Question

Part 1) Education influences attitude and lifestyle. Differences in education are a big factor in the "generation gap." Is the younger generation really better educated? Large surveys of people age 65 and older were taken in n₁ = 40 U.S. cities. The sample mean for these cities showed that x₁ = 15.2% of the older adults had attended college. Large surveys of young adults (age 25 - 34) were taken in n₂ = 37 U.S. cities. The sample mean for these cities showed that x₂ = 18.8% of the young adults had attended college. From previous studies, it is known that ?₁ = 7.2% and ?₂ = 4.2%. Does this information indicate that the population mean percentage of young adults who attended college is higher? Use ? = 0.10.

a) State the null and alternate hypotheses.

H₀: ?₁ = ?₂; H₁: ?₁ < ?₂

H₀: ?₁ = ?₂; H₁: ?₁ ? ?₂    

H₀: ?₁ < ?₂; H₁: ?₁ = ?₂

H₀: ?₁ = ?₂; H₁: ?₁ > ?₂

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

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.    

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.

(c) What is the value of the sample test statistic? (Test the difference ?₁ ? ?₂. Round your answer to two decimal places.)


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

(e) 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 ??

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

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

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

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

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

Fail to reject the null hypothesis, there is insufficient evidence that the mean percentage of young adults who attend college is higher.

Fail to reject the null hypothesis, there is sufficient evidence that the mean percentage of young adults who attend college is higher.    

Reject the null hypothesis, there is sufficient evidence that the mean percentage of young adults who attend college is higher.

Reject the null hypothesis, there is insufficient evidence that the mean percentage of young adults who attend college is higher.

Answer 1

(a)

The level of significance is  ? = 0.05

H0: ?1 = ?2; H1: ?1 < ?2

(b)

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

(c)

Given information:

Test statitics will be

Here test is left tailed so p-value is

P(z < -2.70) = 0.0034

(d)

Since p-value is less than 0.10 so we reject the null hypothesis.

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

(e)

Reject the null hypothesis, there is sufficient evidence that the mean percentage of young adults who attend college is higher.