A study was conducted to determine the proportion of people who dream in black and white instead of color. Among 295 people over the age of 55, 69 dream in black and white, and among 290 people under the age of 25, 11 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. Complete parts (a) through (c) below.

A. Hypotheses:
[tex]\[
\begin{array}{l}
H_0: P_1 \leq P_2 \\
H_1: P_1 \ \textgreater \ P_2
\end{array}
\][/tex]

Identify the test statistic:
[tex]\[
z = \text{(calculated value)}
\][/tex]
(Round to two decimal places as needed.)

Identify the P-value:
[tex]\[
\text{P-value} = \text{(calculated value)}
\][/tex]
(Round to three decimal places as needed.)

Conclusion based on the hypothesis test:
The P-value is less than the significance level of [tex]\(\alpha = 0.01\)[/tex], so 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.



Answer :

Let's go through the process of solving the question step by step.

### Step 1: State the Hypotheses

The null and alternative hypotheses are:

[tex]\[ H_0: P_1 \leq P_2 \][/tex]
[tex]\[ H_1: P_1 > P_2 \][/tex]

where [tex]\( P_1 \)[/tex] is the proportion of people over 55 who dream in black and white, and [tex]\( P_2 \)[/tex] is the proportion of people under 25 who dream in black and white. This is a one-tailed test.

### Step 2: Calculate the Sample Proportions

First, we calculate the sample proportions for each group:
[tex]\[ \hat{p}_1 = \frac{x_1}{n_1} = \frac{69}{295} \approx 0.2339 \][/tex]
[tex]\[ \hat{p}_2 = \frac{11}{290} \approx 0.0379 \][/tex]

### Step 3: Pooled Proportion

Next, we calculate the pooled proportion, assuming the null hypothesis is true:
[tex]\[ \hat{p}_{\text{pool}} = \frac{x_1 + x2}{n_1 + n_2} = \frac{69 + 11}{295 + 290} \approx 0.1368 \][/tex]

### Step 4: Standard Error

We calculate the standard error for the difference between the two sample proportions:
[tex]\[ SE = \sqrt{\hat{p}_{\text{pool}} \cdot (1 - \hat{p}_{\text{pool}}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \][/tex]
[tex]\[ SE \approx \sqrt{0.1368 \cdot (1 - 0.1368) \left( \frac{1}{295} + \frac{1}{290} \right)} \approx 0.0284 \][/tex]

### Step 5: Test Statistic

Calculate the z-score:
[tex]\[ z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \][/tex]
[tex]\[ z = \frac{0.2339 - 0.0379}{0.0284} \approx 6.90 \][/tex]

### Step 6: P-value

Determine the p-value for the z-score in a one-tailed test:
[tex]\[ \text{P-value} \approx 1.32 \times 10^{-12} \][/tex]

### Step 7: Conclusion

Based on the p-value and the significance level of [tex]\( \alpha = 0.01 \)[/tex]:

Since [tex]\( \text{P-value} < \alpha \)[/tex], we reject the null hypothesis.

### Summary

Based on the hypothesis test, we reject the null hypothesis. There is significant 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.