Q1. Hypothesis Testing with a Z test (14 points total)
A research institute examined the number of smart phones and tablets in American households and reported a mean (μ) of 5 devices per household and a standard deviation (σ) of 1.5 devices. But I wonder if the statistics in my neighborhood, Chandler, would be different from the national average. To test this idea, I randomly picked 16 friends of mine living in Chandler and asked them how many smart phones and tablets are in their households. The data set is shown below. (The same data are also provided in an Excel file if you choose to use Excel for your calculations.)

I would like to perform a Z test to see if the numbers of mobile devices in Chandler households (represented by the sample of 16 households) are significantly different from the numbers of mobile devices in the national households. The significance level for my Z test was set at α = .05. The hypotheses should be nondirectional because I am not predicting any specific direction of difference, so the test should be two-tailed.

Subject # (Household) # of mobile devices
1 6
2 8
3 6
4 6
5 5
6 7
7 5
8 8
9 7
10 4
11 4
12 6
13 5
14 7
15 6
16 6

a. What is the dependent variable in this study? (1 point)

b. What should be my null and alternative hypotheses? State each hypothesis using both words and statistical symbol notation. (2 points total. 1 for each hypothesis: .5 for written, .5 for notation)
Note: The hypotheses should be non-directional.

c. Calculate the sample mean. (1 point total: .5 if the process is correct but the result was calculated incorrectly)

d. Calculate standard error using the population standard deviation σ given in the research scenario (SE, which is the standard deviation of the sampling distribution) (2 point total: 1 if the process is correct but the result was calculated incorrectly)

e. Calculate the Z statistic (which indicates where our sample mean is located on the sampling distribution) (2 point total: 1 if the process is correct but the result was calculated incorrectly)

f. Determine the critical Z value(s). Explain how you find the answer. (1 point: .5 for the Z value, .5 for the rationale)

g. Compare the Z statistic with the appropriate critical Z value (is the Z statistic more extreme than the critical Z?), and then draw a conclusion about the result of the hypothesis test (do you “reject” or “fail to reject” the null hypothesis?) (2 points: 1 for Z statistic comparison; 1 for hypothesis test decision)

h. Write 1-2 sentences to answer the research question (you can use the wording from the hypotheses or explain it in another way) (1 point)

i. Calculate the standardized effect size (2 point total: 1 if the process is correct but the result was calculated incorrectly)

Q2. Hypothesis Testing with a Z Test (12 points)
According to the CDC report, the mean life expectancy in the US population is currently (µ) 78.6 years with a standard deviation (σ) of 12. A researcher examined the age of death of 400 people recently recorded in several Arizona hospitals and calculated the mean to be 79.7 years old. He runs a two-tailed Z test with α = .05 to see if Arizona has a significantly different life expectancy compared to the US population. Because the researcher is not predicting a direction, the hypotheses should be non-directional and the test should be two-tailed.

a. Calculate standard error (SE, which is the standard deviation of the sampling distribution) (1 point total: .5 if the process is correct but the result was calculated incorrectly)

b. Calculate the Z statistic (which indicates where our sample mean is located on the sampling distribution) (1 point total: .5 if the process is correct but the result was calculated incorrectly)

c. Determine the critical Z value(s). Explain how you find the answer. (1 point: .5 for the Z value, .5 for the rationale)

d. Compare the Z statistic with the appropriate critical Z value (is the Z statistic more extreme than the critical Z?), and then draw a conclusion about the result of the hypothesis test (do you “reject” or “fail to reject” the null hypothesis?) (1 points: .5 for Z statistic comparison; .5 for hypothesis test decision)

e. Calculate the standardized effect size. (1 point: .5 if the process is correct but the result was calculated incorrectly)

f. Write 1-2 sentences to answer the research question (you can use the wording from the hypotheses or explain it in another way) (1 point)

Answer the following questions based on this alternative scenario:
Because a sample of 400 people is small, it may not represent the state of Arizona adequately. So the researcher continues to collect data until the sample becomes 900. The average life expectancy remains 79.5, the same as the previous scenario.

g. What is the standard error now? (.5 point) How is it different from the hypothesis test with the smaller sample in question (a)? (.5 point)

h. What is the Z statistic now? (.5 point) How is it different from the hypothesis test with the smaller sample in question (b)? (.5 point)

i. What is the conclusion of the hypothesis test now? (.5 point) How is it different from the hypothesis test result with the smaller sample in question (d)? (.5 point)

j. What is the standardized effect size now? (.5 point) How is it different from the effect size with the smaller sample in question (e) above? (.5 point)

k. Based on the answers to questions g-j, discuss the effects of simply increasing the sample size while everything else remains the same. (1 point) Provide one reason why it is a good idea to have a larger sample for a study. (1 point)

Q3. Hypothesis testing using a Z test (14 points)
A professor has been teaching introductory statistics for many years and the final exam performance has been very consistent from class to class and the scores have been normally distributed. Overall, the whole data base (i.e. population) of final exam scores has a mean (μ) of 20 points (out of a maximum of 30 points) and a standard deviation (σ) of 6 points. The professor would like to revise the course design to see if student performance on the final exam could be improved.

The new course design was implemented in the most recent academic year. There were 100 students and the average final exam score was 21.5. The professor would like to run a hypothesis test to see if this sample of students in the recent academic year performed significantly better than the past population. In other words, the hypothesis was a comparison between the population with new course design (represented by the sample of 100 students) with the population with the old course design.

The professor is predicting an increase of final exam score with the new design, so the hypotheses should be directional, and the test should be one-tailed. α = .05.

a. Identify the dependent variable for this study (1 point)

b. Hypothesis testing is always about making inferences about the populations. The recent class of 100 is considered a sample. What is the population from which this sample has been drawn? (1 point)

c. State your null hypothesis and alternative hypothesis using both words and symbol notation
Note: The hypotheses should be directional.
(2 points total. Each hypothesis is 1 point, with .5 for the written and .5 for the notation)

d. Calculate standard error (SE, which is the standard deviation of the sampling distribution) (2 points total: 1 if the process is correct but the result was calculated incorrectly)

e. Calculate the Z statistic (which indicates where our sample mean is located on the sampling distribution) (2 points total: 1 if the process is correct but the result was calculated incorrectly)

f. Determine the critical value for Z. Explain how you come up with the answer. (1 point: .5 for the answer and .5 for the rationale)

g. Compare the Z statistic with the critical Z value (that is, “is the Z statistic more extreme than the critical Z?”), and then draw a conclusion about the result of the hypothesis test (do you “reject” or “fail to reject” the null hypothesis?) (2 points: 1 for Z statistic comparison; 1 for hypothesis test decision)

h. Write 1-2 sentences to answer the research question (you can use the wording from the hypotheses or explain it in another way) (1 point)

i. Calculate the standardized effect size for this test. (2 points total: 1 if the process is correct but the result was calculated incorrectly)

Sample Solution