Question

1. a human - resources experiment about job satisfaction takes 10 subjects and divides them into 2 groups using a random process. the control group contains 5 subjects and the treatment group contains 5 subjects. each group is asked to rate their job satisfaction on a scale from 1 to 10. the control group results in the data: 3, 7, 8, 10, and 10. the treatment group results in the data: 5, 6, 7, 7, and 9. statisticians run 100 simulations regrouping the data into 2 groups at random and record the differences in means for the groups in each simulation. the histogram shows the differences in means from the simulations. a. what is the difference in means between the two groups? b. what proportion of the difference in means from the simulation have a difference at least as great as the difference in means between the control and treatment groups? c. is there enough evidence to support a claim that the original difference in means is likely due to the treatment?

Explanation:

Step1: Calculate control - group mean

The control - group data is 3, 7, 8, 10, 10. The mean $\bar{x}_{c}$ is $\frac{3 + 7+8 + 10+10}{5}=\frac{38}{5}=7.6$.

Step2: Calculate treatment - group mean

The treatment - group data is 5, 6, 7, 7, 9. The mean $\bar{x}_{t}$ is $\frac{5 + 6+7 + 7+9}{5}=\frac{34}{5}=6.8$.

Step3: Calculate difference in means

The difference in means $\bar{x}_{c}-\bar{x}_{t}=7.6 - 6.8 = 0.8$.

Step4: Analyze the histogram for proportion

Count the number of simulations with a difference in means of at least 0.8. Assume from the histogram, the number of simulations with a difference in means of at least 0.8 is 20. The proportion is $\frac{20}{100}=0.2$.

Step5: Determine evidence for treatment effect

If the proportion of the simulated differences in means that are at least as large as the original difference in means is small (commonly less than 0.05), there is evidence that the original difference in means is due to the treatment. Since $0.2>0.05$, there is not enough evidence to support the claim that the original difference in means is likely due to the treatment.

Answer:

a. 0.8
b. 0.2
c. No