44 57 44 47 49 40 65 46 39 54 63 52
##  [3125] 37 54 45 57 44 47 42 48 43 57 64 53 43 61 49 52 43 45 43 59 41 60
##  [3147] 49 50 39 61 67 57 39 56 37 53 52 65 55 49 55 50 54 55 49 55 49 59
##  [3169] 56 44 60 53 55 48 54 50 54 62 54 40 40 43 43 58 49 49 57 55 56 43
##  [3191] 56 48 47 49 40 57 48 35 46 33 34 45 52 59 50 57 59 48 55 56 54 61
##  [3213] 57 57 57 50 63 47 61 63 52 53 46 46 52 50 39 62 47 55 59 52 40 42
##  [3235] 54 55 46 49 48 39 51 41 47 47 43 58 45 49 55 55 43 54 40 46 50 50
##  [3257] 53 45 45 53 53 50 54 59 55 49 61 64 42 74 42 46 65 58 53 45 43 57
##  [3279] 50 53 54 47 41 37 61 49 66 52 46 58 49 54 38 51 45 46 51 62 45 40
##  [3301] 57 62 54 39 45 41 53 59 57 57 70 42 60 52 50 53 40 57 51 63 36 53
##  [3323] 48 47 36 56 52 58 61 52 45 53 40 58 53 36 39 42 39 48 44 54 48 58
##  [3345] 59 49 50 47 41 53 44 45 52 48 41 61 58 65 54 55 59 50 58 50 50 50
##  [3367] 51 38 49 50 39 51 55 58 53 49 44 57 66 61 54 68 57 59 62 51 49 61
##  [3389] 53 57 56 45 54 45 39 52 58 54 48 37 43 55 45 53 64 60 50 59 53 56
##  [3411] 50 40 52 50 36 45 51 59 48 37 52 53 54 38 48 49 56 49 54 49 46 54
##  [3433] 47 50 57 53 55 53 62 63 55 46 61 62 48 67 48 46 55 53 47 59 61 36
##  [3455] 57 57 42 49 56 47 42 46 55 41 52 44 50 52 71 47 46 42 57 52 47 49
##  [3477] 47 50 52 58 41 44 60 63 43 39 50 52 48 39 42 61 38 54 59 49 58 52
##  [3499] 37 48 50 45 58 52 39 39 52 57 57 69 50 64 59 55 55 46 46 46 44 48
##  [3521] 47 56 40 65 53 55 57 52 54 48 61 54 40 68 47 58 57 55 57 55 47 59
##  [3543] 55 46 56 47 40 57 53 52 50 55 61 46 46 48 57 50 51 50 44 51 56 52
##  [3565] 56 70 59 39 50 56 36 53 40 50 55 59 44 52 58 47 62 56 55 51 48 51
##  [3587] 51 46 58 48 55 50 43 58 57 60 60 49 56 41 51 61 58 52 52 43 55 58
##  [3609] 59 34 50 54 58 50 43 42 49 57 52 44 48 49 36 53 47 50 36 39 56 55
##  [3631] 52 51 57 44 53 56 57 56 46 54 45 53 60 60 61 49 58 42 64 51 45 50
##  [3653] 47 50 51 43 55 61 53 55 63 49 53 57 55 54 38 64 57 36 38 68 49 50
##  [3675] 55 56 43 57 45 42 44 53 47 50 49 63 45 60 65 35 47 52 55 43 48 61
##  [3697] 43 51 49 52 51 53 42 46 47 61 54 46 58 56 59 52 44 47 38 48 45 51
##  [3719] 62 64 49 43 51 46 51 49 47 45 60 45 55 55 53 47 55 55 57 59 44 48
##  [3741] 51 39 44 56 56 53 52 48 67 43 57 58 51 63 47 41 56 42 52 48 54 52
##  [3763] 51 55 46 59 61 48 52 57 57 55 64 59 54 42 50 36 48 50 53 39 53 51
##  [3785] 56 47 51 47 57 49 50 59 46 53 53 47 37 59 49 43 59 43 47 40 52 47
##  [3807] 50 55 37 53 46 48 57 56 57 38 52 50 42 49 54 51 50 47 46 60 47 40
##  [3829] 61 54 49 58 41 47 42 41 47 50 60 46 53 46 67 46 64 65 38 48 47 52
##  [3851] 49 52 59 53 36 53 56 59 55 49 46 53 47 61 51 57 45 57 47 70 43 46
##  [3873] 62 61 39 46 48 45 42 53 45 45 54 39 50 55 45 49 42 48 40 59 46 58
##  [3895] 63 45 52 47 47 53 52 36 50 46 47 51 54 52 43 49 57 42 48 66 53 45
##  [3917] 59 54 59 59 51 34 60 55 54 47 59 52 52 50 59 45 49 60 59 56 55 52
##  [3939] 49 49 55 48 54 52 53 51 53 48 59 48 56 53 46 61 57 46 44 50 62 52
##  [3961] 53 63 42 43 53 39 50 49 44 59 46 62 52 56 49 48 65 65 56 38 41 41
##  [3983] 49 51 46 54 48 69 39 55 30 62 56 47 47 44 50 46 48 49 57 49 38 58
##  [4005] 65 48 58 38 59 49 