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. 16. how would you describe the variability you see in the above distribution? please explain.

Explanation:

Step1: Analyze the center of the distribution

In a histogram, the center can be estimated by looking at the peak (mode) or the balance point (mean). Here, visually, the highest frequency of sample proportions occurs around \( \hat{p} = 0.4\).

Step2: Analyze the variability

Variability in a distribution of sample proportions (sampling distribution) can be described by how spread - out the data is. A normal - looking (bell - shaped) histogram like this one (which is typical for sampling distributions of proportions when \(np\geq5\) and \(n(1 - p)\geq5\), here \(n = 50\)) has a certain amount of spread. The spread of a sampling distribution of a proportion is given by the formula \(\sigma_{\hat{p}}=\sqrt{\frac{p(1 - p)}{n}}\). But just from the histogram, we can say that the sample proportions vary around the center (\(p\approx0.4\)) and the distribution has a relatively small spread (since the bars are clustered around \(0.4\) and don't spread out extremely wide, for example, not from \(0\) to \(1\))

Answer:

1. The distribution appears to be centered at approximately \(0.4\). This is because the highest frequency (the tallest bars in the histogram) of sample proportions \(\hat{p}\) occurs around the value of \(0.4\).
2. The variability is relatively small. The sample proportions \(\hat{p}\) are clustered around the center (\(\hat{p}\approx0.4\)) and do not have an extremely wide spread. The shape of the histogram (bell - shaped, which is characteristic of a sampling distribution of a proportion when \(n = 50\) is reasonably large) indicates that the sample proportions follow a normal - like pattern with a moderate amount of spread around the center.