0.1 Sampling from Ames, Iowa

If you have access to data on an entire population, say the size of every house in Ames, Iowa, it’s straight forward to answer questions like, “How big is the typical house in Ames?” and “How much variation is there in sizes of houses?”. If you have access to only a sample of the population, as is often the case, the task becomes more complicated. What is your best guess for the typical size if you only know the sizes of several dozen houses? This sort of situation requires that you use your sample to make inference on what your population looks like.

0.2 Pre-Requistes : Available Libraries

0.3 The data

In the previous lab, ``Sampling Distributions’’, we looked at the population data of houses from Ames, Iowa. Let’s start by loading that data set.

load("more/ames.RData")

In this lab we’ll start with a simple random sample of size 60 from the population. Specifically, this is a simple random sample of size 60. Note that the data set has information on many housing variables, but for the first portion of the lab we’ll focus on the size of the house, represented by the variable Gr.Liv.Area.

population <- ames$Gr.Liv.Area
samp <- sample(population, 60)

0.3.1 Exercise 1

  1. Describe the distribution of your sample. What would you say is the “typical” size within your sample? Also state precisely what you interpreted “typical” to mean.
par(mfrow=c(1,3))
hist(population)
hist(samp)

summary(samp)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     492    1156    1370    1425    1641    2784
uniqv <- unique(samp)
uniqv[which.max(tabulate(match(samp, uniqv)))]
## [1] 1595

  • The distribution of the sample is Unimodal and right skewed. Hence, typical size within the sample is using the median instead of the mean
  • Typical size within the sample is the house size with the mode of the house size value 1502
  • Typical to mean is the house size centered near the mean house size value 1424
ggplot(data.frame(samp),aes(x=samp)) +
  geom_histogram(binwidth=500,color='black',fill='black',alpha=0.3) +
  theme_bw() +
  xlab('Living Area') + 
  ylab('Count') + 
  xlim(0,5000)
## Warning: Removed 2 rows containing missing values (geom_bar).

0.3.2 Exercise 2

  1. Would you expect another student’s distribution to be identical to yours? Would you expect it to be similar? Why or why not?

It is highly unlikely that another student distribution will be identical. Most of the student’s distribution will be similarly normally distributed. This is because of central theorem as all of the students generated samples are from the same population using a simple random sampling method (same dataset with same sample size). However, there will be some students who have more outliers.

0.4 Confidence intervals

One of the most common ways to describe the typical or central value of a distribution is to use the mean. In this case we can calculate the mean of the sample using,

sample_mean <- mean(samp)

Return for a moment to the question that first motivated this lab: based on this sample, what can we infer about the population? Based only on this single sample, the best estimate of the average living area of houses sold in Ames would be the sample mean, usually denoted as \(\bar{x}\) (here we’re calling it sample_mean). That serves as a good point estimate but it would be useful to also communicate how uncertain we are of that estimate. This can be captured by using a confidence interval.

We can calculate a 95% confidence interval for a sample mean by adding and subtracting 1.96 standard errors to the point estimate (See Section 4.2.3 if you are unfamiliar with this formula).

se <- sd(samp) / sqrt(60)
lower <- sample_mean - 1.96 * se
upper <- sample_mean + 1.96 * se
c(lower, upper)
## [1] 1316.014 1533.619

This is an important inference that we’ve just made: even though we don’t know what the full population looks like, we’re 95% confident that the true average size of houses in Ames lies between the values lower and upper. There are a few conditions that must be met for this interval to be valid.

0.4.1 Exercise 3

  1. For the confidence interval to be valid, the sample mean must be normally distributed and have standard error \(s / \sqrt{n}\). What conditions must be met for this to be true?
  • Each sample should be independent. It should have equal probability of fetching.
  • If the population distribution is skewed, then the sample size should be >= 30, hence we can use z distribution. If sample size is less than 30, then we an use the T-distribution (given that the population is normally distributed)
  • For the mean to be Normally Distributed, the sample size must meet the criteria of Central Limit Theorem which means the sample has to be randomly selected and have a sample size greater than 30.
  • If the population distribution is not roughly normally distributed, then we need substantially larger than 30 to get a roughly normal sampling distribution. The greater the skew, the larger the n required to get a roughly normal sampling distribution.

0.5 Confidence levels

0.5.1 Exercise 4

  1. What does “95% confidence” mean? If you’re not sure, see Section 4.2.2.

  • 95% confidence level means that if we assume that the sampling distribution of the mean is normal, the 95% confidence interval gives the interval in which the probability that the population mean is within it is 95%.

  • Another way to say this is that if you took a bunch of samples and their 95% confidence intervals, we would expect the population mean to be within the interval in around 95% of these samples.

In this case we have the luxury of knowing the true population mean since we have data on the entire population. This value can be calculated using the following command:

mean(population)
## [1] 1499.69

0.5.2 Exercise 5

  1. Does your confidence interval capture the true average size of houses in Ames? If you are working on this lab in a classroom, does your neighbor’s interval capture this value?
summary(population)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     334    1126    1442    1500    1743    5642

Yes, the population mean of 1500 is within the 95% confidence interval of 1347-1562.

