Chapter 11, Problem 2RE

Do you view a chocolate bar as delicious or as fattening? Your attitude may depend on your gender. In a study of American college students, Rozin, Bauer, and Catanese (2003) examined the importance of food as a source of pleasure versus concerns about food associated with weight gain and health. The following results are similar to those obtained in the study. The scores are a measure of concern about the negative aspects of eating.

a. Based on these results, is there a significant difference between the attitudes for males and for females? Use a two-tailed test with $\alpha$ = .05.

b. Compute r², the percentage of variance accounted for by the gender difference, to measure effect size for this study.

c. Write a sentence demonstrating how the result of the hypothesis test and the measure of effect size would appear in a research report.

To determine

To find: Whetherthere is a significant difference between the attitudes of males and females using two-tailed testat $\alpha =0.05$.

Answer to Problem 2RE

There is a significant difference between the attitudes of males and females

Explanation of Solution

Given info:

Two samples of males and females each with size 9 and 15 have means 33 and 42 respectively. Similarly, both of them have sum of squared difference (SS) equal to 740 and 1240 respectively.

Calculations:

Step 1:

Stating the testing hypotheses:

Null Hypothesis

$H_0:{\mu }_1-{\mu }_2=0$

Alternate Hypothesis

$H_1:{\mu }_1-{\mu }_2\ne 0$

Step 2: For the given sample, the t-statistics will have degrees of freedom equals:

$\begin{array}{c}df=n_1+n_2-2\\ =9+15-2\\ =22\end{array}$

For a two-tailed test with $\alpha =0.05$ and $df=22$, the critical value (CV)  is obtained from the t-table as $t=\pm 2.074$

Step 3: t-statistics is calculated using formula:

$t=\frac{(M_1-M_2)-({\mu }_1-{\mu }_2)}{s_{M_1-M_2}}$

But before that calculate pooled variance, $s_p^2$, using formula:

$s_p^2=\frac{S{S}_1+S{S}_2}{d{f}_1+d{f}_2}$

Substitute 740for $S{S}_1$, 1240 for $S{S}_2$ and 8 for $d{f}_1$ and 14 $d{f}_2$ in the above formula

$\begin{array}{c}{s}_p^2=\frac{740+1240}{8+14}\\ =90\end{array}$

Now, the formula for estimated standard error of the difference is

$s_{M_1-M_2}=\sqrt{\frac{s_p{}^2}{n_1}+\frac{s_p{}^2}{n_2}}$

Substitute for $s_p^2=90$,9 for $n_1$ and 15 for $n_2$ in the formula:

$\begin{array}{c}{s}_{M_1-M_2}=\sqrt{\frac{90}{9}+\frac{90}{15}}\\ =\sqrt{16}\\ =4\end{array}$

Finally, substitute 4for $s_{M_1-M_2}$, 33 for $M_1$ and 42 for $M_2$ in the t-statistics formula

$\begin{array}{c}t=\frac{33-42}{4}\\ =-2.25\end{array}$

Step 4: Decision rule: Reject when $t-statistics<-2.074$ or when $t-statistics>2.074$

Since $t-statistics(=-2.25)<critical\_value(=-2.074)$.

Step 5: Conclusion: The null hypothesis is rejected. Based on the results of hypothesis test, wehave sufficient evidence to reject the null hypothesis at $\alpha =0.05$

Hence,reject the null hypothesis and conclude that there is a significant difference between the attitudes of males and females

To determine

To Find: The value of $r^2$ to measure effect size for the mean difference 

Answer to Problem 2RE

Value of $r^2$ is 0.187

Explanation of Solution

Calculations:

The formula for $r^2$ is :

$r^2=\frac{t^2}{t^2+df}$

Substitute $t=-2.25$ and $df=22$ in the formula

$\begin{array}{c}{r}^2=\frac{{2.25}^2}{{2.25}^2+22}\\ =0.187\end{array}$

Hence, $r^2$is0.187

To determine

To describe: The outcome of the hypothesis test and the measure of the effect size.

Answer to Problem 2RE

Rejectthe null hypothesis and conclude that there is a significant difference between the attitudes of males and females. The measure of the effect size is–2.25 and 0.187 respectively as calculate in part a and part b.

Explanation of Solution

Justification:

As per the rejection rule when $t-statistics<-2.074$ or when $t-statistics>2.074$ then reject the null hypothesis. Since, $t-statistics(=-2.25)<critical\_value(=-2.074)$ so, reject the null hypothesis. In other words, based on the results of hypothesis test, we have sufficient evidence to reject the null hypothesis at $\alpha =0.05$

Hence, reject the null hypothesis and conclude that there is a significant difference between the attitudes of males and females

Also effect size from part a. and b. can be measured in terms of t statistic and $r^2$. The t statistic and $r^2$ obtained are –2.25 and 0.187 respectively.

Conclusion:

Reject the null hypothesis and conclude that there is a significant difference between the attitudes of males and females. Calculated t statistic and $r^2$ are $-2.25$ and 0.187 respectively as calculate in part a. and part b.