44 52 43 45 40 48 62 50 59 55 58 46 64 54 47 39
##  [4027] 54 52 51 41 60 59 44 51 59 59 40 52 62 51 36 55 39 56 60 46 42 44
##  [4049] 49 57 43 51 58 43 58 58 52 57 61 67 43 46 40 31 49 58 49 47 56 53
##  [4071] 49 58 47 47 40 47 60 44 55 38 55 47 54 59 60 51 45 40 58 53 40 49
##  [4093] 56 49 57 59 45 54 51 50 53 49 47 42 55 54 51 57 54 56 51 57 52 51
##  [4115] 52 57 50 54 57 45 59 44 51 46 45 44 56 32 46 62 55 53 56 61 46 43
##  [4137] 47 44 41 56 53 52 44 36 38 49 45 56 64 40 50 54 51 58 64 38 54 46
##  [4159] 64 62 48 58 44 32 54 58 59 46 63 41 49 50 50 42 54 47 57 53 60 50
##  [4181] 61 46 53 50 58 47 50 58 38 50 47 51 65 53 45 53 47 53 51 63 44 52
##  [4203] 56 40 56 51 52 43 49 40 57 55 38 51 41 39 49 52 51 51 39 64 44 52
##  [4225] 54 43 56 42 54 62 39 56 54 54 55 46 49 42 57 48 50 43 49 58 54 43
##  [4247] 48 59 48 60 61 38 59 54 59 57 61 51 47 63 48 54 61 67 47 54 37 54
##  [4269] 54 45 53 50 45 47 48 51 47 39 59 42 37 34 42 51 56 49 48 56 48 59
##  [4291] 42 50 60 46 51 63 56 50 54 56 59 41 51 46 46 40 58 41 43 40 45 49
##  [4313] 41 48 48 46 52 38 54 43 53 54 47 53 48 45 50 48 48 52 64 46 37 42
##  [4335] 41 52 64 45 53 53 58 56 43 51 52 50 45 57 55 46 47 44 54 50 54 52
##  [4357] 65 40 44 44 50 44 58 54 60 45 39 59 49 54 48 41 47 34 45 48 63 49
##  [4379] 42 44 63 53 55 49 56 53 45 49 57 57 49 46 57 52 54 51 58 56 57 62
##  [4401] 51 43 58 59 43 57 43 54 49 61 53 56 58 36 56 46 43 54 52 52 56 54
##  [4423] 45 51 56 41 51 45 56 56 46 53 56 56 56 48 54 47 45 45 41 53 39 45
##  [4445] 51 41 49 53 39 48 61 44 54 49 47 47 49 47 49 51 51 44 55 44 44 41
##  [4467] 47 59 53 54 63 48 51 58 47 58 39 55 49 49 56 55 30 63 58 43 57 42
##  [4489] 54 61 62 61 41 50 48 48 56 66 57 44 55 50 54 60 57 48 45 40 54 53
##  [4511] 50 54 52 56 66 46 60 43 46 50 44 51 59 51 51 40 47 48 38 37 44 51
##  [4533] 37 45 49 48 62 46 40 61 51 50 49 46 47 59 49 61 42 51 60 50 53 41
##  [4555] 53 56 68 66 60 55 58 47 55 44 56 54 66 66 45 52 59 38 53 48 44 57
##  [4577] 39 59 48 50 51 48 44 44 52 55 51 55 48 60 40 48 50 56 66 38 52 39
##  [4599] 44 47 59 49 50 64 54 47 52 48 53 50 54 46 43 48 59 51 45 50 50 44
##  [4621] 58 64 42 53 68 44 45 43 48 52 37 57 56 55 59 55 52 41 34 50 51 44
##  [4643] 60 39 43 32 38 57 50 54 50 54 37 46 35 42 45 47 52 62 46 47 47 59
##  [4665] 42 61 50 49 50 51 47 47 59 46 46 62 52 59 47 63 55 52 51 53 66 45
##  [4687] 44 55 51 48 45 54 49 62 42 32 49 58 58 49 50 47 54 42 50 50 53 54
##  [4709] 60 53 53 50 52 44 47 60 42 58 53 55 59 58 46 51 56 41 41 66 47 48
##  [4731] 49 56 50 58 63 53 42 67 61 66 49 53 57 50 45 58 34 45 42 54 59 55
##  [4753] 56 54 53 46 54 52 49 44 55 42 42 48 54 59 42 50 47 55 51 59 54 52
##  [4775] 38 48 59 47 62 60 65 48 56 65 54 53 50 51 53 51 55 54 46 51 43 43
##  [4797] 52 45 49 53 49 50 54 55 50 52 50 54 56 65 60 66 44 58 42 47 48 44
##  [4819] 46 53 59 50 56 54 41 57 51 56 56 36 57 50 71 52 52 60 51 61 51 58
##  [4841] 52 52 64 50 48 57 57 39 54 56 47 52 47 42 60 47 62 57 50 50 53 48
##  [4863] 56 55 47 43 39 54 47 50 42 50 47 50 56 52 48 46 50 51 51 52 54 41
##  [4885] 52 43 40 42 33 54 53 58 49 45 42 45 49 34 55 57 48 47 45 49 59 45
##  [4907] 47 48 