 # CLT: Question 8

Now use the applet below, but this time for Population select “Binomial p = .1.” Also select “Show population” (in blue) and “Show sampling distribution” (in green). Click “Show obtained means” (in red) and then “Draw 100 samples” several times for each sample size of N = 5, 25, and 100. Pay special attention to the possible values for the sample means.

What do binomial sampling distributions (p =.1) look like for the various sample sizes?

#### Regardless of sample size, the sampling distributions have the same shape as the population.

You answered that regardless of sample size, the sampling distributions have the same shape as the population. In the population, only two values are possible, 0 or 1. Sampling distributions of means include possible values between 0 and 1, so they do not have the same shape as the population.

#### When N is 25 or larger, the sampling distribution of the mean is always close to normal.

You answered that the sampling distribution for N of 25 or larger is always close to normal. This is wrong because we can see that when we have a binomial distribution with N = 25 and p =.1, the sampling distribution is notably skewed and has a limited set of discrete possible values.

#### The sampling distribution based on N = 100 is closer to normal compared to a sampling distribution based on N = 25.

The sampling distribution based on N = 100 is much closer to a normal distribution than when N = 25 for both p =.5 and p = .1. We noticed that when p = .1, the sampling distribution is not very close to normal even when N = 25.

When dealing with binomial distributions, a rule of thumb is that we can consider the sampling distribution to be approximately normal when Np(1 – p) > 5. When p = .1 and N = 25, this gives a value of (25)(.1)(.9) = 2.25, which is less than 5. Thus, generally it would not be acceptable to assume that the sampling distribution is close to a normal distribution in this case. However, when p = .5 and N = 25, this rule of thumb yields (25)(.5)(.5) = 6.25, a value larger than 5.

With p = .5 and N = 25, the sampling distribution of the mean is close enough to normal that the normal distribution would yield probabilities that would be acceptably accurate for most purposes. We could use the actual values computed from the underlying binomial distribution rather than approximations calculated from the normal distribution if we desire greater accuracy than afforded by the normal distribution.