In Exercises 3–8, find the critical value(s) and rejection region(s) for the type of t-test with level of significance alpha and sample size n.


Right-tailed test, α=0.01, n=31

Verified step by step guidance

1. Determine the degrees of freedom (df) for the t-test. The formula for degrees of freedom is df = n - 1, where n is the sample size. In this case, df = 31 - 1 = 30.
2. Identify the level of significance (α). For this problem, α = 0.01, which represents the probability of rejecting the null hypothesis when it is true.
3. Since this is a right-tailed test, the critical value corresponds to the t-value where the area to the right under the t-distribution curve equals α. Use a t-distribution table or statistical software to find the critical t-value for df = 30 and α = 0.01.
4. Define the rejection region. For a right-tailed test, the rejection region is t > critical t-value. This means any test statistic greater than the critical t-value will lead to rejecting the null hypothesis.
5. Summarize the critical value and rejection region. Clearly state the critical t-value obtained and the corresponding rejection region (e.g., t > critical t-value).

Key Concepts

Critical Value

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the chosen significance level (alpha) and the distribution of the test statistic. For a right-tailed t-test, the critical value corresponds to the point in the t-distribution where the area to the right equals alpha, indicating the cutoff for extreme values.

Rejection Region

The rejection region is the range of values for the test statistic that leads to the rejection of the null hypothesis. In a right-tailed test, this region is located to the right of the critical value. If the calculated test statistic falls within this region, it suggests that the observed data is significantly different from what is expected under the null hypothesis.

T-Test

A t-test is a statistical test used to compare the means of two groups or to assess whether a sample mean significantly differs from a known value. It is particularly useful when the sample size is small (typically n < 30) and the population standard deviation is unknown. The t-test relies on the t-distribution, which accounts for the additional uncertainty introduced by estimating the population standard deviation from the sample.