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A right-tailed test is used to determine if a parameter is significantly greater than a specified value. In the context of a chi-square test, this type of test involves checking the upper tail of the chi-square distribution. Mathematically, it's represented as \[H_0: \text{parameter} \ \text{H}_A: \text{parameter} > \text{specified value} \] To find the critical value for a right-tailed test, you look up the value in the chi-square table using the degrees of freedom and the level of significance α. The degrees of freedom (df) in a chi-square test is typically n-1, where n is the sample size. The critical value corresponds to 1−α in the chi-square table. In our example for a test with 18 degrees of freedom and an α of 0.05, we looked up 0.95 (1-α) in the chi-square table and found the critical value to be approximately 28.869.

A left-tailed test is used to determine if a parameter is significantly less than a specified value. For the chi-square distribution, this involves checking the lower tail. The hypotheses for this test are: \[H_0: \text{parameter} \ \ \text{H}_A: \text{parameter} < \text{specified value}\] To find the critical value, you look up the value corresponding to α in the chi-square table. Here, α is the level of significance. The critical value is where the chi-square value has α of the area to the left of it. For instance, with df = 22 (since n-1 = 23-1) and α = 0.1, we find 0.1 in the chi-square table and obtain a critical value of approximately 13.091.

A two-tailed test checks whether a parameter is significantly different from a specified value, either higher or lower. Thus, it considers both tails of the distribution. The null and alternative hypotheses are: \[H_0: \text{parameter} \ \ H_A: \text{parameter} eq \text{specified value}\] For the chi-square test, you split the level of significance α into two equal parts, α/2, for the upper and lower tails. The critical values are then determined by looking up χ²(α/2, df) and χ²(1-α/2, df) in the table. In our problem, for a sample size of 30 (thus df = 29) and an α of 0.05, we look up 0.025 (0.05/2) and 0.975 (1-0.05/2). The critical values are approximately 16.047 and 45.722, respectively.

Degrees of freedom (df) refer to the number of values that can vary in an analysis without violating any given constraints. In the context of the chi-square test, the degrees of freedom are typically calculated as n-1, where n is the sample size. Degrees of freedom are crucial for determining the shape of the chi-square distribution and for finding the critical values from the chi-square table. The number of degrees of freedom affects the spread and peak of the chi-square distribution curve. For instance:

• In Part (a), with n = 19: df = 18
• In Part (b), with n = 23: df = 22
• In Part (c), with n = 30: df = 29

The level of significance, denoted by α, is the threshold used to determine whether a test statistic is extreme enough to reject the null hypothesis. It represents the probability of making a Type I error, which is rejecting a true null hypothesis. Common levels of significance used in hypothesis testing are 0.01, 0.05, and 0.10. For a right-tailed test, α is used to find the critical value at the upper tail. For a left-tailed test, it defines the critical value at the lower tail. For a two-tailed test, α is divided into two (α/2) for both tails. Clear understanding of the level of significance helps in making informed and statistically sound decisions. It is always chosen before conducting the test to determine the threshold for rejecting H0.