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 = 36 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 = 35 U.S. cities. The sample mean for these cities showed that x₂ = 19.2% of the young adults had attended college. From previous studies, it is known that ?₁ = 6.6% and ?₂ = 5.6%. Does this information indicate that the population mean percentage of young adults who attended college is higher? Use ? = 0.01.

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


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

Part 2) Is the national crime rate really going down? Some sociologists say yes! They say that the reason for the decline in crime rates in the 1980s and 1990s is demographics. It seems that the population is aging, and older people commit fewer crimes. According to the FBI and the Justice Department, 70% of all arrests are of males aged 15 to 34 years†. Suppose you are a sociologist in Rock Springs, Wyoming, and a random sample of police files showed that of 35 arrests last month, 23 were of males aged 15 to 34 years. Use a 5% level of significance to test the claim that the population proportion of such arrests in Rock Springs is different from 70%.

a)What is the value of the sample test statistic? (Round your answer to two decimal places.)


(b) Find the P-value of the test statistic. (Round your answer to four decimal places.)

(c) 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 ? = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

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

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

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

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

There is sufficient evidence at the 0.05 level to conclude that the true proportion of arrests of males aged 15 to 34 in Rock Springs differs from 70%.

There is insufficient evidence at the 0.05 level to conclude that the true proportion of arrests of males aged 15 to 34 in Rock Springs differs from 70%.

Answer 1

1
a)
H0: The population mean percentage of people age 65 and older who attended college is equal to that of young adults.
H1: The population mean percentage of people age 65 and older who attended college is less than that of young adults.

As, we know the population standard deviations, ?1 and ?2 , we can use z test statistic to test the hypothesis.

Standard error of the mean difference, se =

Sample test statistic, z = (x1 - x2) / se

= (15.2 - 19.2) / 1.451206

= -2.7563

(b)

For one tail test,

P-value = P[z < -2.7563] = 0.0029

As, p-value is less than the significance levele of 0.01, we reject the null hypothesis H0 and conclude that there is significant evidence that population mean percentage of people age 65 and older who attended college is less than that of young adults.