Question

two college students, chris and angie, each survey a random sample of peers, from the same population, in order to estimate the proportion of students who ride a bicycle to campus. chris surveys a random sample of ( n = 150 ) students and angie surveys a random sample of ( n = 600 ) students. which resulting sampling distribution of sample proportions will have more variability from sample to sample?
its impossible to answer this question without knowing the actual population proportion.
whoevers sample proportion happens to fall closer to 0.50 will have the sampling distribution that is most variable.
the sampling distribution based on samples of ( n = 150 ) will have more variability.
the sampling distribution based on samples of ( n = 600 ) will have more variability.
both sampling distributions will have approximately the same amount of variability because chris and angie are each sampling from the same population.

Explanation:

Step1: Recall the formula for the standard deviation of the sampling distribution of the sample proportion

The standard deviation of the sampling distribution of the sample proportion \(\hat{p}\) is given by \(\sigma_{\hat{p}}=\sqrt{\frac{p(1 - p)}{n}}\), where \(p\) is the population proportion and \(n\) is the sample size.

Step2: Analyze the effect of sample size on variability

Since \(p\) (the population proportion) is the same for both Chris and Angie (they are sampling from the same population), the standard deviation (a measure of variability) of the sampling distribution of the sample proportion is inversely related to the square - root of the sample size. That is, \(\sigma_{\hat{p}}\propto\frac{1}{\sqrt{n}}\).

When \(n_1 = 150\) and \(n_2=600\), and \(n_1\lt n_2\), we have \(\sqrt{n_1}\lt\sqrt{n_2}\), so \(\frac{1}{\sqrt{n_1}}\gt\frac{1}{\sqrt{n_2}}\)

Answer:

The sampling distribution based on samples of \(n = 150\) will have more variability.