Question

A report included the following information on the heights (in.) for non-Hispanic white females.

Age	Sample Size	Sample Mean	Std. Error Mean
20–39	863	65.7	0.09
60 and older	939	64.1	0.11

(a)

Calculate a confidence interval at confidence level approximately 95% for the difference between population mean height for the younger women and that for the older women. (Use μ_20–39 − μ_{60 and older}.) ,

We are 95% confident that the true average height of younger women is greater than that of older women by an amount within the confidence interval.We are 95% confident that the true average height of younger women is greater than that of older women by an amount outside the confidence interval.    We cannot draw a conclusion from the given information.We are 95% confident that the true average height of younger women is less than that of older women by an amount within the confidence interval.Interpret the interval.

(b)

Let

μ₁

denote the population mean height for those aged 20–39 and μ₂ denote the population mean height for those aged 60 and older. Interpret the hypotheses

H₀: μ₁ − μ₂ = 1

and

H_a: μ₁ − μ₂ > 1.

The null hypothesis states that the true mean height for older women is 1 inch higher than for younger women. The alternative hypothesis states that the true mean height for older women is more than 1 inch higher than for younger women.The null hypothesis states that the true mean height for younger women is more than 1 inch higher than for older women. The alternative hypothesis states that the true mean height for younger women is 1 inch higher than for older women.    The null hypothesis states that the true mean height for younger women is 1 inch higher than for older women. The alternative hypothesis states that the true mean height for younger women is more than 1 inch higher than for older women.The null hypothesis states that the true mean height for older women is more than 1 inch higher than for younger women. The alternative hypothesis states that the true mean height for older women is 1 inch higher than for younger women.

Carry out a test of these hypotheses at significance level 0.001. Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.)

z=

P-value=

(c)

Based on the P-value calculated in (b) would you reject the null hypothesis at any reasonable significance level? Explain your reasoning.

Reject H₀. The data suggests that the difference in the true average heights exceeds 1.Fail to reject H₀. The data suggests that the difference in the true average heights exceeds 1.    Reject H₀. The data does not suggest that the difference in the true average heights exceeds 1.Fail to reject H₀. The data does not suggest that the difference in the true average heights exceeds 1.

(d)

What hypotheses would be appropriate if μ₁ referred to the older age group, μ₂ to the younger age group, and you wanted to see if there was compelling evidence for concluding that the population mean height for younger women exceeded that for older women by more than 1 in.?

H₀: μ₁ − μ₂ = 1
H_a: μ₁ − μ₂ < 1H₀: μ₁ − μ₂ = −1
H_a: μ₁ − μ₂ < −1    H₀: μ₁ − μ₂ = 1
H_a: μ₁ − μ₂ > 1H₀: μ₁ − μ₂ = −1
H_a: μ₁ − μ₂ > −1

You may need to use the appropriate table in the Appendix of Tables to answer this question.

Answer 1

a) At 95% confidence level, the critical value is z_0.025 = 1.96

The 95% confidence interval is

We are 95% confident that the true average height of younger women is greater than that of older women by an amount within the confidence interval.

b) The null hypothesis states that the true mean height for younger women is 1 inch higher than for older women.

The alternative hypothesis states that the true mean height for younger women is more than 1 inch higher than for older women.

   

  

P-value = P(Z > 11.26)

             = 1 - P(Z < 11.26)

             = 1 - 1 = 0

c) Since the P-value is less than the significance level(0 < 0.05), so we should reject the null hypothesis.

Reject H₀. The data suggests that the difference in the true average heights exceeds 1.

d)