Question

A random sample of n1 = 16 communities in western Kansas gave the following information for people under 25 years of age.

x1: Rate of hay fever per 1000 population for people under 25 96 92 120 130 91 123 112 93 125 95 125 117 97 122 127 88 A random sample of n2 = 14 regions in western Kansas gave the following information for people over 50 years old.

x2: Rate of hay fever per 1000 population for people over 50 95 111 102 97 112 88 110 79 115 100 89 114 85 96

(i) Use a calculator to calculate x₁, s₁, x₂, and s₂. (Round your answers to two decimal places.)

x1	=
s1	=
x2	=
s2	=

(ii) Assume that the hay fever rate in each age group has an approximately normal distribution. Do the data indicate that the age group over 50 has a lower rate of hay fever? Use α = 0.10.
(a) What is the level of significance?

State the null and alternate hypotheses.

H₀: μ₁ = μ₂; H₁: μ₁ > μ₂

H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂

H₀: μ₁ > μ₂; H₁: μ₁ = μ₂

H₀: μ₁ = μ₂; H₁: μ₁ < μ₂

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

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

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

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

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

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

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

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 = 0.10 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.10 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.10 level, we reject the null hypothesis and conclude the data are statistically significant.

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

Reject the null hypothesis, there is sufficient evidence that the mean rate of hay fever is lower for the age group over 50.

Fail to reject the null hypothesis, there is insufficient evidence that the mean rate of hay fever is lower for the age group over 50.

Fail to reject the null hypothesis, there is sufficient evidence that the mean rate of hay fever is lower for the age group over 50.

Reject the null hypothesis, there is insufficient evidence that the mean rate of hay fever is lower for the age group over 50.

Answer 1

(i)

1 = 109.5625

s1 = 15.5990

2 = 99.5

s2 = 11.6603

(ii)

(a)

Level of significance = = 0.10

Correct option:

H0:

H1:

(b)

Correct option:

The Student's t. We assume both populations are approximately normal with unknown standard deviation

Test statistic is:

t = (109.5625 - 99.5)/5.0899 = 10.0625/5.0899 = 1.977

(c)

ndf = 16 + 14 - 2 = 28

By Technology, p - value = 0.0290

(d)
Correct option:

At = 0.10 level, we fail to reject the null hypothesis and conclude that the data are not statistically significant.

(e)

Correct option:

Fail to reject the null hypothesis. there is insufficient evidence that the mean rate of hay fever is lower for the age group over 50.