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Chapter 4: Problem 78

Interpreting a P-value In each case, indicate whether the statement is a proper interpretation of what a p-value measures. (a) The probability the null hypothesis \(H_{0}\) is true. (b) The probability that the alternative hypothesis \(H_{a}\) is true. (c) The probability of seeing data as extreme as the sample, when the null hypothesis \(H_{0}\) is true. (d) The probability of making a Type I error if the null hypothesis \(H_{0}\) is true. (e) The probability of making a Type II error if the alternative hypothesis \(H_{a}\) is true.

Short Answer

Expert verified
Statements (a), (b), (d) and (e) are not proper interpretations of a P-value. Only statement (c) correctly interprets what a P-value measures.

Step by step solution

01

Understanding P-values and their interpretation

A P-value is probabilistic evidence against a null hypothesis. It is the probability of obtaining the observed data or data more extreme than that observed, assuming the null hypothesis is true. Therefore, the smaller the P-value, the stronger the evidence against the null hypothesis.
02

Determine whether statement (a) is true

Statement (a) claims that a P-value is 'The probability that the null hypothesis \(H_{0}\) is true.' This is however incorrect. The P-value does not measure the probability that \(H_{0}\) is true. It is the probability of the observed data assuming that \(H_{0}\) is true.
03

Determine whether statement (b) is true

Statement (b) claims that a P-value is 'The probability that the alternative hypothesis \(H_{a}\) is true.' This is also incorrect. The P-value does not measure the probability that \(H_{a}\) is true.
04

Determine whether statement (c) is true

Statement (c) claims that a P-value is 'The probability of seeing data as extreme as the sample, when the null hypothesis \(H_{0}\) is true.' This is a correct interpretation of a P-value. This statement accurately describes what a P-value measures.
05

Determine whether statement (d) is true

Statement (d) claims that a P-value is 'The probability of making a Type I error if the null hypothesis \(H_{0}\) is true.' However, this is not correct. A Type I error is the rejection of the null hypothesis when it is true. The P-value is not a measure of the probability of making a Type I error.
06

Determine whether statement (e) is true

Statement (e) claims that a P-value is 'The probability of making a Type II error if the alternative hypothesis \(H_{a}\) is true.' This is incorrect. A Type II error is failing to reject the null hypothesis when the alternative hypothesis is true. The P-value does not give a measure of the probability of making this mistake.

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