(Solution) - Suppose that the actual proportion of students at a particular -(2025 Original AI-Free Solution)

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Suppose that the actual proportion of students at a particular college who use public transportation to travel to campus is 0.15. In a study of parking needs at the campus, college administrators would like to estimate this proportion. They plan to take a random sample of 75 students and use the sample proportion who use public transportation, ? , as an estimate of the population proportion.
a. Show that the standard deviation of ? p is equal to ? ? = 0.0412.
b. If for a different sample size, ? ? = 0.0319, would you expect more or less sample- to- sample variability in the sample proportions than for when n = 75?
c. Is the sample size that resulted in ? ? p = 0.0319 larger than 75 or smaller than 75? Explain your reasoning.