As people in our society live longer, there is a growing need for older adults to maintain a healthy lifestyle, part of which is their physical fitness. One study evaluated a program designed to increase older adults' level of fitness using

weight and strength training (Doll, 2009). In the study, eight adults (mean age \(=75.60\) years) received training on a weight machine over an 8 -week period, doing exercises such as bench presses and abdominal crunches. Before and after beginning the training, each adult was measured on a number of functional tasks, one of which was the number of times they could lift a 5 -pound weight over their heads. The descriptive statistics for the number of overhead lifts were the following: pretest \((M=45.60, s=15.00)\) and posttest \((M=53.90, s=18.60)\). Furthermore, the mean difference score \(\left(\bar{X}_{D}\right)\) was -8.30 ; the standard deviation of the difference scores \(\left(s_{\bar{D}}\right)\) was 9.84 . Test the difference in overhead lifts between the pretest and posttest.

a. State the null and alternative hypotheses \(\left(\mathrm{H}_{0}\right.\) and \(\left.\mathrm{H}_{1}\right)\).

b. Make a decision about the null hypothesis.
(1) Calculate the degrees of freedom \((d f)\).
(2) Set alpha ( \(\alpha\) ), identify the critical values (draw the distribution), and state a decision rule.
(3) Calculate a value for the \(t\)-test for dependent means.
(4) Make a decision whether to reject the null hypothesis.
(5) Determine the level of significance.

c. Draw a conclusion from the analysis.

d. What conclusions might the researchers draw regarding the effectiveness of a weight and strength training program designed for older adults?