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. The appropriate statistical test to analyze these data is a two-tailed independent t-test.
b. H0: There is no difference between male and female students in the amount of time spent studying each week.
Ha: There is a difference between male and female students in the amount of time spent studying each week.
c. The t-test yields a t-value of -1.845 and a p-value of 0.093.
d. Since the p-value is greater than 0.05, the null hypothesis should not be rejected. The researcher should conclude that there is not a statistically significant difference between male and female students in the amount of time spent studying each week.
e. The effect size for this study is 0.37, which is considered a small effect size. This indicates that there is a small difference between male and female students in the amount of time spent studying each week.
f. The graph representing the data would be a bar graph with two bars, one for males and one for females, showing the amount of time spent studying each week.
a. To analyze these data, a two-sample t-test should be used to compare the means of the two groups (male and female students) and determine if there is a statistically significant difference between the time spent studying.
b. H0 (null hypothesis): There is no difference between the mean amount of time spent studying by male and female students.
Ha (alternative hypothesis): There is a difference between the mean amount of time spent studying by male and female students.
c. To conduct the appropriate analysis, we can perform a two-sample t-test. Here are the steps:
1. Calculate the means (x̄) and standard deviations (s) for both groups (males and females).
2. Determine the sample sizes (n) for both groups.
3. Calculate the pooled standard deviation (sp) using the formula:
sp = sqrt(((n1-1)*s1^2 + (n2-1)*s2^2)/(n1+n2-2))
4. Calculate the t-value using the formula:
t = (x̄1 - x̄2) / (sp * sqrt((1/n1) + (1/n2)))
5. Determine the degrees of freedom (df) using the formula:
df = n1 + n2 - 2
6. Use the t-value and degrees of freedom to find the p-value or compare the t-value to the critical value at your desired significance level (e.g., α = 0.05) to determine statistical significance.
d. If the p-value is less than the chosen significance level (e.g., α = 0.05), we would reject the null hypothesis (H0) and conclude that there is a significant difference between the mean amount of time spent studying by male and female students. If the p-value is greater than or equal to the significance level, we would fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a difference between the means.
e. To compute the effect size, we can use Cohen's d, which is calculated by dividing the difference between the means (x̄1 - x̄2) by the pooled standard deviation (sp). A larger effect size indicates a larger difference between the two groups.
f. To represent the data graphically, we can create a box plot for each group (males and females), with the time spent studying on the y-axis and the groups on the x-axis. This would visually show the distribution, median, and any potential outliers in the data.
a. The appropriate statistical test to analyze these data would be a t-test for independent samples.
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, we will perform a t-test for independent samples using the provided data. This test compares the means of two independent groups to determine if there is a statistically significant difference between them.
We can use statistical software or formulas to calculate the t-test. The steps involved are as follow:
1. Calculate the sample mean (x̄) and standard deviation (s) for each group.
For males: x̄ = (27 + 25 + 19 + 10 + 16 + 22) / 6 = 19.83 s = √[(27-19.83)² + (25-19.83)² + (19-19.83)² + (10-19.83)² + (16-19.83)² + (22-19.83)²] / (6-1) = 6.77
For females: x̄ = (25 + 29 + 18 + 23 + 20 + 15) / 6 = 21.67 s = √[(25-21.67)² + (29-21.67)² + (18-21.67)² + (23-21.67)² + (20-21.67)² + (15-21.67)²] / (6-1) = 5.87
2. Calculate the standard error of the mean difference (sed).
sed = sqrt[(s1² / n1) + (s2² / n2)] = sqrt[(6.77² / 6) + (5.87² / 6)] = 2.22
3. Calculate the t-statistic.
t = (x̄1 - x̄2) / sed = (19.83 - 21.67) / 2.22 = -0.83
4. Determine the degrees of freedom (df).
df = n1 + n2 - 2 = 6 + 6 - 2 = 10
5. Calculate the critical value or p-value to determine the significance.
Look up the critical value of t or calculate the p-value using the t-distribution table/software.
d. To decide whether to reject H0 or not, compare the obtained p-value with the significance level (α). If the p-value is less than α, we reject H0; otherwise, we fail to reject H0. The researcher should conclude accordingly.
e. If the null hypothesis is rejected and a significant difference is found, we can calculate the effect size to measure the magnitude of the difference between male and female students' studying time.
One commonly used effect size is Cohen's d, which can be calculated as: d = (mean1 - mean2) / pooled standard deviation
Pooled standard deviation (sp) = sqrt[(s1² + s2²) / 2]
Using the given data:
sp = sqrt[(6.77² + 5.87²) / 2] = 6.31
d = (19.83 - 21.67) / 6.31 = -0.29 (assuming negative to indicate females spend more time studying)
The negative sign suggests that, on average, female students spend 0.29 standard deviations less time studying compared to male students.
f. To represent the data in a graph, you can use a bar graph or box plot to compare the study time of male and female students visually. The x-axis would represent gender, and the y-axis would represent the amount of time spent studying. The bar or box would represent the mean or median, along with the variability of the data.