Do students at various colleges differ in how sociable they are? Twenty-five
students were randomly selected from each of three colleges in a particular region
and were asked to report on the amount of time they spent socializing each
day with other students. The results for College X was a mean of 5 hours and
an estimated population variance of 2 hours; for College Y, M = 4, S2 = 1.5;
and for College Z, M = 6, S2 = 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 (a) and (b) to someone who has never
had a course in statistics
Statistics help needed.
"college" is not a subject
Ho: μX = μY = μZ
Ho: μX ≠ μY ≠ μZ
Use three two-tailed tests (XY, XZ and YZ) for difference between means or an ANOVA. (Remember with the three tests, the probability of getting any one of them significant is now .15 [.05 + .05 + .05].)
Your explanation will probably deal with probability and chance expectations.
For you to explain it to someone without a statistics background will mean that you really understand what is happening. In my teaching, I found that, to teach a concept, I really had to understand it myself. Now you need to understand the concepts yourself. It takes a good amount of effort. Are you willing to expend the effort?
I hope this helps.
To analyze the sociability differences among colleges, we can follow the steps of hypothesis testing:
a) Steps of Hypothesis Testing:
Step 1: State the hypotheses:
- Null Hypothesis (H0): There is no significant difference in sociability among the three colleges.
- Alternative Hypothesis (Ha): There is a significant difference in sociability among the three colleges.
Step 2: Set the criteria for a decision:
Since the significance level is given as .05, we will use this value as the threshold to determine the decision.
Step 3: Collect and analyze sample data:
The sample data provided includes the means (M) and estimated population variances (S2) for each college.
Step 4: Compute the test statistic:
For comparing means from three independent samples, the appropriate test is the analysis of variance (ANOVA). ANOVA will determine if the differences observed among the means are statistically significant or if they could have occurred by chance.
Step 5: Make a decision:
Using the obtained test statistic and comparing it to the critical value (using the significance level of .05), we can either reject or fail to reject the null hypothesis. If the test statistic is greater than the critical value, we reject the null hypothesis and conclude that there is a significant difference in sociability among the three colleges. If the test statistic does not exceed the critical value, we fail to reject the null hypothesis, indicating that there is no significant difference in sociability among the colleges.
b) Effect Size Calculation:
Effect size measures the magnitude of the observed differences. In our case, we can use eta-squared (η^2), which represents the proportion of total variation in the sociability scores attributed to differences between colleges.
To calculate eta-squared, we use the formula: η^2 = SSbetween / SStotal, where SSbetween represents the sum of squares between samples and SStotal represents the total sum of squares.
c) Explaining the answers (a) and (b):
To someone without a statistics background, you can explain that hypothesis testing is a structured way to evaluate if the observed differences in sociability among the colleges are likely due to chance or if they are statistically significant.
By following the steps of hypothesis testing, we can make a decision based on the collected data and determine if there is evidence to support the alternative hypothesis (that there is a significant difference in sociability among the colleges).
In addition, effect size (η^2) helps us understand the practical significance of the observed differences. It quantifies the proportion of variation in sociability that can be attributed to the differences between colleges.
Overall, by conducting hypothesis testing and examining the effect size, we can draw conclusions about the sociability differences among the three colleges and better understand the practical importance of these differences.