8. Sampling Distributions & Confidence Intervals: Proportion
Sampling Distribution of Sample Proportion
2:07 minutesProblem 6.3.17b
Textbook Question
MCAT The Medical College Admissions Test (MCAT) is used to help screen applicants to medical schools. Like many such tests, the MCAT uses multiple-choice questions with each question having five choices, one of which is correct. Assume that you must make random guesses for two such questions. Assume that both questions have correct answers of “a.â€
b. Find the mean of the sampling distribution of the sample proportion.
Verified step by step guidance1
Step 1: Understand the problem. The problem involves finding the mean of the sampling distribution of the sample proportion when guessing answers to two multiple-choice questions, each with five options. The sample proportion refers to the proportion of correct answers in the sample (in this case, two questions).
Step 2: Recall the formula for the mean of the sampling distribution of the sample proportion. The mean is given by \( \mu_{\hat{p}} = p \), where \( p \) is the population proportion of success (the probability of guessing a correct answer).
Step 3: Determine the value of \( p \). Since there are five choices for each question and only one correct answer, the probability of guessing correctly is \( p = \frac{1}{5} \).
Step 4: Substitute the value of \( p \) into the formula for the mean of the sampling distribution. This gives \( \mu_{\hat{p}} = \frac{1}{5} \).
Step 5: Conclude that the mean of the sampling distribution of the sample proportion is equal to the probability of guessing a correct answer, which is \( \frac{1}{5} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sampling Distribution
A sampling distribution is the probability distribution of a statistic obtained from a larger population, formed by taking multiple samples. In this context, it refers to the distribution of sample proportions derived from random guesses on the MCAT questions. Understanding this concept is crucial for determining how sample proportions behave and vary when drawn from a population.
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Sampling Distribution of Sample Proportion
Sample Proportion
The sample proportion is the ratio of the number of successes in a sample to the total number of observations in that sample. In the MCAT scenario, if a correct answer is 'a,' the sample proportion would represent the fraction of correct guesses made when answering the questions. This concept is essential for calculating the mean of the sampling distribution.
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Mean of the Sampling Distribution
The mean of the sampling distribution of the sample proportion is the expected value of the sample proportion across all possible samples. It is calculated as the population proportion, which in this case is the probability of guessing the correct answer. For the MCAT questions, if the correct answer is 'a,' the mean would reflect the likelihood of randomly guessing 'a' correctly, providing insight into the overall performance of random guessing.
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