. Do students at various universities differ in how sociable they are? Twenty-five
students were randomly selected from each of three universities in a region and
were asked to report on the amount of time they spent socializing each day with
other students. The result for University X was a mean of 5 hours and an estimated
population variance of 2 hours; for University Y, ; and for University
Z, . What should you conclude? Use the .05 level.
(a) Use the steps of hypothesis testing, (b) figure the effect size for the study;
and (c) explain your answers to parts (a) and (b) to someone who has never had
a course in statistics.
Here are a few hints:
Check out a one-way ANOVA for this kind of problem. Do the calculations to find the F-ratio to compare to the critical or cutoff value from the F-distribution table (used for ANOVA tests). Find the critical or cutoff value at .05 level of significance using the table to reject the null hypothesis (which would state that all population means are equal). If the null is rejected in favor of the alternate hypothesis (which would state that all population means are not equal), then you can conclude a difference.
To answer this question, we need to follow the steps of hypothesis testing. Here's how you can do it:
Step 1: State the Null Hypothesis (H0) and Alternative Hypothesis (Ha):
- Null Hypothesis (H0): There is no difference in the sociability of students among the three universities.
- Alternative Hypothesis (Ha): There is a difference in the sociability of students among the three universities.
Step 2: Choose the significance level (α):
For this question, the significance level is given as 0.05.
Step 3: Determine the test statistic:
Since we are comparing means from three different samples, we can use an analysis of variance (ANOVA) test.
Step 4: Calculate the test statistic and p-value:
For ANOVA, we calculate the F-statistic and its associated p-value. The formula for the F-statistic is:
F = (MSB / MSE)
where MSB is the "between-groups mean square" and MSE is the "within-groups mean square". The p-value is then calculated using the F-distribution.
Step 5: Make a decision:
We compare the obtained p-value with the significance level (α) to make a decision. If the p-value is less than α (0.05), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Now let's find the effect size (Cohen's d) for the study.
To calculate Cohen's d, we use the formula:
d = (M1 - M2) / SDpooled
where M1 - M2 is the difference in means between two groups and SDpooled is the pooled standard deviation.
Based on the information given, we don't have the necessary data to calculate Cohen's d, as we are only provided with the means and estimated population variances. To calculate Cohen's d, we would need the standard deviations of the samples, which are not given.
In conclusion, by following the steps of hypothesis testing, we would conduct an ANOVA test to analyze if there is a significant difference in sociability among the three universities. Unfortunately, we cannot calculate the effect size (Cohen's d) in this case due to insufficient data.