Question

A study was conducted to determine the proportion of people who dream in black and white instead of color. Among 293 people over the age of​ 55, 69 dream in black and​ white, and among 305 people under the age of​ 25,18 dream in black and white. Use a 0.05 significance level to test the claim that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25. Complete parts​ (a) through​ (b)

a. Consider the first sample to be the sample of occupants not wearing seat belts and the second sample to be the sample of occupants wearing seat belts. What are the null and alternative hypotheses for the hypothesis​ test?

Identify the test statistic.

Identify the​ P-value.

b. Test the claim by constructing an appropriate confidence interval.

Answer 1

a)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p1 = p2
Alternate Hypothesis, Ha: p1 > p2

p1cap = X1/N1 = 69/293 = 0.2355
p1cap = X2/N2 = 18/305 = 0.059
pcap = (X1 + X2)/(N1 + N2) = (69+18)/(293+305) = 0.1455

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.2355-0.059)/sqrt(0.1455*(1-0.1455)*(1/293 + 1/305))
z = 6.12

P-value Approach
P-value = 0
As P-value < 0.05, reject the null hypothesis.

b)

Here, , n1 = 293 , n2 = 305
p1cap = 0.2355 , p2cap = 0.059

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.2355 * (1-0.2355)/293 + 0.059*(1-0.059)/305)
SE = 0.0282

For 0.95 CI, z-value = 1.96
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.2355 - 0.059 - 1.96*0.0282, 0.2355 - 0.059 + 1.96*0.0282)
CI = (0.1212 , 0.2318)

As the confidence interval does not contain 0 reject the null hypothesis