Sure, let's go through the step-by-step solution for this hypothesis test. Here's a detailed explanation:
### a. Stating the Hypotheses
We need 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.
- [tex]\( p_1 \)[/tex]: Proportion of people over the age of 55 who dream in black and white.
- [tex]\( p_2 \)[/tex]: Proportion of people under the age of 25 who dream in black and white.
The null and alternative hypotheses can be set as follows:
- Null hypothesis ([tex]\( H_0 \)[/tex]): [tex]\( p_1 \leq p_2 \)[/tex]
- Alternative hypothesis ([tex]\( H_1 \)[/tex]): [tex]\( p_1 > p_2 \)[/tex]
Therefore, the correct answer for the hypotheses is (A):
- [tex]\( H_0: p_1 \leq p_2 \)[/tex]
- [tex]\( H_1: p_1 > p_2 \)[/tex]
### Test Statistic
The test statistic used for comparing proportions is calculated as a z-score.
Given:
- Sample size for people over 55 ([tex]\( n_1 \)[/tex]) = 295
- Sample size for people under 25 ([tex]\( n_2 \)[/tex]) = 290
- Number of people over 55 who dream in black and white ([tex]\( x_1 \)[/tex]) = 69
- Number of people under 25 who dream in black and white ([tex]\( x_2 \)[/tex]) = 11
The proportions are:
- [tex]\( \hat{p}_1 = \frac{69}{295} \)[/tex]
- [tex]\( \hat{p}_2 = \frac{11}{290} \)[/tex]
Next, we calculate the pooled proportion ([tex]\( \hat{p} \)[/tex]):
[tex]\[ \hat{p} = \frac{69 + 11}{295 + 290} = \frac{80}{585} \][/tex]
The standard error (SE) for the difference in proportions is:
[tex]\[ SE = \sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{295} + \frac{1}{290} \right)} \][/tex]
The z-value (test statistic) is calculated as:
[tex]\[ z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \][/tex]
Based on the calculations (your result):
[tex]\[ z = 6.90 \][/tex]
### [tex]\( P \)[/tex]-value
To find the [tex]\( P \)[/tex]-value, we consider the fact that this is a one-tailed test (since [tex]\( H_1 \)[/tex] states [tex]\( p_1 > p_2 \)[/tex]).
Given:
[tex]\[ z = 6.90 \][/tex]
Using the cumulative distribution function (CDF) for the normal distribution to find the tail probability:
[tex]\[ P \text{-value} = 1 - \Phi(z) \][/tex]
Where [tex]\( \Phi(z) \)[/tex] is the standard normal cumulative distribution function.
For [tex]\( z = 6.90 \)[/tex], the [tex]\( P \)[/tex]-value is extremely small (close to 0) because it is highly significant.
Based on the calculations (your result):
[tex]\[ P \text{-value} = 0.000 \][/tex]
### Conclusion
Since the [tex]\( P \)[/tex]-value is less than the significance level ([tex]\( \alpha = 0.01 \)[/tex]), 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.
