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.
