Question

In this problem, assume that the distribution of differences is approximately normal. Note: For degrees of freedom d.f. not in the Student's t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more "conservative" answer.

A survey of houses and traditional hogans was made in a number of different regions of the modern Navajo Indian Reservation. The following table is the result of a random sample of eight regions on the Navajo Reservation.

Area on Navajo Reservation	Number of Inhabited Houses	Number of Inhabited Hogans
Bitter Springs	18	13
Rainbow Lodge	16	12
Kayenta	66	44
Red Mesa	9	32
Black Mesa	11	15
Canyon de Chelly	28	47
Cedar Point	50	17
Burnt Water	50	18

Does this information indicate that the population mean number of inhabited houses is greater than that of hogans on the Navajo Reservation? Use a 5% level of significance. (Let d = houses − hogans.)

(a) What is the level of significance?


State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test?

H₀: μ_d = 0; H₁: μ_d ≠ 0; two-tailedH₀: μ_d = 0; H₁: μ_d < 0; left-tailed    H₀: μ_d = 0; H₁: μ_d > 0; right-tailedH₀: μ_d > 0; H₁: μ_d = 0; right-tailed

(b) What sampling distribution will you use? What assumptions are you making?

The standard normal. We assume that d has an approximately normal distribution.The Student's t. We assume that d has an approximately uniform distribution.    The standard normal. We assume that d has an approximately uniform distribution.The Student's t. We assume that d has an approximately normal distribution.

What is the value of the sample test statistic? (Round your answer to three decimal places.)


(c) Find (or estimate) the P-value.

P-value > 0.2500.125 < P-value < 0.250    0.050 < P-value < 0.1250.025 < P-value < 0.0500.005 < P-value < 0.025P-value < 0.005

Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

Answer 1

The statistical software output for this problem is:

Paired T hypothesis test: HD = 11 - 12 : Mean of the difference between Number of Inhabited Houses and Number of Inhabited Hogans Ho: PD=0 HA: PD > 0 Hypothesis test results: Difference Mean Std. Err. DF T-Stat P-value Number of Inhabited Houses - Number of Inhabited Hogans 6.25 7.5964042 7 0.82275769 0.2189

Hence,

a) Level of significance = 0.05

Hypotheses: H₀: μ_d = 0; H₁: μ_d > 0; right-tailed

b) The Student's t. We assume that d has an approximately normal distribution.

Sample test statistic = 0.823

c) 0.125 < P-value < 0.250

Graph A is correct.

d) At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

d)