Most likely, my neighbor’s interval also captures this value. However, the fact that there is a 95% confidence interval, there are 5% chance of outlikers as well which are far from population mean. So, there is some chance that my neighbor’s interval does not include the population mean.

lower <= mean(population)
## [1] TRUE
upper >= mean(population)
## [1] TRUE

0.5.3 Exercise 6

  1. Each student in your class should have gotten a slightly different confidence interval. What proportion of those intervals would you expect to capture the true population mean? Why? If you are working in this lab in a classroom, collect data on the intervals created by other students in the class and calculate the proportion of intervals that capture the true population mean.
samp <- sample(population, 60)
se <- sd(samp) / sqrt(60)
lower <- sample_mean - 1.96 * se
upper <- sample_mean + 1.96 * se
c(lower, upper)
## [1] 1325.958 1523.675
mean(population)
## [1] 1499.69

As can be seen from above, mean (1500) falls between the lower interval (1398) and upper interval (1640). Thus true mean falls in that specific confidence interval

Lets run the code multiple times and see how many times the mean falls between the random intervals.

confidence_interval_includes_population_mean <- rep(NA,times=1000)
for(i in 1:1000)
{
  mysample <- sample(population,60) # Obtain a random sample
  lower <- mean(mysample) - 1.96 * (sd(mysample)/sqrt(60)) # Calculate lower interval using sample's mean and standard deviation
  upper <- mean(mysample) + 1.96 * (sd(mysample)/sqrt(60)) # Calculate upper interval using sample's mean and standard deviation
  confidence_interval_includes_population_mean[i] <- lower <= mean(population) & upper >= mean(population) # Use the intervals to calculate many confidence intervals
}
table(confidence_interval_includes_population_mean) # shows how many of randaom samples fall within the range
## confidence_interval_includes_population_mean
## FALSE  TRUE 
##    48   952

As can be seen from above by 1000 sample trials, population mean falls between the lower and upper interval 948 times which is close to 95% confidence level (~94.8% to be exact)

Using R, we’re going to recreate many samples to learn more about how sample means and confidence intervals vary from one sample to another. Loops come in handy here (If you are unfamiliar with loops, review the Sampling Distribution Lab).

Here is the rough outline:

  • Obtain a random sample.
  • Calculate and store the sample’s mean and standard deviation.
  • Repeat steps (1) and (2) 50 times.
  • Use these stored statistics to calculate many confidence intervals.

But before we do all of this, we need to first create empty vectors where we can save the means and standard deviations that will be calculated from each sample. And while we’re at it, let’s also store the desired sample size as n.

samp_mean <- rep(NA, 50)
samp_sd <- rep(NA, 50)
n <- 60

Now we’re ready for the loop where we calculate the means and standard deviations of 50 random samples.

for(i in 1:50){
  samp <- sample(population, n) # obtain a sample of size n = 60 from the population
  samp_mean[i] <- mean(samp)    # save sample mean in ith element of samp_mean
  samp_sd[i] <- sd(samp)        # save sample sd in ith element of samp_sd
}

Lastly, we construct the confidence intervals.

lower_vector <- samp_mean - 1.96 * samp_sd / sqrt(n) 
upper_vector <- samp_mean + 1.96 * samp_sd / sqrt(n)

Lower bounds of these 50 confidence intervals are stored in lower_vector, and the upper bounds are in upper_vector. Let’s view the first interval.

c(lower_vector[1], upper_vector[1])
## [1] 1394.322 1656.344

0.6 On your own

library(dplyr)
## Warning: package 'dplyr' was built under R version 3.5.3
## 
## Attaching package: 'dplyr'
## The following object is masked _by_ '.GlobalEnv':
## 
##     contains
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

  • Using the following function (which was downloaded with the data set), plot all intervals. What proportion of your confidence intervals include the true population mean? Is this proportion exactly equal to the confidence level? If not, explain why.

    plot_ci(lower_vector, upper_vector, mean(population))

    interval <- data.frame(lower_vector,upper_vector)
true_population_mean <- mean(population)

interval <- mutate(interval,in_true_value=lower_vector<true_population_mean & upper_vector<true_population_mean)

filter(interval,in_true_value==TRUE)
    ##   lower_vector upper_vector in_true_value
## 1     1257.896     1492.537          TRUE
## 2     1275.729     1462.071          TRUE
    is_above_lower <- (lower_vector <= mean(population))
is_below_upper <- (upper_vector >= mean(population))

qty_lower <- table(is_above_lower)["TRUE"]
qty_lower_ex <- length(is_above_lower) - table(is_above_lower)["TRUE"]

qty_upper <- table(is_below_upper)["TRUE"]
qty_upper_ex <- length(is_below_upper) - table(is_below_upper)["TRUE"]

within<-(qty_lower+qty_upper)
within/100
    ## TRUE 
## 0.96
    #c(qty_lower, qty_lower_ex)
#c(qty_upper, qty_upper_ex)

    As per above, the most of the values are in the 95% confidence interval. Only 3 values are not in that interval and hence resulting in 97% confidence.

  • Pick a confidence level of your choosing, provided it is not 95%. What is the appropriate critical value?

    confidence_level <- 50
alpha <- 1 - (confidence_level/100)
criticalp <- 1 - (alpha/2)
criticalz <- qnorm(criticalp)
paste("Assuming a confidence level of 50% the appropriate critical value is ",criticalz)
    ## [1] "Assuming a confidence level of 50% the appropriate critical value is  0.674489750196082"

  • Calculate 50 confidence intervals at the confidence level you chose in the previous question. You do not need to obtain new samples, simply calculate new intervals based on the sample means and standard deviations you have already collected. Using the plot_ci function, plot all intervals and calculate the proportion of intervals that include the true population mean. How does this percentage compare to the confidence level selected for the intervals?

    lower_interval <- samp_mean - criticalz * samp_sd / sqrt(50) 
upper_interval <- samp_mean + criticalz * samp_sd / sqrt(50)
plot_ci(lower_interval, upper_interval, mean(population))

    confidence_interval_includes_population_mean <- lower_interval <= mean(population) & upper_interval >= mean(population)
table(confidence_interval_includes_population_mean)
    ## confidence_interval_includes_population_mean
## FALSE  TRUE 
##    25    25

    As can be seen from above, the confidence level (50%) is pretty close to the intervals