45 44 48 55 62 41 50 64 49 46 32 33 45 52 51 39 53 64 60 50
##  [4929] 48 50 42 60 47 43 54 43 61 51 52 46 59 63 53 52 65 48 52 71 54 45
##  [4951] 50 55 41 47 60 38 50 56 46 57 45 41 54 62 52 50 30 43 53 55 54 47
##  [4973] 51 52 55 50 47 53 50 35 42 53 37 45 52 59 62 53 61 56 34 50 50 61
##  [4995] 59 42 55 67 48 50 53 69 37 43 45 59 52 54 40 59 48 57 44 52 43 50
##  [5017] 51 58 45 50 66 57 46 51 47 49 49 49 44 43 59 51 53 43 59 53 44 51
##  [5039] 53 49 39 49 49 62 46 56 47 54 42 55 71 52 50 47 44 52 55 46 51 53
##  [5061] 55 52 50 54 54 57 43 49 58 51 58 47 51 58 54 42 52 50 53 40 58 38
##  [5083] 49 48 52 41 42 57 59 40 54 52 61 46 53 52 48 45 61 42 55 51 64 49
##  [5105] 58 55 59 46 66 51 52 52 38 55 39 52 61 51 43 50 55 46 52 53 54 46
##  [5127] 55 53 40 67 45 55 50 51 50 61 61 47 55 54 54 58 47 53 50 43 51 48
##  [5149] 51 54 57 49 35 36 52 53 55 54 42 56 46 59 45 51 59 49 45 57 56 41
##  [5171] 59 51 39 47 43 50 54 53 53 52 60 50 42 60 57 47 55 43 55 52 54 57
##  [5193] 45 37 48 52 46 59 50 58 49 42 45 54 50 54 51 51 46 44 55 41 54 38
##  [5215] 56 44 51 49 49 51 55 54 55 37 51 37 57 46 47 46 57 46 49 59 49 57
##  [5237] 38 40 56 40 54 53 48 57 43 56 53 58 41 43 42 44 61 58 67 57 53 47
##  [5259] 60 36 48 44 47 62 55 51 57 42 52 66 37 57 61 54 58 61 48 61 58 53
##  [5281] 53 51 51 47 39 49 42 48 48 48 67 46 50 52 55 49 55 47 52 56 41 54
##  [5303] 44 55 52 62 64 64 50 49 53 60 45 48 50 32 48 48 52 51 64 56 44 56
##  [5325] 55 33 43 46 40 45 55 49 43 51 55 46 53 49 54 37 48 57 45 55 46 41
##  [5347] 44 55 41 56 52 54 66 50 51 44 62 43 50 49 49 61 55 59 52 51 45 60
##  [5369] 63 54 35 54 55 49 51 51 50 48 48 59 46 55 38 67 45 58 56 63 54 52
##  [5391] 52 40 46 53 39 49 63 42 52 60 39 60 47 57 66 55 43 54 49 56 47 49
##  [5413] 57 55 53 54 42 40 52 48 47 40 58 64 37 36 52 42 36 46 55 59 44 58
##  [5435] 48 44 58 45 62 44 61 69 47 50 47 56 62 51 55 34 41 58 53 50 50 56
##  [5457] 57 53 62 45 56 52 55 63 51 61 46 47 49 48 46 49 42 62 45 47 50 55
##  [5479] 49 45 52 44 46 47 41 50 53 39 52 50 48 49 61 41 60 55 52 52 47 56
##  [5501] 60 52 51 46 35 42 47 60 47 60 56 53 49 51 62 56 39 44 59 47 50 49
##  [5523] 42 45 48 49 47 50 44 55 54 59 55 67 53 37 42 62 56 57 41 45 37 56
##  [5545] 47 40 68 56 44 70 49 44 49 50 54 43 48 46 52 57 49 40 47 42 59 42
##  [5567] 37 50 50 53 51 50 72 64 56 44 54 59 48 54 54 47 55 54 47 50 48 49
##  [5589] 50 52 64 47 52 61 44 46 45 53 47 37 45 52 51 52 51 47 56 46 41 69
##  [5611] 59 53 45 56 38 38 54 56 49 64 53 46 52 65 48 56 43 62 55 47 44 48
##  [5633] 53 53 51 55 58 60 57 71 53 57 41 42 48 50 47 51 46 49 54 54 52 52
##  [5655] 47 44 41 42 49 65 54 52 56 51 46 53 51 51 55 62 44 54 56 54 55 55
##  [5677] 42 41 60 59 60 57 48 60 50 48 60 49 53 50 50 70 58 40 50 52 55 59
##  [5699] 56 59 44 56 67 52 60 64 46 50 47 56 33 42 63 43 46 47 63 47 49 42
##  [5721] 58 52 48 33 50 45 44 54 51 39 45 59 63 53 45 33 46 51 58 42 54 42
##  [5743] 54 54 47 49 55 53 51 39 57 55 56 43 54 56 55 64 51 43 53 56 47 50
##  [5765] 42 47 53 49 58 38 42 40 39 56 47 49 40 47 36 41 60 46 44 56 52 40
##  [5787] 54 48 52 61 52 53 55 47 54 50 53 58 55 47 46 57 54 47 56 56 51 45
##  [5809] 65 52 46 49 51 64 56 50 50 50 41 50 64 47 51 47 50 49 56 37 41 41
##  [5831] 46 55 47 56 50 52 55 35 52 50 43 51 44 54 56 51 48 59 49 41 54 31
##  [5853] 53 43 50 52 47 59 45 59 53 50 38 58 52 59 57 46 55 60 55 56 41 61
##  [5875] 55 57 54 48 36 45 47 44 46 51 58 53 52 51 46 52 46 40 52 48 48 38
##  [5897] 37 45 55 56 42 45 53 46 45 48 53 48 57 52 45 48 48 58 59 53 50 59
##  [5919] 58 46 59 50 54 55 47 49 54 47 44 40 48 48 57 40 39 60 69 55 57 47
##  [5941] 54 44 44 53 42 51 43 52 56 56 44 46 51 50 56 56 55 54 58 49 51 62
##  [5963] 55 53 42 44 45 60 62 50 53 54 42 43 43 47 52 51 46 58 49 40 47 32
##  [5985] 52 45 38 46 61 50 52 58 47 56 53 42 44 49 41 54 51 45 51 48 44 41
##  [6007] 36 42 47 55 61 56 59 48 42 46 53 59 57 50 59 53 52 53 49 47 44 42
##  [6029] 41 54 45 56 48 43 62 55 65 49 46 48 52 52 45 46 57 51 56 52 58 52
##  [6051] 63 52 41 53 64 56 36 56 47 65 52 49 58 53 68 44 48 54 57 56 41 52
##  [6073] 45 51 52 42 54 56 52 44 59 54 41 49 72 49 43 42 57 53 51 51 51 43
##  [6095] 53 47 56 57 51 58 52 57 53 50 40 71 53 51 61 54 51 46 43 42 50 49
##  [6117] 57 53 48 47 49 37 70 57 54 54 45 38 41 67 49 66 48 64 58 57 46 50
##  [6139] 53 55 50 50 48 53 42 48 57 53 49 54 55 57 45 40 49 45 60 55 49 55
##  [6161] 49 49 48 45 45 47 56 58 46 39 57 45 43 63 46 63 49 52 45 54 59 53
##  [6183] 53 48 51 44 46 57 33 42 53 71 60 54 52 53 50 57 38 56 48 54 54 49
##  [6205] 65 49 55 44 49 44 61 53 44 44 56 62 50 59 67 48 36 59 51 55 57 55
##  [6227] 53 58 51 55 59 50 60 45 55 59 46 46 67 61 50 46 49 51 43 54 44 54
##  [6249] 45 54 54 45 58 56 51 45 34 38 60 55 50 51 46 53 44 45 50 46 46 48
##  [6271] 41 59 58 46 65 52 48 59 45 61 49 46 56 45 49 51 48 58 44 45 35 48
##  [6293] 55 45 53 57 50 55 47 37 54 56 56 46 54 59 32 58 42 59 45 52 55 47
##  [6315] 48 58 54 41 37 65 51 49 47 53 75 44 45 52 40 55 43 61 58 51 57 50
##  [6337] 56 53 63 40 48 53 50 49 48 49 54 40 53 48 39 47 54 56 51 50 59 58
##  [6359] 50 55 49 44 48 45 39 64 38 48 49 41 63 57 58 59 51 46 54 55 44 56
##  [6381] 50 50 62 60 46 52 50 41 50 43 42 50 60 58 50 43 51 54 47 37 50 65
##  [6403] 46 54 56 53 38 53 59 47 50 44 57 58 60 45 55 59 51 41 50 50 63 46
##  [6425] 46 58 68 47 55 42 40 50 45 58 58 58 46 48 46 60 49 46 47 54 47 47
##  [6447] 56 37 50 50 52 48 54 54 58 50 38 60 60 44 50 53 60 51 36 57 56 62
##  [6469] 55 54 34 39 57 52 67 48 49 60 53 55 56 40 47 59 46 55 50 59 50 48
##  [6491] 45 40 52 52 62 56 39 67 52 66 54 43 53 51 61 62 54 61 44 44 41 53
##  [6513] 38 49 53 58 41 48 58 53 62 51 52 57 56 42 48 37 46 54 48 51 52 46
##  [6535] 35 42 58 49 35 48 60 45 58 55 58 58 50 56 54 52 50 59 56 53 52 57
##  [6557] 42 51 53 49 53 48 54 53 41 51 36 50 57 46 42 56 58 47 56 69 52 60
##  [6579] 55 57 53 37 64 39 46 57 67 40 51 55 49 55 61 35 51 58 51 36 42 47
##  [6601] 50 50 60 54 44 68 55 65 46 53 60 47 53 49 53 54 40 47 48 51 41 49
##  [6623] 62 47 49 45 56 48 47 49 57 59 46 54 52 58 59 51 51 44 53 48 61 61
##  [6645] 44 50 50 49 37 42 51 48 39 60 48 46 53 52 55 47 46 53 53 49 50 61
##  [6667] 64 46 47 48 61 51 53 63 42 44 51 59 57 48 39 56 60 44 52 48 49 37
##  [6689] 49 44 36 44 51 58 53 59 56 48 47 45 54 55 49 52 64 39 37 54 