Question

A study was conducted to determine the proportion of people who dream in black and white instead of color. Among 298 people over the age of​ 55, 61 dream in black and​ white, and among 290 people under the age of​ 25, 14 dream in black and white. Use a 0.01 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. Find the test statistic Z to two or more decimal places, and the P-value to three or more decimal places

Answer 1

p1cap = X1/N1 = 61/298 = 0.2047
p1cap = X2/N2 = 14/290 = 0.0483
pcap = (X1 + X2)/(N1 + N2) = (61+14)/(298+290) = 0.1276

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

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.2047-0.0483)/sqrt(0.1276*(1-0.1276)*(1/298 + 1/290))
z = 5.68

P-value Approach
P-value = 0.0000
As P-value < 0.01, reject the null hypothesis.

There is sufficient evidence to conclude that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25.