Question

(Logic)

Bookstores like education, because national data show that 71% of college graduates have read a book in the past year, compared to 54% of the

general population age 18 and over. The data also show the nationwide average educational level to be 13 years of schooling completed, with an SD of about 3 years, for

persons age 18 and over. A bookstore is doing a market survey in a certain county, and takes a simple random sample of 1,000 people age 18 and over. They find the average

educational level to be 14 years, and the SD is 5 years.

a. Use a z - test to test for significance. What are z and p?

b. Compute the 68% -95%- and 99.7%-confidence intervals.

c.Can the difference in average educational level between the sample and the nation be explained by chance variation? If not, what other explanation can you give?

Answer 1

Data given is:

Sample size, n = 1000

Sample mean m = 14

Population standard deviation, S = 3

Sample standard deviation s = 5

(a)

We have been given standard deviations of both sample and the population, but since we have to make use of a z-test here, so we will be using the population standard deviation.

z = (m-13)/(S/(n^0.5)) = (14-13)/(3/(1000^0.5)) = 10.54

Since this is an extremely small value of z, so the value of p is essentially zero.

(b)

The 68% CI is calculated as:

m-(0.99*(S/(n^0.5))) < < m+(0.99*(S/(n^0.5)))

Put values:

14-(0.99*(3/(1000^0.5))) < < 14+(0.99*(3/(1000^0.5)))

13.906 < < 14.094

The 95% CI is calculated as:

m-(1.96*(S/(n^0.5))) < < m+(1.96*(S/(n^0.5)))

Put values:

14-(1.96*(3/(1000^0.5))) < < 14+(1.96*(3/(1000^0.5)))

13.814 < < 14.186

The 99.7% CI is calculated as:

m-(2.97*(S/(n^0.5))) < < m+(2.97*(S/(n^0.5)))

Put values:

14-(2.97*(3/(1000^0.5))) < < 14+(2.97*(3/(1000^0.5)))

13.718 < < 14.282

(c)

In this case since the p-value is essentially close to zero, this means that this particular difference is not a chance variation. It can be that different countries have different level of education and the country in which the survey is being done is different from the one for which the population data is given.