Question

A random sample of n₁ = 18 winter days in Denver gave a sample mean pollution index x₁ = 43. Previous studies show that σ₁ = 13. For Englewood (a suburb of Denver), a random sample of n₂ = 17 winter days gave a sample mean pollution index of x₂ = 48. Previous studies show that σ₂ = 15. Assume the pollution index is normally distributed in both Englewood and Denver. Do these data indicate that the mean population pollution index of Englewood is different (either way) from that of Denver in the winter? Use a 1% level of significance.

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

A. The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.

B. The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.  

C. The standard normal. We assume that both population distributions are approximately normal with known standard deviations.

D. The Student's t. We assume that both population distributions are approximately normal with known standard deviations

(b) What is the value of the sample test statistic? (Test the difference μ₁ − μ₂. Round your answer to two decimal places.)


(c) Find (or estimate) the P-value. (Round your answer to four decimal places)

(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?

A. At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.

B. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.    

C. At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

D. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

A. Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.

B. Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.

C. Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver

D. .Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.

Answer 1

a)

C. The standard normal. We assume that both population distributions are approximately normal with known standard deviations.

b)

Pooled Variance
sp = sqrt(s1^2/n1 + s2^2/n2)
sp = sqrt(169/18 + 225/17)
sp = 4.7565

Test statistic,
z = (x1bar - x2bar)/sp
z = (43 - 48)/4.7565
z = -1.05

c)

P-value Approach
P-value = 0.2937
As P-value >= 0.01, fail to reject null hypothesis.

d)

D. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

e)

A. Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.