Given an experiment with H0: μ=35, Ha: μ<35, and a possible correct value of 32, you obtain a sa

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Given an experiment with H0: μ=35, Ha: μ<35, and a possible...

Given an experiment with $H_{0}: \mu=35, H_{\mathrm{a}}: \mu<35,$ and a possible correct value of $32,$ you obtain a sample statistic of $\sigma_{\bar{x}_{1}-\bar{x}_{2}}$. After doing analysis, you realize that the sample size $n$ is actually larger than you first thought. Which of the following results from reworking with the increase in sample size? (A) Decrease in probability of a Type I error; decrease in probability of a Type II error, decrease in power (B) Increase in probability of a Type I error, increase in probability of a Type II error, decrease in power (C) Decrease in probability of a Type I error; decrease in probability of a Type II error; increase in power (D) Increase in probability of a Type I error, decrease in probability of a Type II error, decrease in power (E) Decrease in probability of a Type I error; increase in probability of a Type II error; increase in power

Given an experiment with $H_{0}: \mu=35, H_{\mathrm{a}}: \mu<35,$ and a possible correct value of $32,$ you obtain a sample statistic of $\sigma_{\bar{x}_{1}-\bar{x}_{2}}$. After doing analysis, you realize that the sample size $n$ is actually larger than you first thought. Which of the following results from reworking with the increase in sample size?
(A) Decrease in probability of a Type I error; decrease in probability of a Type II error, decrease in power
(B) Increase in probability of a Type I error, increase in probability of a Type II error, decrease in power
(C) Decrease in probability of a Type I error; decrease in probability of a Type II error; increase in power
(D) Increase in probability of a Type I error, decrease in probability of a Type II error, decrease in power
(E) Decrease in probability of a Type I error; increase in probability of a Type II error; increase in power

Key Concepts

Power of a Test

The power of a test is defined as the probability of correctly rejecting a false null hypothesis (i.e., 1-?). An increase in sample size typically increases the power of a test because a larger sample provides more information about the population, leading to a greater ability to detect an effect when one truly exists.

Effect of Sample Size on Hypothesis Testing

Increasing the sample size generally decreases the standard error of the sampling distribution, which in turn makes the test statistic more sensitive to departures from the null hypothesis. This results in a lower probability of a Type II error and consequently higher power. The nominal Type I error rate, however, is maintained at the predetermined level, assuming the test is conducted at the proper significance level.

Type II Error

A Type II error happens when a false null hypothesis is not rejected. Its probability, denoted by beta (?), is inversely related to the sensitivity of the test; a decrease in ? indicates that the test is less likely to miss a true effect. When sample size increases, the test statistic is estimated with greater precision, thereby reducing the probability of a Type II error.

Type I Error

A Type I error occurs when a true null hypothesis is incorrectly rejected. In hypothesis testing, the probability of committing a Type I error is denoted by alpha (?) and is typically chosen by the researcher before the test is conducted; as such, it is set at a desired significance level (for example, 0.05) and does not inherently change with sample size when the test procedure is properly implemented.

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