Introduction to hypothesis testing：練習問題

In a statistical hypothesis test, which of the following terms refers to the probability of rejecting the null hypothesis \( H_0 \) when it is actually true?

In a hypothesis test for the mean of a normal distribution with a known variance \(\sigma^2 = 16\), a sample of size \(n=25\) is taken. If the null hypothesis is \(H_0: \mu = 50\), calculate the standard error of the mean used in the test statistic.

A hypothesis test is conducted with a significance level of \(\alpha = 0.01\). If the probability of a Type II error (\(\beta\)) is 0.15, what is the power of the test and what does it represent?

In statistical hypothesis testing, which of the following best describes the occurrence of a Type I error?

A biologist tests a null hypothesis \(H_0: p = 0.4\) against \(H_1: p \neq 0.4\) at a 5% significance level. The p-value is calculated to be 0.032. Which of the following is the most appropriate conclusion?

A researcher is testing a hypothesis about the population mean \(\mu\) using a large sample. Define the term statistic as it relates to this process.

Define the term standard error in the context of sampling and state how it relates to the population parameter and sample size \(n\) for a sample mean.

A researcher is conducting a hypothesis test at the 5% significance level. They define a Type II error as failing to reject the null hypothesis when it is false. Explain the mathematical relationship between the power of the test and the probability of a Type II error, \(\beta\).

A large production plant manufactures light bulbs. The probability that a bulb is defective is given by \( p \). Historically, this value has been 0.08. A new manufacturing process is introduced, and a manager believes it has reduced the proportion of defective bulbs. To test this claim, a random sample of 60 bulbs is inspected, and 2 are found to be defective.

(a) State the null and alternative hypotheses to test the manager's belief. (1 point)
(b) Using a 5% significance level, find the critical region for this test. (2 points)
(c) State the conclusion of the test in context. (2 points)
(d) Find the maximum number of defective bulbs that could have been found in the sample for the manager's claim to be supported at the 1% significance level. (2 points)

A manufacturer claims that more than 80% of their components last over 1,000 hours. A random sample of 20 components is tested, and 19 are found to last over 1,000 hours.

(a) Define a suitable test statistic and distribution for a hypothesis test.
(b) Conduct the test at a 5% significance level and state the p-value.
(c) Explain the consequence of a Type I error in this context.