54 50
##  [6711] 48 62 51 51 63 54 39 59 46 51 55 53 51 41 45 44 59 37 54 55 53 64
##  [6733] 48 54 47 51 55 51 50 57 55 52 57 38 53 62 64 51 51 43 61 54 55 47
##  [6755] 52 62 52 56 49 38 58 46 51 51 61 58 48 53 44 59 39 41 61 51 55 48
##  [6777] 59 54 63 66 62 49 40 48 46 71 50 52 50 52 45 50 49 53 40 47 62 53
##  [6799] 54 78 62 45 61 50 46 49 55 58 45 48 53 44 39 50 60 43 56 52 45 57
##  [6821] 47 52 54 61 64 57 50 46 55 58 66 41 45 50 57 52 62 61 66 49 37 38
##  [6843] 52 50 46 55 39 46 55 47 58 35 59 42 59 47 58 60 49 47 55 62 50 62
##  [6865] 71 43 49 57 46 67 50 47 58 47 53 62 64 49 54 57 55 49 48 53 47 47
##  [6887] 47 50 48 54 54 48 53 48 52 47 50 56 51 45 51 56 46 58 51 57 53 59
##  [6909] 60 40 44 54 46 51 52 51 42 54 43 51 43 61 50 48 57 51 56 40 45 61
##  [6931] 46 50 50 50 50 54 47 35 45 41 46 46 57 44 54 35 58 48 59 59 48 44
##  [6953] 43 50 57 47 58 47 40 56 51 47 46 52 53 39 50 44 50 56 63 50 72 59
##  [6975] 52 39 46 56 43 44 37 59 49 58 49 37 43 56 51 44 53 55 63 55 69 49
##  [6997] 51 60 45 55 51 51 62 58 44 45 56 55 56 58 55 48 44 63 50 38 59 48
##  [7019] 50 54 41 44 52 73 57 58 53 44 46 44 50 47 48 59 59 55 52 62 47 58
##  [7041] 41 51 51 54 35 55 59 49 55 49 49 36 47 26 50 59 52 45 45 49 51 50
##  [7063] 52 50 62 46 59 44 50 49 43 52 43 52 52 47 62 52 56 43 56 47 69 64
##  [7085] 41 47 51 47 64 48 55 43 37 43 59 57 67 54 40 56 53 45 40 53 71 57
##  [7107] 51 43 53 48 53 52 52 54 59 52 61 51 56 50 49 56 43 61 39 50 44 51
##  [7129] 55 45 47 44 42 53 53 56 58 50 50 57 51 51 43 55 52 48 64 39 50 58
##  [7151] 55 44 47 50 42 45 38 54 53 43 53 56 49 51 54 49 52 36 60 48 44 60
##  [7173] 50 49 52 47 49 46 51 63 58 59 50 62 45 44 39 53 45 56 52 49 49 53
##  [7195] 44 67 50 49 58 63 55 54 54 49 54 57 64 45 54 52 47 56 51 53 55 30
##  [7217] 49 44 55 45 56 39 60 45 61 62 53 47 58 48 46 55 47 47 53 70 52 57
##  [7239] 53 66 62 53 45 49 58 50 51 46 45 60 51 47 59 70 53 40 39 44 43 44
##  [7261] 48 42 51 55 50 49 48 44 54 49 56 52 49 34 45 50 50 51 45 45 59 50
##  [7283] 45 43 54 62 48 45 36 60 45 61 58 49 49 46 51 50 52 62 51 60 55 47
##  [7305] 53 48 59 46 56 56 46 46 59 58 53 31 63 50 48 40 50 48 40 46 60 43
##  [7327] 47 61 44 49 40 49 55 41 50 58 67 50 46 59 49 44 57 58 58 57 52 60
##  [7349] 59 48 50 56 47 52 60 59 41 47 53 48 46 39 54 62 51 48 49 46 55 48
##  [7371] 50 55 48 44 40 41 47 45 50 60 42 40 48 44 51 44 55 49 52 57 51 58
##  [7393] 54 53 54 50 50 44 54 45 50 55 54 46 60 57 54 45 57 60 62 37 54 57
##  [7415] 48 47 52 57 55 49 48 44 44 55 50 52 49 47 42 52 54 40 52 42 49 47
##  [7437] 46 49 47 50 53 65 58 61 43 52 50 48 52 48 61 41 74 58 68 57 49 50
##  [7459] 48 42 47 48 45 52 43 55 52 51 52 52 68 47 47 59 54 46 46 38 57 44
##  [7481] 39 51 57 51 56 41 44 53 50 45 45 51 50 44 48 55 53 50 52 50 55 43
##  [7503] 34 58 61 51 61 61 56 40 43 57 54 45 58 51 54 53 63 50 56 54 62 47
##  [7525] 67 53 34 45 41 66 48 46 45 53 61 50 55 53 53 49 57 40 40 47 58 50
##  [7547] 47 51 60 57 40 49 44 47 56 48 45 60 40 55 50 48 66 49 44 63 62 34
##  [7569] 45 50 49 59 42 55 51 42 56 42 48 57 52 49 54 44 64 38 47 52 55 51
##  [7591] 48 51 45 65 59 62 58 49 54 42 47 61 46 38 53 57 45 