a. Determine the critical value for a right-tailed test of a population mean at the α = 0.01 level of significance with 22 degrees of freedom.

Verified step by step guidance

1. Identify the type of test and significance level: This is a right-tailed test with a significance level of \( \alpha = 0.01 \).
2. Determine the degrees of freedom (df): Given as 22, which will be used to find the critical value from the t-distribution.
3. Understand that the critical value corresponds to the t-score where the area to the right under the t-distribution curve is \( \alpha = 0.01 \).
4. Use a t-distribution table or statistical software to find the t-value that corresponds to \( \alpha = 0.01 \) in the right tail with 22 degrees of freedom.
5. The critical value is this t-score, which will be positive since it is a right-tailed test.

Key Concepts

Right-Tailed Test

A right-tailed test is a hypothesis test where the critical region is in the right tail of the probability distribution. It is used when the alternative hypothesis states that the population parameter is greater than a certain value. The critical value marks the boundary beyond which we reject the null hypothesis.

Level of Significance (α)

The level of significance, denoted by α, is the probability of rejecting the null hypothesis when it is actually true (Type I error). For α = 0.01, there is a 1% risk of this error. It determines the size of the critical region in the tail(s) of the distribution.

Degrees of Freedom and t-Distribution

Degrees of freedom (df) relate to the number of independent values in a calculation, often n - 1 for a sample size n. When the population standard deviation is unknown and the sample size is small, the t-distribution is used instead of the normal distribution. The critical value depends on both α and df.