Section: Math 0 Likes
Micha and Rana each selected a random sample of students at their (May 2023 U.S.)
Micha and Rana each selected a random sample of students at their school and asked how many soft drink servings each student had consumed the previous week. Micha estimated that the mean number of soft drink servings was 7.1, with an associated margin of error of 1.2.
Rana estimated that the mean number of soft drink servings was 8.3, with an associated margin of error of 0.8.
Assuming the margins of error were calculated in the same way, which of the following best explains why Rana obtained a smaller margin of error than Micha?
A. Rana’s sample contained more students than Micha’s sample contained.
B. Rana’s sample contained more students who drink soft drinks than Micha’s sample contained.
C. Rana’s sample contained more students who drank exactly seven soft drink servings than Micha’s sample contained.
D. Rana’s sample contained more students who drank exactly eight soft drink servings than Micha’s sample contained.
Solution
Correct answer: A. Rana’s sample contained more students than Micha’s sample contained.
The margin of error in a sample mean estimate is inversely proportional to the square root of the sample size. This means that as the sample size increases, the margin of error decreases.
Specifically, the formula for the margin of error (ME) is:
ME = z * σ / √n
Where:
z is the critical value from the standard normal distribution (e.g., 1.96 for a 95% confidence level)
σ is the population standard deviation
n is the sample size
Since Rana's sample had a smaller margin of error than Micha's, it must be the case that Rana's sample size was larger than Micha's sample size, all else being equal.
Therefore, the best explanation is that Rana's sample contained more students than Micha's sample, which led to the smaller margin of error.
Exam Edition: May 2023 U.S.
Exam Year: 2023
Related topics: Sampling and Estimation, Hypothesis Testing and Inference, Probability and Statistics