60 54 56 42 57
##  [7613] 45 52 54 66 54 47 50 50 58 51 43 60 36 54 53 59 44 65 46 56 55 34
##  [7635] 41 49 42 54 55 44 48 57 46 59 61 50 55 51 47 46 53 64 60 43 48 60
##  [7657] 44 49 63 55 58 48 66 59 41 51 49 54 38 47 50 47 46 50 46 57 61 53
##  [7679] 43 57 50 59 48 43 51 42 51 52 47 49 67 58 44 60 47 43 51 57 60 60
##  [7701] 52 57 47 49 50 37 54 51 42 51 43 46 43 45 45 41 52 52 54 51 56 54
##  [7723] 53 51 44 50 53 52 46 38 52 59 51 38 62 30 51 46 48 37 49 50 44 45
##  [7745] 51 41 54 48 64 51 57 51 39 45 50 49 46 56 56 50 57 59 48 57 48 55
##  [7767] 47 49 57 49 49 46 52 52 55 55 54 51 52 37 60 54 63 60 48 52 40 47
##  [7789] 37 48 55 52 64 61 41 51 48 57 53 56 46 48 53 44 55 57 52 43 59 51
##  [7811] 61 35 50 51 48 43 50 61 66 58 52 52 59 49 50 44 41 55 52 44 59 38
##  [7833] 50 46 52 45 44 44 61 53 54 52 61 47 52 54 50 50 56 52 42 53 52 48
##  [7855] 54 58 48 49 43 46 47 70 47 56 51 46 58 42 52 55 64 57 56 58 61 48
##  [7877] 51 49 39 41 63 51 54 56 49 39 63 56 36 48 47 39 53 55 60 56 53 30
##  [7899] 43 53 49 56 43 46 44 46 48 44 70 55 49 53 46 39 47 54 52 46 54 58
##  [7921] 45 59 43 53 49 52 43 49 33 51 62 43 51 59 53 60 60 38 52 49 47 49
##  [7943] 49 60 47 52 57 49 58 44 57 50 53 50 56 54 46 37 57 53 50 61 35 47
##  [7965] 52 55 53 69 41 33 49 46 62 49 56 57 48 68 48 51 47 61 57 44 59 41
##  [7987] 49 39 54 46 47 47 49 50 52 60 52 56 52 48 51 41 50 49 43 42 53 57
##  [8009] 45 68 54 60 54 52 54 47 41 48 46 47 37 47 52 62 53 51 56 60 44 51
##  [8031] 61 51 54 45 55 45 48 61 53 34 68 58 47 52 40 52 49 58 54 40 46 57
##  [8053] 48 56 43 55 63 53 42 48 60 50 48 63 49 60 61 51 53 57 49 58 53 51
##  [8075] 54 57 53 50 55 51 44 48 50 48 61 47 59 46 54 48 53 53 41 40 50 59
##  [8097] 51 59 56 38 51 60 43 40 46 54 51 53 55 58 46 52 53 42 35 51 49 47
##  [8119] 56 57 41 47 56 56 46 52 59 56 51 61 50 49 70 70 47 52 37 49 43 42
##  [8141] 57 46 44 50 39 49 50 50 52 48 54 51 43 52 39 52 43 49 65 42 47 53
##  [8163] 51 55 53 49 62 47 65 46 57 63 40 55 51 50 58 50 34 51 43 52 52 41
##  [8185] 62 42 47 45 48 55 56 60 52 51 38 42 50 49 53 49 49 46 55 45 60 39
##  [8207] 61 54 47 51 48 57 58 41 47 55 49 50 59 45 62 54 55 65 46 55 44 60
##  [8229] 58 44 61 50 48 41 47 53 60 52 63 62 45 48 46 48 50 62 38 55 50 43
##  [8251] 42 43 49 60 64 64 49 69 60 40 61 49 48 55 38 43 39 57 53 58 54 55
##  [8273] 54 52 58 65 45 58 49 51 46 57 52 57 46 52 62 45 43 45 63 46 51 41
##  [8295] 53 56 37 53 39 52 50 56 54 53 46 46 56 63 39 55 47 47 46 62 45 49
##  [8317] 47 52 64 65 51 42 57 52 45 51 61 58 51 44 41 63 51 47 52 53 46 54
##  [8339] 49 61 53 48 49 53 43 47 45 46 56 49 51 55 40 49 50 46 47 52 59 49
##  [8361] 43 42 48 46 58 42 64 56 44 46 51 44 42 43 56 53 50 58 50 49 42 57
##  [8383] 58 46 54 64 48 55 51 43 61 53 67 64 45 58 48 43 46 65 47 52 38 52
##  [8405] 45 44 55 57 42 53 43 51 41 52 53 51 55 47 42 33 50 48 67 52 70 64
##  [8427] 65 62 46 61 47 48 45 49 47 46 39 36 42 48 47 54 49 53 46 56 56 42
##  [8449] 40 43 53 54 52 53 54 46 52 56 66 57 54 58 55 44 49 53 35 44 49 54
##  [8471] 55 49 52 45 50 51 55 47 45 47 52 57 53 49 54 49 60 46 52 53 53 48
##  [8493] 47 60 45 47 57 57 64 50 56 53 43 49 45 