Chapter 11, Problem 21P

In the Chapter Preview we described a study showing that students had more academic problems following nights with less than average sleep compared to nights with more than average sleep (Gillcn-O’Neel. Huynh, & Fuligni, 2013). Suppose a researcher is attempting to replicate this study using a sample of $n=8$ college freshmen. Each student record the amount of study time and amount of sleep each night and reports the number of academic problems each day. The following data show the results from the study.

Number of Academic Problems
Student	Following Nights

with Above

Average SleepFollowing Nights

with Below

Average Sleep A 10 13 B 8 6 C 5 9 D 5 6 E 4 6 F 10 14 G 11 13 H 3 5

a. Treat the data as if the scores are from an independent-measures study using two separate samples, each with $n=8$ participants. Compute the pooled variance, the estimated standard error for the mean difference, and the independent-measures t statistic. Using a two-tailed test with $\alpha =.05$ , is there a significant difference between the two sets of scores?

b. Now assume that the data are from a repeated-measures study using the same sample of is $n=8$ participants in both treatment conditions. Compute the variance for the sample of difference scores, the estimated standard error for the mean difference and the repealed-measures t statistic. Using a two-tailed test with $\alpha =.05$ , is there a significant difference between the two sets of scores? (You should find that the repeated-measures design substantially reduces the variance and increases the likelihood of rejecting $H_0$ .)

n=8 n=8

To determine

Is there a significant difference between the two sets of scores by using an independent-measures design and then by using repeated measures design.

Answer to Problem 21P

1. Using an independent-measures design:

STEP 1: State the hypotheses. In symbols, the null and alternative hypotheses are:

$\begin{array}{l}{H}_0:{\mu }_1={\mu }_2\\ H_a:{\mu }_1\ne {\mu }_2\end{array}$

STEP 2: Locate the critical region. The critical value of t values for the critical region are $t=\pm 2.145$.

STEP 3: Compute the test statistic. The test statistic is $t=-1.15$.

STEP 4: Make a decision about the null hypothesis. We cannot conclude that there is a significant difference between the two sets of scores by using an independent-measures design.

2. Using repeated measures design:

STEP 1: State the hypotheses. In symbols, the null and alternative hypotheses are:

$\begin{array}{l}{H}_0:{\mu }_D=0\\ H_a:{\mu }_D\ne 0\end{array}$

STEP 2: Locate the critical region. The critical value of t values for the critical region are $t=\pm 2.365$.

STEP 3: Compute the test statistic. The test statistic is $t=2.94$.

STEP 4: Make a decision about the null hypothesis. We can conclude that there is a significant difference between the two sets of scores by using repeated measures design.

Explanation of Solution

We are given to use a two-tailed test. That's why alternative hypothesis has $\ne$ sign.

The critical value of t values for the critical region are $t=\pm 2.145$. So, reject null hypothesis if $t<-2.145$ or $t>2.145$.

The critical value of t values for the critical region are $t=\pm 2.365$. So, reject null hypothesis if $t<-2.365$ or $t>2.365$.

Given:

The table that summarizes the scores for a sample of $n=8$ participants and $\alpha =0.05$.

Formula used:

$df=n_1+n_2-2$$S_P^2=\frac{S{S}_1+S{S}_2}{n_1+n_2-2}$$SE=\sqrt{\frac{S_P^2}{n_1}+\frac{S_P^2}{n_2}}$$t=\frac{M_1-M_2}{SE}$

$\begin{array}{l}df=n-1\\ S_D=\sqrt{\frac{SS}{n-1}}\\ t=\frac{M_D-{\mu }_D}{S_D/\sqrt{n}}\end{array}$

Calculation:

1. Using an independent-measures design:

STEP 1: State the hypotheses. The null hypothesis states that there is no difference between the two sets of scores. In symbols:

$H_0:{\mu }_1={\mu }_2$

The alternative hypothesis states that there is a significant difference between the two sets of scores. In symbols:

$H_a:{\mu }_1\ne {\mu }_2$

STEP 2: Locate the critical region. Degree of freedom is:

$\begin{array}{c}df=n_1+n_2-2\\ =8+8-2\\ =14\end{array}$ From the t distribution table, for a two-tailed test with $\alpha =0.05$ for $df=14$, the critical value of t values for the critical region are $t=\pm 2.145$.

