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 with p = .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 for N = 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 for N = 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

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