Consider all current OSU students to be a population. In this population, the proportion of students who prefer drinking smoothies instead of milkshakes is 0.47. Imagine that Sal and Jane each survey a random sample from this population. Sal surveys a random sample of n = 30 OSU students about whether they prefer to drink smoothies instead of milkshakes, and Jane surveys a random sample of n = 50 OSU students. Based on what you know about sampling distributions, which one of the following statements is correct?

Question

Consider all current OSU students to be a population.  In this population, the proportion of students who prefer drinking smoothies instead of milkshakes is 0.47.  Imagine that Sal and Jane each survey a random sample from this population.  Sal surveys a random sample of n = 30 OSU students about whether they prefer to drink smoothies instead of milkshakes, and Jane surveys a random sample of n = 50 OSU students.  Based on what you know about sampling distributions, which one of the following statements is correct?

BlackTom AI · Accepted Answer

We begin by restating the scenario and the options to be evaluated. In the population of all OSU students, the true proportion who prefer smoothies over milkshakes is p = 0.47. Sal takes a random sample of n = 30 students, while Jane takes a random sample of n = 50 students. We are asked to assess the truth of each statement about the sampling distributions of the sample proportions.

Option 1: 'Sal is more likely than Jane to observe a sample proportion that is 0.40 or smaller.' This can be analyzed by comparing the variability of the sampling distributions. The standard deviation of a sample proportion p-hat is sqrt[p(1 − p)/n]. With p = 0.47, p(1 − p) = 0.47 × 0.53 ≈ 0.2491. For Sal (n = 30), SD ≈ sqrt(0.2491/30) ≈ sqrt(0.008303) ≈ 0.0911. For Jane (n = 50), SD ≈ sqrt(0.2491/50) ≈ sqrt(0.004982) ≈ 0.0706. The smaller SD for Jane means a tighter distribution around 0.47, making extreme low values like 0.40 (which is 0.07 below 0.47) far less likely than for Sal. Therefore Sal is more likely than Jane to observe p-hat ≤ 0.40. This statement is consistent with the increased sampling variability for the smaller sample size.

Option 2: 'Sal is less likely than Jane to observe a sample proportion that is 0.40 or smaller.' Given the calculations above, this is inconsistent with the greater variability at n = 30. Since Sal has more dispersion, he is not less likely to observe such an extreme value; he is more likely. Hence this option is incorrect.

Option 3: 'Because Sal and Jane are each sampling from the same population, they each have exactly the same probability of observing a sample proportion that is 0.40 or smaller.' Although they share the same population proportion p, the sampling variability depends on n. The two samples have different SDs, so the probabilities are not the same. This option ignores the impact of sample size on dispersion and is incorrect.

Option 4: 'If Sal and Jane are each sampling randomly, they should each observe sample proportions equal to exactly 0.47.' In practice, sample proportions vary around the true p, and due to sampling error they will not be exactly 0.47 in general. This statement wrongly assumes no sampling error and is false.

Option 5: 'It’s impossible to answer this question since the means and the standard deviations of the sampling distributions based on samples of size n = 30 and n = 50 are not given.' Even without explicit numeric means and standard deviations provided in the prompt, we can compute the standard deviations from p and n using SD = sqrt[p(1 − p)/n]. The mean of the sampling distribution of p-hat is p = 0.47, and the standard deviations differ as shown above. Therefore the question can be answered with the given information, making this statement incorrect.

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