in this movie we will continue our discussion of the behavior of sample proportions. In particular, we're going to investigate this question: How does the size of the sample impact the spread and the sample proportions? To investigate this question, we're going to return to the familiar context of the previous movie and look at the population of all part-time college students. We will assume as we did before that 60 percent of this population is female. Now, in this movie we're going to be comparing two different distributions of sample proportions. Both distributions will be generated by collecting thousands of random samples. But they will differ in the sample size that we use. We will return to the sampling distribution we generated in the previous movie when we collected random samples of 25 students at a time and then we will generate a second sampling distribution using a larger sample size. We will look at samples of size 100. The question, then, that we are interested in is whether or not this increase in sample size will have an impact on the spread. Now, before we actually begin to collect random samples, let's remind ourselves what we observed previously. When we collected samples of 25 students at a time, we saw that the sample proportions were normally distributed and centered at 0.6. In other words, when I took all of the p-hat values collected from the thousands of samples and calculated the mean, I got 0.6, the population proportion. When I calculated the standard deviation of the p-hat values, I got about 0.1. So for samples of size 25, we saw that typical values fell between about 0.5 and 0.7, within one standard deviation of the mean. Now, we're interested in what happens if we increase the sample size to 100. Will we expect to see more variability, variability that is roughly the same, or much less variability? In particular, we're interested in finding out what happens to the standard deviation. Before I begin to collect random samples of 100 students, let me get you oriented what I have here on the screen. Here you will see each sample and the one hundred individuals in it. We'll be able to tell if they are male or female. On the right is a table that summarizes what happened in a sample. For example, here I had 67 females out of the 100 randomly selected part time college students. That's a p-hat value of 0.67. The p-hat value is graphed above. Here is where we will generate the sampling distribution. Let's see what happens now as we collect a random sample. Here I have 100 randomly selected part-time college students. For this particular sample, I get a sample proportion of 0.65. I've graphed that above. I've now turned the animation off so that we can collect samples more quickly. What we're interested in is what is happening to the spread that we're seeing in the sample proportions. We're expecting that many of the sample proportions will be close to the population proportion of 0.6, but do we think we will see more variability, less variability, or variability that's about the same as we saw previously? Pause the movie and investigate this question: What do you think will be the standard deviation of the sample proportions when the sample size is 100? Let's see what happened. Is your intuition correct? Here on the right I have the sampling distribution generated from taking thousands of random samples where each sample had 100 students in it. I can see visually that there is much less variability in this sampling distribution than in the previous one. If I compare the standard deviations I can get a better sense of what happened. When I had a sample size of 25, the standard deviation of sample proportions was about 0.1. When I increased the sample size by a factor of four, the standard deviation decreased to about half of what it was previously. So, I can conclude that larger samples do have less variability. I also notice that the mean of the sampling distribution stayed at 0.6. So the mean does not seem to be impacted by sample size. I also see that the distribution is normal for this case. Now we're going to move back into the text where we will see what advanced probability theory has to tell us about calculating the standard deviation of sample proportions. We will also find later that the distribution will not always be normal.