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, s^2=1.5 ; and for University
Z,M = 6 , s^2 = 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.
(a) Steps of hypothesis testing:
Step 1: State the hypotheses:
- Null Hypothesis (H0): There is no difference in sociability between the three universities.
- Alternative hypothesis (Ha): There is a difference in sociability between at least two of the universities.
Step 2: Set the significance level (α):
We are given the level of significance as 0.05.
Step 3: Compute the test statistic:
We will perform an Analysis of Variance (ANOVA) test to compare the means of the three universities. The formula for the test statistic in ANOVA is F = (Between-group variability) / (Within-group variability).
Step 4: Determine the critical region:
To find the critical region, we compare the calculated F-value to the critical F-value from the F-distribution table.
Step 5: Make a decision:
If the calculated F-value falls into the critical region, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 6: Interpret the results:
Based on the decision made in Step 5, we conclude whether there is a significant difference in sociability between the universities or not.
(b) Effect size for the study:
Effect size measures the strength or magnitude of the relationship between variables. For ANOVA, the effect size is typically measured using eta-squared (η²).
η² = (Between-group variability) / (Total variability)
(c) Explanation:
(a) Hypothesis testing helps us determine if there is a significant difference in sociability between the three universities. By comparing the means and variances of the sampled students in each university, we can evaluate if any differences observed are statistically significant or simply due to random chance.
(b) Effect size (η²) helps us understand the magnitude of the difference between the universities. A larger effect size indicates a stronger relationship between the universities and their sociability levels.
It should be noted that statistical knowledge is not required to understand the main concepts of hypothesis testing and effect size. Our analysis aims to investigate if there is a meaningful difference in sociability between the universities and to quantify the size of that difference.
To answer this question, we will follow the steps of hypothesis testing. Hypothesis testing allows us to make conclusions about a population based on a sample. In this case, we will compare the means of the three universities to determine if there are significant differences in sociability.
Step 1: State the hypotheses
Our null hypothesis (H0) states that there are no differences in sociability between the universities. The alternative hypothesis (Ha) states that there are differences.
H0: µX = µY = µZ
Ha: At least one µ is different
Step 2: Set the significance level
The significance level (alpha) is the probability of rejecting the null hypothesis when it is true. In this case, we are using α = 0.05.
Step 3: Compute the test statistic
We will use the analysis of variance (ANOVA) test to compare the means. The ANOVA test calculates the F-statistic, which measures the ratio of between-group variability to within-group variability.
Step 4: Determine the critical value
We need to compare the calculated F-statistic to the critical value from the F-distribution table. The critical value depends on the degrees of freedom and the significance level.
Step 5: Make a decision
If the calculated F-statistic is greater than the critical value, we reject the null hypothesis and conclude that there are significant differences in sociability between the universities. Otherwise, we fail to reject the null hypothesis.
Now, let's move on to calculating the effect size (part b). Effect size measures the strength of the relationship between variables.
We will use Cohen's d as the effect size measure. Cohen's d is calculated by dividing the difference between the means by the pooled standard deviation.
Finally, let's explain the answers to someone who has never had a course in statistics (part c):
(a) In hypothesis testing, we compare the means of different groups to see if there is enough evidence to conclude that they are significantly different. We set a significance level (alpha) to decide how strong the evidence needs to be. Based on the data we collected from the three universities, we perform an analysis using the ANOVA test to compare the means. If the calculated test statistic is greater than the critical value, we conclude that there are significant differences in sociability between the universities.
(b) Effect size measures the magnitude or strength of the difference between groups. In this study, we calculate the effect size using Cohen's d, which tells us how much the means differ in standard deviation units. A larger effect size indicates a stronger difference between the means.
In summary, based on the hypothesis testing and the effect size calculation, we can conclude that there are significant differences in sociability between the universities.