Question

1. In your study of 210 armed robbers in Detroit, you find the proportion that is male is .82. Test the null hypothesis that your sample came from a population in which the proportion that is male is .70 against the alternative hypothesis that the population proportion is different than .70 Use alpha=.05.

a. Formally state the null and the alternative hypothesis for this test (3 pts)

b. What is the critical value(s) for this test? (3 pts)

c. Compute and report the test statistic. (3 pts)

d) Interpret your decision regarding the null hypothesis (3 pts)

e) If necessary, compute 95% confidence intervals for the population proportion and interpret

Answer 1

a)

Below are the null and alternative Hypothesis,
Null Hypothesis: p = 0.7
Alternative Hypothesis: p ≠ 0.7

b)

Rejection Region
This is two tailed test, for α = 0.05
Critical value of z are -1.96 and 1.96.
Hence reject H0 if z < -1.96 or z > 1.96

c)

Test statistic,
z = (pcap - p)/sqrt(p*(1-p)/n)
z = (0.82 - 0.7)/sqrt(0.7*(1-0.7)/210)
z = 3.79

d)

Reject the null hyothesis

e)

sample proportion, = 0.82
sample size, n = 210
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.82 * (1 - 0.82)/210) = 0.0265

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 * 0.0265
ME = 0.0519

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.82 - 1.96 * 0.0265 , 0.82 + 1.96 * 0.0265)
CI = (0.7681 , 0.8719)