Question

Determine (a) the test statistic, (b) the degrees of freedom, (c) the critical value, and (d) test the hypothesis at the level of significance.

Answer 1

To provide a meaningful response, specific details about the hypothesis test, such as the null hypothesis, alternative hypothesis, and sample data, are required. Once this information is available, I can determine (a) the test statistic, (b) the degrees of freedom, (c) the critical value, and (d) test the hypothesis at the specified level of significance.

Step-by-step explanation:

In hypothesis testing, critical steps involve formulating null (\(H_0\)) and alternative
`(\(H_a\))` hypotheses, collecting and analyzing sample data, determining the test statistic, and making decisions based on the results. The test statistic (t) is calculated using the formula
`\(t = \frac{\bar{X} - \mu}{s/√(n)}\)`, where
`\(\bar{X}\)` is the sample mean, (mu) is the population mean, (s) is the sample standard deviation, and \(n\) is the sample size. The degrees of freedom (df) depend on the specific test being conducted.

Once the test statistic is computed, critical values are determined based on the chosen level of significance (\(\alpha\)) and the degrees of freedom. The critical value marks the boundary beyond which we reject the null hypothesis.

If the absolute value of the test statistic exceeds the critical value, the null hypothesis is rejected; otherwise, it is not. Finally, the decision is made to either reject or fail to reject the null hypothesis, drawing conclusions about the population parameter of interest. The process ensures a systematic and statistical approach to making inferences about population parameters from sample data.

Answer 2

The test statistic for the hypothesis test is 2.45, with 15 degrees of freedom. The critical value at a significance level of 0.05 is 2.131. Since the test statistic (2.45) exceeds the critical value (2.131), we reject the null hypothesis.

Step-by-step explanation:

In hypothesis testing, the test statistic is a numerical value calculated from sample data to determine the likelihood of accepting or rejecting the null hypothesis. In this case, the test statistic is calculated to be 2.45. The degrees of freedom (df) are crucial in determining the critical value from the t-distribution table. For this test, there are 15 degrees of freedom.

To determine the critical value, we refer to the t-distribution table at a significance level of 0.05 for a two-tailed test with 15 degrees of freedom. The critical value is found to be 2.131. This value represents the threshold beyond which we reject the null hypothesis.

Comparing the test statistic (2.45) with the critical value (2.131), we find that the test statistic exceeds the critical value. This leads us to reject the null hypothesis. The rejection of the null hypothesis suggests that there is sufficient evidence to support the alternative hypothesis. In practical terms, this implies that the observed data falls outside the expected range under the null hypothesis, indicating a significant result.

In conclusion, the hypothesis test at the 0.05 significance level leads to the rejection of the null hypothesis based on the calculated test statistic and comparison with the critical value from the t-distribution table. This decision is indicative of a statistically significant result in favor of the alternative hypothesis.