Question

Use a 0.05 signific ance level to find the following: (a) The mean value of the ditternces d for the paired sample dats d ˉ = (b) The standard deviation of the differences d for the paired sample dats s d ​ = (c) The t test statistic t= (d) The positive critical value t= (e) The negative critical value. t= (4) Does the test statistic fall in the critical region? A. No B. Yes (a) Construct a 95\% conkcence interval for the Dooulation mean of ail dilferences x−y Note: You can ears partial credit on thit problem

Answer 1

To conduct paired sample data analysis with a significance level of 0.05, calculate the mean difference, standard deviation of differences, t-test statistic, critical values, and finally, construct a 95% confidence interval for the population mean difference.

Step-by-step explanation:

To address the student's question regarding the paired sample data analysis, we need to perform several steps, assuming that all distributions are normal, the population standard deviations are approximately the same, and the data were collected independently and randomly. Moreover, we will use a significance level of 0.05.

Steps for Paired Sample T-Test

1. Calculate the mean difference (ïd¨š) for the paired sample data.
2. Compute the standard deviation of the differences (sd) for the paired sample data.
3. Find the t-test statistic (t) using the formula: t = (d¨š - ý0) / (sd/√n), where d¨š is the mean difference, sd is the standard deviation of the differences, ý0 is the hypothesized mean difference (often 0), and n is the number of pairs.
4. Determine the positive critical value (t) and the negative critical value (t) by looking them up in the t-distribution table for a two-tailed test with n-1 degrees of freedom or by using statistical software.
5. Compare the t-test statistic with the critical values to see if it falls within the critical region.
6. Construct a 95% confidence interval for the population mean of all differences using the mean difference, the standard deviation of the differences, and the appropriate t-value for the degrees of freedom associated with the sample size.

If the test statistic falls outside the critical values, then it lies in the critical region, indicating that there is a significant difference between the paired samples. If the test statistic falls within the critical values, we fail to reject the null hypothesis, indicating that the study produced no evidence of a significant difference.