Question

homework
question 4, 9.1.11-t
part 4 of 7
hw score: 59.38%, 4.75 of 8 points
points: 0 of 1
a study was conducted to determine the proportion of people who dream in black and white instead of color. among 288 people over the age of 55, 64 dream in black and white, and among 299 people under the age of 25, 11 dream below.
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 (c).
a. test the claim using a hypothesis test.
consider the first sample to be the sample of people over the age of 55 and the second sample to be the sample of people under the age of 25. what are the null and alternative hypotheses for the hypothesis test?
a.
h₀: p₁ ≠ p₂
h₁: p₁ = p₂
b.
h₀: p₁ = p₂
h₁: p₁ ≠ p₂
c.
h₀: p₁ ≤ p₂
h₁: p₁ ≠ p₂
d.
h₀: p₁ ≥ p₂
h₁: p₁ ≠ p₂
e.
h₀: p₁ = p₂
h₁: p₁ > p₂
f.
h₀: p₁ = p₂
h₁: p₁ < p₂
identify the test statistic.
z = 6.72
(round to two decimal places as needed.)
identify the p - value.
p - value = 0.000
(round to three decimal places as needed.)
what is the conclusion based on the hypothesis test?
the p - value is
the significance level of α = 0.05, so
the null hypothesis. there is
evidence to support the claim that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25.

Explanation:

Step1: Identify Hypotheses

The claim is that the proportion of people over 55 (\(p_1\)) who dream in black and white is greater than that for those under 25 (\(p_2\)). So, the null hypothesis \(H_0\) is \(p_1 = p_2\), and the alternative hypothesis \(H_1\) is \(p_1 > p_2\). This matches option E.

Step2: Analyze P - value and Conclusion

The P - value is 0.000, which is less than \(\alpha=0.05\). When \(P - value<\alpha\), we reject the null hypothesis. Since we reject \(H_0\) (\(p_1 = p_2\)) and the alternative is \(p_1 > p_2\), there is sufficient evidence to support the claim.

Answer:

Part a (Hypotheses)

E. \(H_0:p_1 = p_2\), \(H_1:p_1>p_2\)

Part c (Conclusion)

The P - value (0.000) is less than the significance level of \(\alpha = 0.05\), so we reject the null hypothesis. There is sufficient evidence to support the claim that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25.