Chapter 27, Problem 27.1AYK

(a)

To determine

To find the median minutes spent standing or walking for each group and which group appears more active.

Answer to Problem 27.1AYK

  $421.444$ is the median minutes spent standing or walking for each group and lean subjects appears more active.

Explanation of Solution

In the question, it is given the table for the subjects spent standing or walking over a ten-day period. It concerned a study comparing the level of physical activity of lean and mildly obese people who do not exercise. The data are a bit irregular but not distinctly non-Normal. Let us use the Wilcoxon test for comparison with the two-sample t test. Now, to find the median minutes spent standing or walking for each group, let us arrange the whole data in ascending order and the bold ones are lean subjects. Then the data will be as:

260.244
267.344
319.212
347.375
358.65
367.138
374.831
410.631
413.667
416.531
426.356
464.756
504.7
511.1
543.388
555.656
578.869
584.644
607.925
677.188

Now, the median is calculated as:

  $\begin{array}{l}=\frac{416.531+426.356}{2}\\ =421.444\end{array}$

And as we can see that the last part of the data contains the lean subjects more so their physical activities are more than the obese subjects. Thus, lean subjects are more active.

(b)

To determine

To arrange all twenty observations in order and find the ranks.

Explanation of Solution

In the question, it is given the table for the subjects spent standing or walking over a ten-day period. It concerned a study comparing the level of physical activity of lean and mildly obese people who do not exercise. The data are a bit irregular but not distinctly non-Normal. Let us use the Wilcoxon test for comparison with the two-sample t test. Now let us arrange all data in order and gave rank by using excel by clicking on the sort icon in the home tab. The data with the ranks will be as:

Rank	Subjects
1	260.244
2	267.344
3	319.212
4	347.375
5	358.65
6	367.138
7	374.831
8	410.631
9	413.667
10	416.531
11	426.356
12	464.756
13	504.7
14	511.1
15	543.388
16	555.656
17	578.869
18	584.644
19	607.925
20	677.188

(c)

To determine

To find out what is the value of W and if null hypothesis is true what are the mean and standard deviation of W .

Answer to Problem 27.1AYK

The value W is $142$ and mean and standard deviation are $105$ and $13.229$ .

Explanation of Solution

In the question, it is given the table for the subjects spent standing or walking over a ten-day period. It concerned a study comparing the level of physical activity of lean and mildly obese people who do not exercise. The data are a bit irregular but not distinctly non-Normal. Let us use the Wilcoxon test for comparison with the two-sample t test.The ranks are given in part (b), so let us sum the ranks for the subjects. Then we have,

Subjects	Sum
Lean subjects	142
Obese subjects	68

Now, the value Wis $142$ . The hypotheses will be defined as:

Null hypothesis: There is no difference in the distribution of subjects.

Alternative hypothesis: Lean subjects are higher than the obese subjects.

And the mean and standard deviation is calculated as:

  $\begin{array}{l}{\mu }_W=\frac{n_1(N+1)}{2}\\ =\frac{10(20+1)}{2}\\ =105\\ {\sigma }_W=\sqrt{\frac{n_1{n}_2(N+1)}{12}}\\ =\sqrt{\frac{10\times 10(20+1)}{12}}\\ =13.229\end{array}$

Now let us calculate the P-value as:

  $\begin{array}{l}P(W\ge 142)=P(Z\ge \frac{142-105}{13.229})\\ =P(Z\ge 2.80)\\ =1-0.99744\\ =0.00256\end{array}$

As we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected, so we have,

  $P<0.05\Rightarrow \text{Reject }{H}_0$

Thus, we have sufficient evidence to prove that the lean subjects spent more minutes standing or walking than the obese subjects.

(d)

To determine

To explain does comparing W with the mean and standard deviation suggest that the lean subjects are more active than the obese subjects.

Answer to Problem 27.1AYK

No, comparing W with the mean and standard deviation does not suggest that the lean subjects are more active than the obese subjects.

Explanation of Solution

In the question, it is given the table for the subjects spent standing or walking over a ten-day period. It concerned a study comparing the level of physical activity of lean and mildly obese people who do not exercise. The data are a bit irregular but not distinctly non-Normal. Let us use the Wilcoxon test for comparison with the two-sample t test. The ranks are given in part (b), so let us sum the ranks for the subjects. Then we have,

Subjects	Sum
Lean subjects	142
Obese subjects	68

And the mean and standard deviation is calculated as:

  $\begin{array}{l}{\mu }_W=\frac{n_1(N+1)}{2}\\ =\frac{10(20+1)}{2}\\ =105\\ {\sigma }_W=\sqrt{\frac{n_1{n}_2(N+1)}{12}}\\ =\sqrt{\frac{10\times 10(20+1)}{12}}\\ =13.229\end{array}$

Thus, comparing W with the mean and standard deviation does not suggest that the lean subjects are more active than the obese subjects because there is less difference between the value of W from the mean of W . so, we do not have a strong evidence from this that the lean subjects are more active than the obese subjects.