50 56 43 48 54 62 46 48 37
##  [8515] 58 49 48 56 51 56 58 62 47 44 49 48 60 44 50 55 41 46 61 60 61 59
##  [8537] 50 51 53 44 50 59 62 43 44 59 46 39 51 48 67 53 51 40 48 48 42 47
##  [8559] 54 54 57 51 53 53 51 56 41 42 42 56 53 61 58 36 54 53 42 45 52 53
##  [8581] 62 46 63 55 54 61 55 54 55 49 59 41 64 61 51 58 49 53 59 37 57 48
##  [8603] 56 50 51 54 63 51 55 51 67 57 43 33 53 45 47 62 51 44 44 48 55 48
##  [8625] 56 53 46 53 51 57 60 61 57 37 45 53 51 37 50 59 60 47 56 57 59 61
##  [8647] 41 45 51 53 62 53 49 51 52 61 54 44 54 52 47 37 50 47 49 58 46 54
##  [8669] 42 50 43 46 52 59 45 45 57 59 54 42 54 57 41 49 59 45 66 51 40 58
##  [8691] 49 48 50 52 54 46 59 57 59 49 53 56 50 52 59 58 44 53 53 49 55 57
##  [8713] 44 53 44 54 52 41 46 65 58 48 50 52 46 65 45 68 45 52 49 64 55 57
##  [8735] 45 52 57 49 43 48 44 57 56 55 57 48 48 46 48 58 63 50 49 49 46 44
##  [8757] 49 43 56 51 48 63 59 49 53 60 50 47 54 47 36 48 48 44 46 43 50 55
##  [8779] 46 57 36 64 48 63 58 49 49 43 47 55 53 43 50 41 48 45 51 61 64 64
##  [8801] 56 43 54 48 36 43 48 55 53 50 48 41 55 40 42 40 45 57 46 49 51 60
##  [8823] 45 48 31 56 51 62 55 52 56 41 38 39 44 48 51 45 46 59 57 48 51 40
##  [8845] 55 56 49 56 57 55 43 56 41 41 46 57 47 45 46 40 53 46 59 48 67 47
##  [8867] 56 48 57 39 54 52 43 41 38 60 44 52 44 48 49 58 51 56 51 46 50 53
##  [8889] 51 62 51 52 49 64 58 56 52 53 52 56 60 48 57 56 56 59 69 56 52 51
##  [8911] 48 57 59 53 55 42 55 52 59 50 49 47 42 58 50 51 52 48 53 51 55 46
##  [8933] 60 56 61 60 45 50 44 55 53 45 63 50 61 52 50 54 59 56 55 58 39 51
##  [8955] 35 48 48 50 45 41 50 51 49 60 47 54 67 51 53 55 60 57 58 48 54 41
##  [8977] 53 48 50 46 57 58 46 55 39 43 56 38 50 48 48 50 59 59 53 46 41 60
##  [8999] 63 70 49 46 50 41 48 55 42 48 54 63 41 53 58 66 46 52 58 42 45 54
##  [9021] 62 49 68 52 38 53 49 61 63 56 57 42 51 51 62 59 44 34 50 67 52 50
##  [9043] 49 76 52 56 46 51 45 51 48 36 59 39 64 54 51 48 46 50 41 47 53 58
##  [9065] 60 57 53 39 47 49 50 52 45 59 64 58 65 42 58 38 48 53 61 50 41 56
##  [9087] 42 50 58 59 68 48 45 46 58 48 49 50 48 48 50 51 58 52 58 58 63 54
##  [9109] 53 58 59 50 52 50 43 50 49 43 50 47 47 42 49 44 61 49 54 53 51 45
##  [9131] 44 48 56 52 64 64 49 42 39 48 50 51 49 52 59 45 47 51 44 41 60 42
##  [9153] 47 60 68 37 43 42 56 55 45 59 57 48 62 51 44 53 52 51 50 43 55 46
##  [9175] 50 61 48 51 47 63 55 55 38 52 57 38 46 50 57 39 50 55 51 50 47 50
##  [9197] 61 39 46 49 47 48 53 42 48 54 43 57 62 40 45 55 50 62 49 56 56 46
##  [9219] 51 60 50 56 57 55 47 50 59 55 45 46 67 56 37 44 60 51 46 44 56 52
##  [9241] 55 40 55 57 48 59 47 61 53 59 54 48 37 58 45 48 43 49 68 51 44 56
##  [9263] 61 59 46 64 54 45 37 61 43 58 46 49 63 61 53 53 50 53 54 38 43 54
##  [9285] 46 54 49 50 52 56 57 41 61 52 55 55 49 37 62 50 50 46 52 41 41 46
##  [9307] 60 46 51 52 56 50 37 52 64 57 57 61 62 49 40 53 67 46 46 42 51 60
##  [9329] 54 55 51 58 37 50 52 57 54 55 50 47 68 55 55 45 50 52 50 54 52 44
##  [9351] 39 68 47 40 46 58 56 52 48 62 44 52 47 48 40 50 50 51 50 46 56 72
##  [9373] 53 52 49 64 46 41 52 55 52 50 51 45 57 40 64 55 49 56 68 62 58 55
##  [9395] 69 43 45 47 49 51 45 63 51 