STEP 3: Compute the test statistic. The data are as follows:

Participant	Above Average Sleep $\left(X_1\right)$	Below Average Sleep $\left(X_2\right)$	${\left(X_1-M_1\right)}^2$	${\left(X_2-M_2\right)}^2$
A	10	13	9	16
B	8	6	1	9
C	5	9	4	0
D	5	6	4	9
E	4	6	9	9
F	10	14	9	25
G	11	13	16	16
H	3	5	16	16
	${\displaystyle \sum X_1}=56$	${\displaystyle \sum X_2}=72$	$S{S}_1=\text{68}$	$S{S}_2=100$

The mean of the above average sleep group is:

$\begin{array}{c}{M}_1=\frac{{\displaystyle \sum X_1}}{n_1}\\ =\frac{56}{8}\\ =7\end{array}$

The mean of the above below sleep group is:

$\begin{array}{c}{M}_2=\frac{{\displaystyle \sum X_2}}{n_2}\\ =\frac{72}{8}\\ =9\end{array}$

The pooled variance is:

$\begin{array}{c}{S}_P^2=\frac{S{S}_1+S{S}_2}{n_1+n_2-2}\\ =\frac{68+100}{8+8-2}\\ =12\end{array}$

The standard error for the mean difference is:

$\begin{array}{c}SE=\sqrt{\frac{S_P^2}{n_1}+\frac{S_P^2}{n_2}}\\ =\sqrt{\frac{12}{8}+\frac{12}{8}}\\ =\sqrt{3}\\ =1.732\end{array}$

The test statistic is:

$\begin{array}{c}t=\frac{M_1-M_2}{SE}\\ =\frac{7-9}{1.732}\\ =-1.15\end{array}$

STEP 4: Make a decision about the null hypothesis. Since test statistic does not fall outside the critical region, fail to reject the null hypothesis. We cannot conclude that there is a significant difference between the two sets of scores by using an independent-measures design.

2. Using repeated measures design:

STEP 1: State the hypotheses. The null hypothesis states that there is no difference between the two sets of scores. In symbols:

$H_0:{\mu }_D=0$

The alternative hypothesis states that there is a significant difference between the two sets of scores. In symbols:

$H_a:{\mu }_D\ne 0$

STEP 2: Locate the critical region. Degree of freedom is:

$\begin{array}{c}df=n-1\\ =8-1\\ =7\end{array}$ From the t distribution table, for a two-tailed test with $\alpha =0.05$ for $df=7$, the critical value of t values for the critical region are $t=\pm 2.365$.

STEP 3: Compute the test statistic. The data are as follows:

Participant	Above Average Sleep $\left(X_1\right)$	Below Average Sleep $\left(X_2\right)$	Difference $D=X_2-X_1$	${\left(D-M_D\right)}^2$
A	10	13	3	1
B	8	6	-2	16
C	5	9	4	4
D	5	6	1	1
E	4	6	2	0
F	10	14	4	4
G	11	13	2	0
H	3	5	2	0
			${\displaystyle \sum D}=\text{16}$	$SS=26$

$\begin{array}{c}{M}_D=\frac{{\displaystyle \sum D}}{n}\\ =\frac{16}{8}\\ =2\end{array}$

The standard deviation of the difference between the two sets of scores is:

$\begin{array}{c}{S}_D=\sqrt{\frac{SS}{n-1}}\\ =\sqrt{\frac{26}{8-1}}\\ =\sqrt{3.7143}\\ =1.927\end{array}$

The test statistic is:

$\begin{array}{c}t=\frac{M_D-{\mu }_D}{S_D/\sqrt{n}}\\ =\frac{2-0}{1.927/\sqrt{8}}\\ =2.94\end{array}$

STEP 4: Make a decision about the null hypothesis. Since test statistic falls outside the critical region, reject the null hypothesis. We can conclude that there is a significant difference between the two sets of scores.

Conclusion:

1. We cannot conclude that there is a significant difference between the two sets of scores by using an independent-measures design.
2. We can conclude that there is a significant difference between the two sets of scores by using repeated measures design.