Question

Use Table 4 in Appendix I to find the following critical values: a. An upper one-tailed rejection region with \(\alpha=.05\) and \(11 d f\). b. A two-tailed rejection region with \(\alpha=.05\) and \(7 d f\). c. A lower one-tailed rejection region with \(\alpha=.01\) and \(15 d f\).

Short answer

Question: Find the critical values for the following cases using a t-distribution table. Mention whether it's an upper one-tailed, two-tailed, or lower one-tailed test. a. Significance level = 0.05, degrees of freedom = 11 b. Significance level = 0.05, degrees of freedom = 7 c. Significance level = 0.01, degrees of freedom = 15 Answer: a. Upper one-tailed: 1.796 b. Two-tailed: -2.365 and 2.365 c. Lower one-tailed: -2.602

Step-by-step solution

Find the significance level and degrees of freedom for each case

Since we need to find critical values for three different cases, we can list down the significance level (\(\alpha\)) and degrees of freedom (\(d f\)) for each case: a. \(\alpha = 0.05\), \(d f = 11\) (Upper one-tailed) b. \(\alpha = 0.05\), \(d f = 7\) (Two-tailed) c. \(\alpha = 0.01\), \(d f = 15\) (Lower one-tailed)

Use Table 4 in Appendix I to find the critical values for each case

To find the critical values, we need to use the information from the table provided in Appendix I, which contains t-distribution critical values for given significance levels and degrees of freedom. a. Upper one-tailed rejection region with \(\alpha =0.05\), \(d f = 11\): Find the row for \(11 d f\) in Table 4 and look at the column for the given significance level (\(\alpha=0.05\)). The critical value for this case is \(1.796\). b. Two-tailed rejection region with \(\alpha=0.05\), \(d f = 7\): Firstly, find the row for \(7 d f\) in Table 4 and look at the column for the given significance level (\(\alpha=0.05\)). Since it's a two-tailed test, we need to find the critical value for \(\alpha/2 = 0.025\). The critical value in this case is \(±2.365\). Keep in mind that since this is two-tailed, the negative value is the lower bound, and the positive value is the upper bound. c. Lower one-tailed rejection region with \(\alpha=0.01\), \(d f = 15\): Find the row for \(15 d f\) in Table 4 and look at the column for the given significance level (\(\alpha=0.01\)). The critical value for this case is \(-2.602\). Since it's a lower one-tailed test, we should use the negative value as the critical value. So, the critical values for each case are: a. Upper one-tailed: \(1.796\) b. Two-tailed: \(-2.365\) and \(2.365\) c. Lower one-tailed: \(-2.602\)

Key concepts

Critical Value

The critical value is a key concept in hypothesis testing that helps determine the cutoff point for deciding if a test statistic is significant. When conducting a t-test, the critical value indicates the threshold beyond which we reject the null hypothesis. It is closely associated with the significance level, denoted as \(\alpha\), which reflects the probability of making a Type I error (rejecting a true null hypothesis).

• For an upper one-tailed test with \(\alpha = 0.05\) and \(d f = 11\), the critical value is found at the 0.05 significance level, giving us \(1.796\).
• In a two-tailed test with \(\alpha = 0.05\) and \(d f = 7\), the critical value is split between the tails; hence we use \(\alpha/2 = 0.025\), resulting in critical values of \(\pm 2.365\).
• For a lower one-tailed test with \(\alpha = 0.01\) and \(d f = 15\), the critical value is found at \(-2.602\), being a negative value because we are considering the lower end of the t-distribution.

Understanding how to determine the critical value is crucial as it directly influences the decision about the null hypothesis.

Degrees of Freedom

Degrees of freedom ( **df**) play a significant role in calculations involving statistical tests, such as the t-distribution. They determine the shape of the distribution and depend primarily on the sample size. In general, degrees of freedom are calculated as the sample size minus one ( **df = n - 1** ). They essentially represent the number of values that are free to vary in a statistical calculation.

• In the original exercise, we dealt with samples where degrees of freedom were 11, 7, and 15.
• Each of these determines the critical regions which are based on how much data we have available for estimating population parameters.
• The more degrees of freedom, the closer the distribution of the test statistic resembles the standard normal distribution.

Getting comfortable with understanding and manipulating degrees of freedom will enhance your capability to conduct accurate statistical tests and make more informed decisions.

One-tailed Test

A one-tailed test is a statistical hypothesis test in which the alternative hypothesis specifies a direction. This means that the rejection region is located on only one side of the probability distribution curve. One-tailed tests can either be upper or lower-tailed based on where the extreme value you are testing for falls.

• In an **upper one-tailed test**, the critical value determines when to reject the null hypothesis if the test statistic is too high (e.g., \(\alpha =0.05\), df = 11 leads to a critical value of \(1.796\)).
• In contrast, a **lower one-tailed test** determines if the test statistic falls below a certain negative critical value (e.g., \(\alpha =0.01\), df = 15 with a critical value of \(-2.602\)).
• These tests are most effective when the research hypothesis predicts a change or effect in a specific direction.

One-tailed tests offer greater statistical power to detect an effect in one direction at the cost of not being able to detect effects in the opposite direction.

Two-tailed Test

A two-tailed test is used when you want to assess whether there is an effect in either direction, without specifying whether it is higher or lower. This makes two-tailed tests very versatile for scenarios where a non-directional change or difference is expected or possible.

• In a **two-tailed test**, such as the one with \(\alpha = 0.05\) and df = 7, both tails of the distribution are considered, resulting in rejection regions in both extremes (\(\pm 2.365\)).
• The significance level \(\alpha\) is divided equally between the two tails, making them more conservative since you need stronger evidence to reject the null as compared to one-tailed tests.
• This test is ideal when no prior expectation exists concerning the direction of the effect being tested, which aligns with the formulation of the alternative hypothesis as "not equal to."

A two-tailed test covers all bases in terms of potential outcomes, ensuring that conclusions drawn are robust to variations on either side of the spectrum.