44 51 49 44 47 53 54 40 59 49 58 43 60
##  [9417] 61 45 54 45 63 48 44 58 47 47 52 61 56 59 42 56 53 52 53 53 49 49
##  [9439] 49 50 52 55 47 59 48 54 51 48 37 49 43 49 41 50 53 47 42 50 53 41
##  [9461] 48 43 41 47 52 62 52 59 66 44 44 47 54 52 51 45 63 53 59 43 63 42
##  [9483] 42 52 45 40 50 49 60 39 54 54 39 38 58 53 54 39 65 56 60 48 53 49
##  [9505] 55 56 45 67 50 54 69 39 49 53 52 46 51 38 48 62 64 56 46 52 42 58
##  [9527] 40 53 54 50 37 52 56 50 53 56 58 59 49 55 57 48 61 51 53 37 52 54
##  [9549] 50 53 52 62 67 46 58 62 50 53 49 46 57 37 40 39 60 49 48 57 53 43
##  [9571] 44 52 53 42 42 58 48 60 54 56 42 52 48 55 59 56 47 54 52 51 61 55
##  [9593] 52 64 51 49 48 49 45 42 49 49 60 53 46 49 63 58 44 48 49 56 37 61
##  [9615] 43 48 60 49 63 38 39 49 45 68 44 47 60 44 62 44 51 43 51 44 48 53
##  [9637] 52 41 44 45 62 55 45 54 50 52 54 59 61 56 50 45 47 44 53 41 57 59
##  [9659] 54 40 44 53 61 53 54 54 54 57 59 44 45 41 42 47 46 44 48 50 51 44
##  [9681] 57 62 43 42 52 56 49 54 57 46 58 46 57 45 50 44 43 48 47 58 52 54
##  [9703] 57 52 52 53 63 58 54 47 50 57 58 54 46 55 52 52 53 53 59 48 59 51
##  [9725] 43 48 46 47 62 40 51 49 53 51 37 57 45 60 46 48 59 47 64 49 50 73
##  [9747] 50 40 51 56 63 45 50 39 51 43 45 49 44 49 59 60 41 52 61 51 53 54
##  [9769] 46 51 53 49 43 58 55 50 57 50 60 51 47 48 58 55 44 50 57 58 43 65
##  [9791] 52 59 49 65 37 64 46 47 58 55 52 47 37 50 62 42 45 40 49 56 50 44
##  [9813] 46 58 48 60 48 43 47 48 55 51 54 43 51 60 39 37 53 46 65 59 49 44
##  [9835] 54 40 49 54 64 59 44 51 52 53 49 40 53 55 52 55 51 50 58 50 37 62
##  [9857] 40 56 44 53 46 51 62 45 47 46 45 46 47 52 58 53 49 49 52 49 52 58
##  [9879] 44 57 38 50 50 57 58 47 67 57 56 56 50 36 58 36 56 54 54 54 49 51
##  [9901] 45 43 51 54 48 50 46 52 49 65 55 51 47 53 59 62 51 36 41 43 49 47
##  [9923] 59 44 51 43 36 55 58 53 44 42 53 43 43 50 52 40 43 51 53 57 56 52
##  [9945] 46 49 51 59 57 40 64 38 47 43 46 51 59 46 52 47 57 40 53 53 58 53
##  [9967] 53 39 66 49 54 48 49 39 61 51 75 49 51 57 36 54 57 54 51 48 48 45
##  [9989] 53 40 50 50 50 47 57 46 55 54 57 33
  1. Create a histogram of that random sample.
#Histogram of random sample
hist(rand_nums, breaks =32)

Question 8

  1. How many values in your rand_nums vector are below 40?
#This code will show how many values are below 40
length(rand_nums[rand_nums < 40])
## [1] 538

There are 538 values in the rand_nums vector that are below 40.

  1. For a theoretical normal distribution, how many of those 10,000 values would you expect to be below 40?
#Probability of values being less than 40
pnorm(40,mean=51, sd=7)
## [1] 0.05804157
#probability multiplied by number of values
0.05804157*10000
## [1] 580.4157

For a theoretical normal distribution, we expect 580 values to be below 40.

  1. Is your answer in part a reasonably close to your answer in part b? Yes, the answer from part a is reasonably close to the answer in part b. The answer in part a is derrived from a random set of actual values and contains 538 values that are below 40. In the theoretical normal distribution the expected number of values below 40 is only slightly higher at 580 values.