Question

1. The amount of income spent on housing is an important component of the cost of living. The total costs of housing for homeowners might include mortgage payments, property taxes, and utility costs (water, heat, electricity). An economist selected a sample of 20 homeowners in New England and then calculated these total housing costs as a percent of monthly income, five years ago and now. The information is reported below. At the .05 significance level, is it reasonable to conclude the percent is less now than five years ago?

Really important: Use Excel as described in “How to perform two-sample hypothesis tests” in the content area to do this problem. You can copy the table below into an Excel file—no need to type it. Use the same Excel file that you used for the previous problem, but put each problem on a separate worksheet (you move to another worksheet by clicking the tabs at the bottom). Please name each worksheet by right-clicking on the tab, selecting Rename, and typing the problem number. This is chapter 11, problem 4, so call it “11-4.”

Homeowner	% Five Years Ago	% Now
1	17	10
2	20	39
3	29	37
4	43	27
5	36	12
6	43	41
7	45	24
8	19	26
9	49	28
10	49	26
11	35	32
12	16	32
13	23	21
14	33	12
15	44	40
16	44	42
17	28	22
18	29	19
19	39	35
20	22	12

a. State the null and alternate hypotheses.

b. Select alpha.

c. Select the test statistic.

d. Formulate the decision rule.

e. What is the value of the test statistic?

f. Determine and interpret the effect size and p-value.

g. Draw conclusions based on statistical and practical significance.

Answer 1

(a)

The null hypothesis H0: The total housing costs as a percent of income now is the same as the total housing costs as a percent of income five years ago.

The alternate hypothesis HA: The total housing costs as a percent of income now is less than the total housing costs as a percent of income five years ago.

(b)

Alpha level = significance level = 0.05

(c)

The statistic used for the analysis of this data is the F-statistic.

(d)

Decision rule: If the F-statistic for this data is more than the critical F-statistic for the given significance level given the degrees of freedom, then we reject the null hypothesis. But, if the F-statistic is less than the critical F-statistic, then the null hypothesis cannot be rejected.

(e)

The F-statistic obtained for this data is 3.441. The critical F-statistic F(dfN=1, dfD=38) for a significance level of 0.05 is 4.098.

(f)

P-value = 0.0714. As this is greater than the significance level, we cannot reject the null hypothesis.

(g)

As the F-statistic is less than the critical F-statistic for the given significance level given the degrees of freedom, then we cannot reject the null hypothesis. There is no evidence to show that the total housing costs as a percent of income now is less than that five years ago.

Practically, this means that even if the income levels have increased in five years, we do not have the evidence to show that the housing has become more affordable.