 # CLT: Activity 2

## Activity 2: Sampling from a Non-normal Population

Up to now, we’ve been working with a normally distributed population. It’s probably no surprise to you that sample means from normal populations yielded normal sampling distributions. But, what happens with non-normal population distributions? The next activity will illustrate what happens when the population distribution is a binomial distribution.

#### What is the Binomial Distribution?

A binomial distribution results when only two outcomes are possible (e.g., 0 or 1) andN independent cases are selected, where the probability = p for a specific outcome for each case. For example, there are only two possible outcomes when we flip a coin: heads or tails. If the coin is fair, p = .5 is the probability of getting a head on each toss, and the outcome from each toss is independent of outcomes from other tosses. The number of heads on each toss is either 0 or 1. If we toss the coin N = 10 times, we can observe 0 to 10 heads. The probabilities of all possible numbers of heads define a binomial probability distribution.

With a binomial distribution where p = .5, you have equal chances of observing either outcome on each observation. If one of the outcomes is assigned a value of 0 and the other outcome a value of 1, then the population mean is .5. The binomial population distribution for N = 1 is very non-normal because only two outcomes are possible, 0 or 1, one at each end of the distribution.

In this activity, we are interested in voter support in Ourtown for Proposition A on an upcoming ballot. Suppose support for Proposition A is evenly split in Ourtown, such that half of the population would say “yes” and half would say “no” when asked their view. We conduct a survey of randomly selected Ourtown voters and code their responses as “Yes” = 1 and “No” = 0. Thus, in this example p(Y = 1) = .50, or simply p = .50.

Using the applet below, we will examine the possible outcomes of surveys with three different sample sizes, N. For each sample size, you will be guided through a series of steps and observations. We begin with a sample of N = 5.

##### Q1. How do we represent a binomial population withp= .5?
• Set the population to “Binomial p = .5.”
• Set the sample size to “N=5.”
• Select “Show sample data” (in black).
• Select “Show population” (in blue).
• Why are the blue bars equal in height? Answer.
##### Q2. In each sample, how many people support the proposition and how many don’t (i.e., how many 1’s and 0’s are there)?
• Click “Draw a sample.”
• Observe the relative number of 0s and 1s in the sample.
• Calculate the percentage of people who support the proposition. (Give me a hint)
• Compare this percentage to the sample mean.
• What does the sample mean indicate in this activity?  Answer.
• Click “Draw a sample” again and verify the relationship between the sample mean and the percentage of people who support the proposition for this sample.
##### Q3. What are the possible values for the sample mean for this sample size,N= 5?
• Click “Draw a sample” 10 times and record the mean for each sample below:

• Click “Show obtained means” (in red) to see the distribution of your sample means.
• Click “Draw 100 samples” several times and pay attention to the possible sample means.
• Describe the possible values for the sample means. Answer for N = 5
##### Q4. Is the distribution of possible sample means an example of a normal distribution?
• Select “Show sampling distribution of the mean” (in green) to see the actual distribution of possible means.
• How is this distribution similar to a normal curve and how does it differ? Answer for N = 5
##### Q5. What are the possible values for the sample mean forN= 100?
• De-select “Show sampling distribution of the mean” (in green).
• Change sample size to N = 100.
• Click “Draw a sample” 10 times and record the mean for each sample below:

• Click “Draw 100 samples” several times (be sure that “Show obtained means” is selected).
• Describe the possible values for the sample means. Answer for N = 100
##### Q6. Is the distribution of possible sample means an example of a normal distribution?
• Select “Show sampling distribution of the mean” (in green) to see the actual distribution of possible means.
• How is this distribution similar to a normal curve and how does it differ? Answer for N = 100
##### Q7. What are the possible values for the sample mean forN= 25?
• De-select “Show sampling distribution of the mean” (in green).
• Change sample size to N = 25.
• Click “Draw a sample” 10 times and record the mean for each sample below:

• Click “Draw 100 samples” several times (be sure that “Show obtained means” is selected).
• Describe the possible values for the sample means. Answer for N = 25
##### Q8. Is the distribution of possible sample means an example of a normal distribution?
• Select “Show sampling distribution of the mean” (in green) to see the actual distribution of possible means.
• How is this distribution similar to a normal curve and how does it differ? Answer for N = 25 