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.
Data for the other two universities are not given. Test for the difference between means.
Z = (mean1 - mean2)/(standard error of difference between means)
SE diff. = sq.rt. (SE1^2 + SE2^2)
SE = standard deviation/sq.rt. (n-1)
SD = sq. rt. of variance
I'll leave the explanation up to you. From my experience in teaching, once you can explain the process to someone else, you will really understand it yourself. Good luck.
To analyze the sociability differences among students at different universities, we will proceed with the following steps of hypothesis testing:
Step 1: State the null and alternative hypotheses:
The null hypothesis (H0): There is no difference in the mean amount of time students spend socializing among the three universities.
The alternative hypothesis (Ha): There is a difference in the mean amount of time students spend socializing among the three universities.
Step 2: Determine the significance level:
The significance level (denoted as α) is given as .05, which means we are willing to accept a 5% chance of making a Type I error (rejecting the null hypothesis when it is true).
Step 3: Conduct the statistical test:
We will use an analysis of variance (ANOVA) test to compare the means of the three universities.
Step 4: Calculate the F-statistic and p-value:
Using the values given in the question, we calculate the F-statistic to compare the variances among the universities. The formula for F-statistic is:
F = (SSB / dfB) / (SSW / dfW)
Where SSB is the sum of squares between groups, dfB is the degrees of freedom for between groups, SSW is the sum of squares within groups, and dfW is the degrees of freedom for within groups.
Step 5: Determine the critical value or p-value:
To determine if the result is statistically significant, we compare the obtained p-value to the significance level (α). If the p-value is less than α (0.05), we reject the null hypothesis.
In terms of effect size, we can calculate the eta-squared (η²) or Cramer's V coefficient. Eta-squared is the proportion of variance explained by the group differences, while Cramer's V is a measure of association.
To explain these steps in simpler terms to someone without a statistics background:
(a) Hypothesis testing involves evaluating two competing hypotheses: the null hypothesis (assuming no difference) and the alternative hypothesis (assuming a difference). By analyzing the data, we determine whether the evidence supports the null hypothesis or suggests there is a difference between the universities in terms of sociability.
(b) Effect size measures the magnitude of the difference between the universities. It helps us understand the practical significance of the results, beyond statistical significance. Calculating effect size allows us to gauge the impact of the university variable on the sociability of students.
In summary, to answer the question of whether students at various universities differ in sociability, we need to conduct an ANOVA test and compare the p-value obtained to the significance level. Additionally, calculating the effect size will help us understand the practical significance of the results.