## 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) and*N* **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