Question

A statewide census examined the number of beds in households and reported a mean (μ) of 2.25 beds and standard deviation (σ) of 1.9 beds per household. But, since I live in a neighborhood with larger families, I have a hunch that the average number of beds in households will be higher in my neighborhood. To test this idea, I randomly picked 25 families in my neighborhood and surveyed them on the number of beds in their home. I would like to perform a Z test to see if the average number of beds in households in my neighborhood is significantly higher than the statewide average. The significance level for my Z test was set at α = .10.

Household #	# of beds
1	3
2	2
3	3
4	2
5	1
6	3
7	2
8	3
9	2
10	4
11	2
12	4
13	3
14	4
15	1
16	2
17	3
18	6
19	3
20	2
21	3
22	4
23	3
24	2
25	4

a) What is the dependent variable in this study? b) What should be my null and alternative hypotheses? State each hypothesis using both words and statistical notation. Hint: I am interested in the idea of my neighbors having more beds per household than the state average, so the hypotheses would be directional. c) Calculate the sample mean. d) Calculate standard error (SE, which is the standard deviation of the sampling distribution) e) Calculate the Z statistic (which indicates where our sample mean is located on the sampling distribution) f) Specify whether the hypothesis test should be a two-tailed or a one-tailed test, and explain the rationale for the choice. g) Determine the critical value for Z h) Compare obtained Z and critical Z and then make a decision about the result of the hypothesis test: Explicitly state “reject” or “fail to reject” the null hypothesis i) Write a 1-2 sentence conclusion interpreting the results (you can simply restate the accepted hypothesis or explain it in another way) j) Calculate the raw and standardized effect sizes k) If the test was done with α level of .05, using the same directional hypotheses, what would be the critical Z value from the Z table? What would be the result of the hypothesis test (in terms of rejecting or failing to reject the null hypothesis)? l) Compare the hypothesis tests result when α = .05 and when α = .10. Were the results the same? Why or why not?

Answer 1

(a)

The dependent variable in this study : the number of beds in households

(b)

H0: Null Hypothesis: = 2.25

HA: Alternative Hypothesis: > 2.25 (Claim)

(c)

Sample mean = = 71/25 = 2.84

(d)

SE = /

= 1.9/

= 0.38

(e)

Z = (2.84 - 2.25)/0.38

= 1.5526

(f)

The hypothesis test should be a one-tailed test, because the Alternative Hypothesis contains > symbol instead of symbol.

(g)

For = 0.10, from Table, critical value of Z = 1.28

(h)

Since calculated value of Z = 1.5526 is greater than critical value of Z = 1.28, the difference is significant. Reject Null Hypothesis.

(i) The data support the claim that the number of beds in households in my neighborhood is significantly higher than the statewide average.

(j)

Raw Effect size = 2.84 - 2.25 = 0.59

Standardized Effect Size = 0.59/1.9 = 0.3105

(k)

For = 0.05, from Table, critical value of Z = 1.645

Since calculated value of Z = 1.5526 is less than critical value of Z = 1.645, the difference is not significant. Fail to reject Null Hypothesis.

(l)

Results are not same because as the confidence level increases from = 0.05 to = 0.10, the width of the confidence interval increases.