Question

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 = 38 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 = 34 U.S. cities. The sample mean for these cities showed that x₂ = 18.1% of the young adults had attended college. From previous studies, it is known that σ₁ = 7.2% and σ₂ = 5.4%. Does this information indicate that the population mean percentage of young adults who attended college is higher? Use α = 0.05.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: μ₁ = μ₂; H₁: μ₁ > μ₂H₀: μ₁ < μ₂; H₁: μ₁ = μ₂     H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂H₀: μ₁ = μ₂; H₁: μ₁ < μ₂

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

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.)


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

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

Answer 1

a)
5% level of significance
null and alternate hypotheses
H0: μ1 = μ2; H1: μ1 < μ2

b)
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
value of the sample test statistic

We have given

Sample 1	Sample 2
n1 = 38	n2 = 34
X1 = 15.2	X2 = 18.1
σ1 = 7.2	σ2 = 5.4

We will calculate test statistics using below formulae

=(15.2-18.1)/sqrt(7.2^2/38+5.4^2/34)

=-1.9455

c)

p-value

one tail test hence

p value= p(Z<-1.9455)

=0.025855

The P-Value is .025857.
The result is significant at p < .05.

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

d)

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level

e)

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