4. A college student is interested in whether there is a difference between male and female students in the amount of time spent studying each week. The student gathers information from a random sample of male and female students on campus. Amount of time spent studying is normally distributed. The data follow:
Males Females
27 25
25 29
19 18
10 23
16 20
22 15
a. What statistical test should be used to analyze these data?
b. Identify H0 and Ha for this study.
c. Conduct the appropriate analysis.
d. Should H0 be rejected? What should the researcher conclude?
e. If significant, compute the effect size and interpret this.
f. If significant, draw a graph representing the data.
a. To analyze these data, a two-sample t-test should be used. This test compares the means of two independent groups to determine if there is a significant difference between them.
b. H0 (null hypothesis): There is no difference between male and female students in the amount of time spent studying each week.
Ha (alternative hypothesis): There is a difference between male and female students in the amount of time spent studying each week.
c. To conduct the analysis, follow these steps:
1. Calculate the means and standard deviations of the two groups (males and females).
2. Check if the assumptions for a t-test are met, including normality and equal variances between groups.
3. Determine the significance level (usually α = 0.05) for hypothesis testing.
4. Use the formula for a two-sample t-test to calculate the t-statistic.
5. Compare the calculated t-statistic with the critical t-value from the t-distribution table.
6. Calculate the p-value associated with the t-statistic.
7. Compare the p-value with the significance level. If the p-value is less than the significance level, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
d. Based on the analysis, if H0 is rejected, it means there is evidence to support the alternative hypothesis. In this case, it would indicate that there is a significant difference between male and female students in the amount of time spent studying each week. If H0 is not rejected, it means there is not enough evidence to support the alternative hypothesis, suggesting that there is no significant difference between the two groups.
e. To compute the effect size, various measures can be used, such as Cohen's d. It quantifies the difference between the means of the two groups, taking into account the variability within each group. The formula for Cohen's d is: d = (mean1 - mean2) / pooled standard deviation. The effect size can be interpreted as small (d = 0.2), medium (d = 0.5), or large (d = 0.8), indicating the magnitude of the difference between the groups.
f. To draw a graph representing the data, you can create a side-by-side boxplot or a bar chart. In the boxplot, the two groups (males and females) are represented on the x-axis, and the amount of time spent studying is represented on the y-axis. The boxes show the interquartile range, with the median line inside. The whiskers extend to the minimum and maximum values, excluding outliers. The bar chart would have the two groups on the x-axis and the mean or median values on the y-axis, with error bars indicating the variability or confidence interval. This visual representation helps to compare the distributions and see if there are any noticeable differences.