Question

2.
One in four Americans (25%) surveyed in a poll in 2016 said that they didn’t read a single book within the last 12 months. When Pew Research surveyed 1,520 adults living in all 50 U.S. states and the District of Columbia in 2016, they learned that the number of respondents that didn’t read a book within the last year did not budge from 2015 figures. A recent sample of 120 people indicated that 41 of them didn’t read a book within the last year.

1. Based on the sample results, construct a 95% confidence interval for the recent population proportion for those who did not read a book within the last year.
2. Based on the sample results, test if the recent population proportion is greater than the population proportion in 2016. Use the critical value approach with the 5% significance level.
3. Repeat the preceding hypothesis test using the p-value approach.

Answer 1

a)

sample proportion, = 0.3417
sample size, n = 120
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.3417 * (1 - 0.3417)/120) = 0.0433

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

Margin of Error, ME = zc * SE
ME = 1.96 * 0.0433
ME = 0.0849

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.3417 - 1.96 * 0.0433 , 0.3417 + 1.96 * 0.0433)
CI = (0.257 , 0.427)

b)
Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p = 0.25
Alternative Hypothesis, Ha: p > 0.25

Test statistic,
z = (pcap - p)/sqrt(p*(1-p)/n)
z = (0.3417 - 0.25)/sqrt(0.25*(1-0.25)/120)
z = 2.32

Rejection Region
This is right tailed test, for α = 0.05
Critical value of z is 1.64.
Hence reject H0 if z > 1.64

reject the null hypothesis.

c)

P-value Approach
P-value = 0.0102
As P-value < 0.05, reject the null hypothesis.