Question

the histogram below shows you what happens when we survey thousands of random samples of size ( n = 50 ) from the population of college students and then determine the proportion from each sample who were initially undecided about what major to purse. what is being displayed in the histogram is many sample proportions. 15. at what value does it appear the above distribution is centered? please explain. the distribution appears to be centered at approximately 0.4/0.41 because the center of the sampling distribution should be approximately equal to the population proportion. 16. how would you describe the variability you see in the above distribution? please explain.

Explanation:

Step1: Analyze the spread of the histogram

The spread of a distribution (variability) can be described by looking at how wide or narrow the histogram is. A narrow histogram means less variability (sample proportions are close to each other), and a wide histogram means more variability (sample proportions are more spread out).

Step2: Observe the given histogram

Looking at the histogram of sample proportions (with sample size \(n = 50\)), we can see that the sample proportions \(\hat{p}\) are clustered around the center (approximately \(0.4 - 0.41\)) and the spread of the sample proportions (the range of \(\hat{p}\) values from the left - most to the right - most bar) is from roughly \(0.2\) to \(0.6\).

Answer:

The variability of the distribution is moderate. The sample proportions \(\hat{p}\) are not extremely tightly clustered (so not very low variability) but also not extremely spread out (so not very high variability). The range of \(\hat{p}\) values (from approximately \(0.2\) to \(0.6\)) indicates that there is some spread around the center (around \(0.4 - 0.41\)), which is typical for a sampling distribution of sample proportions with \(n=50\).