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, M=4, SƒU=1.5; and for University Z, M=6, s2ƒU=2.5. 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.
Indicate your subject in the "School Subject" box, so those with expertise in the area will respond to the question.
We do not do your work for you. Once you have answered your questions, we will be happy to give you feedback on your work. Although it might require more time and effort, you will learn more if you do your own work. Isn't that why you go to school?
To start you out:
Ho: mean X = mean Y = mean Z
(a) Steps of hypothesis testing:
Step 1: State the null hypothesis (H0) and alternative hypothesis (Ha).
- Null hypothesis (H0): There is no difference in the average amount of time spent socializing among the three universities.
- Alternative hypothesis (Ha): There is a difference in the average amount of time spent socializing among the three universities.
Step 2: Set the level of significance (α).
- In this case, the level of significance is given as .05.
Step 3: Collect sample data and calculate the test statistic.
- Given the sample means and estimated population variances, we can calculate the test statistic using the formula for independent samples t-test.
Step 4: Determine the critical value or p-value.
- With the given level of significance (α) and the degrees of freedom, we can find the critical value or p-value using the t-distribution table or statistical software.
Step 5: Compare the test statistic with the critical value or p-value.
- If the test statistic falls in the critical region (reject H0), we conclude that there is a significant difference in the average amount of time spent socializing among the three universities. If the test statistic does not fall in the critical region (fail to reject H0), we conclude that there is not enough evidence to support a difference.
(b) Effect Size:
- The effect size measures the magnitude of the difference between the groups. In this case, we can use Cohen's d as the effect size measure.
- Cohen's d is calculated using the formula: d = (M1 - M2) / SDpooled, where M1 and M2 are the means of the two groups and SDpooled is the pooled standard deviation.
(c) Explanation to someone with no statistics background:
(a) Hypothesis testing involves setting up competing hypotheses, collecting data, and evaluating the evidence to determine if there is a significant difference. In this case, the null hypothesis assumes there is no difference in the average time spent socializing among the three universities, while the alternative hypothesis suggests there is a difference. Using the collected data, we calculate a test statistic and compare it to a critical value or p-value to make a conclusion.
(b) Effect size is a measure of the magnitude of the difference between groups. In this study, we can use Cohen's d as the effect size measure. It tells us how much the means of the groups differ relative to their standard deviations. By calculating Cohen's d, we can quantify the size of the difference in the average time spent socializing among the universities.
Overall, based on the hypothesis testing results and the effect size measure, we can make conclusions about the differences in sociability among students at the three universities in the region.
(a) Steps of hypothesis testing:
1. State the null hypothesis (H0) and alternative hypothesis (Ha):
- Null hypothesis: There is no difference in sociability between students at the three universities.
- Alternative hypothesis: There is a difference in sociability between students at the three universities.
2. Set the significance level (α): In this case, α is given as .05.
3. Collect and analyze the data:
- Compute the means and variances for each university.
- Compare the means and variances to determine if there is a significant difference.
4. Perform a statistical test:
- One-way ANOVA (Analysis of Variance) can be used to test for significant differences between the means of three or more groups.
5. Calculate the test statistic and p-value:
- The test statistic for ANOVA is called the F-statistic.
- The p-value is the probability of obtaining results as extreme as the observed data, assuming the null hypothesis is true.
6. Make a decision:
- If the p-value is less than the significance level (α), reject the null hypothesis.
- If the p-value is greater than the significance level (α), fail to reject the null hypothesis.
(b) Effect size for the study:
To calculate the effect size, one commonly used measure is eta-squared (η²). It represents the proportion of variance in the dependent variable (amount of time spent socializing) that can be explained by the independent variable (university).
η² = SSB / SST
Where SSB represents the sum of squares between groups and SST represents the total sum of squares.
(c) Explanation:
In hypothesis testing, we start by stating the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis assumes there is no difference in sociability between students at the three universities, while the alternative hypothesis suggests that there is a difference. We collect and analyze data to determine if there is enough evidence to reject the null hypothesis.
To perform the hypothesis test, we choose a significance level (α), which determines how much evidence is required to reject the null hypothesis. In this case, α is set to .05, meaning there is a 5% chance of wrongly rejecting the null hypothesis.
We perform a one-way ANOVA test to compare the means of the three groups (universities). ANOVA calculates a test statistic called the F-statistic, and we look at the associated p-value. The p-value indicates the probability of obtaining results as extreme as the observed data, assuming the null hypothesis is true. If the p-value is smaller than α, we reject the null hypothesis and conclude that there is a significant difference in sociability between the universities.
In addition to testing for statistical significance, it is also important to consider the effect size. The effect size tells us how much of the variance in the dependent variable can be explained by the independent variable. In this case, we can calculate eta-squared (η²) as a measure of effect size.
By explaining these steps and concepts to someone without a statistics background, we can help them understand the process of hypothesis testing and how to draw conclusions based on the data.