Answer :
### a. Formulating the Hypotheses
To test the claim that the proportion of people over 55 who dream in black and white is greater than the proportion of those under 25, we need to set up our null and alternative hypotheses. These hypotheses will be:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The proportion of people over 55 who dream in black and white is equal to the proportion of those under 25. Mathematically, this is:
[tex]\[ H_0: p_1 = p_2 \][/tex]
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The proportion of people over 55 who dream in black and white is greater than the proportion of those under 25. Mathematically, this is:
[tex]\[ H_1: p_1 > p_2 \][/tex]
By matching the options provided, the correct answer is:
D. [tex]\(H_0: p_1 = p_2\)[/tex]; [tex]\(H_1: p_1 > p_2\)[/tex]
### b. Calculate the Test Statistic
Let's compute the required values step-by-step based on the results we need to achieve.
1. Sample Proportions Calculation:
For the first sample (over 55):
[tex]\[ p_1 = \frac{65}{296} \approx 0.2196 \][/tex]
For the second sample (under 25):
[tex]\[ p_2 = \frac{18}{292} \approx 0.0616 \][/tex]
2. Pooled Proportion Calculation:
The pooled proportion [tex]\( p_{pool} \)[/tex] is calculated as:
[tex]\[ p_{pool} = \frac{65 + 18}{296 + 292} \approx 0.1412 \][/tex]
3. Standard Error Calculation:
The standard error (SE) of the difference between two sample proportions is given by:
[tex]\[ SE = \sqrt{ p_{pool} \left(1 - p_{pool}\right) \left(\frac{1}{296} + \frac{1}{292}\right) } \approx 0.0287 \][/tex]
4. Test Statistic (z) Calculation:
The test statistic [tex]\( z \)[/tex] for the hypothesis test is calculated by the formula:
[tex]\[ z = \frac{(p_1 - p_2)}{SE} \approx \frac{0.2196 - 0.0616}{0.0287} \approx 5.5000 \][/tex]
Therefore, the test statistic is approximately 5.5000.
### c. Calculate the P-value
The next step is to calculate the p-value for the test statistic obtained. Since we're performing a one-tailed test, we look at the area to the right of the test statistic in the standard normal distribution.
Based on a standard normal distribution:
- The p-value corresponding to [tex]\( z = 5.5000 \)[/tex] is extremely small.
- Specifically, [tex]\( p-value \approx 1.8989 \times 10^{-8} \)[/tex].
### Conclusion
Since the p-value ([tex]\( \approx 1.8989 \times 10^{-8} \)[/tex]) is much smaller than our significance level ([tex]\( \alpha = 0.05 \)[/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 of those under 25.
To summarize:
Hypotheses:
[tex]\[ H_0: p_1 = p_2 \\ H_1: p_1 > p_2 \][/tex]
Test Statistic:
[tex]\[ z \approx 5.5000 \][/tex]
P-value:
[tex]\[ p-value \approx 1.8989 \times 10^{-8} \][/tex]
Decision:
Reject the null hypothesis.
Conclusion:
The proportion of people over 55 who dream in black and white is significantly greater than the proportion of people under 25 who do.
To test the claim that the proportion of people over 55 who dream in black and white is greater than the proportion of those under 25, we need to set up our null and alternative hypotheses. These hypotheses will be:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The proportion of people over 55 who dream in black and white is equal to the proportion of those under 25. Mathematically, this is:
[tex]\[ H_0: p_1 = p_2 \][/tex]
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The proportion of people over 55 who dream in black and white is greater than the proportion of those under 25. Mathematically, this is:
[tex]\[ H_1: p_1 > p_2 \][/tex]
By matching the options provided, the correct answer is:
D. [tex]\(H_0: p_1 = p_2\)[/tex]; [tex]\(H_1: p_1 > p_2\)[/tex]
### b. Calculate the Test Statistic
Let's compute the required values step-by-step based on the results we need to achieve.
1. Sample Proportions Calculation:
For the first sample (over 55):
[tex]\[ p_1 = \frac{65}{296} \approx 0.2196 \][/tex]
For the second sample (under 25):
[tex]\[ p_2 = \frac{18}{292} \approx 0.0616 \][/tex]
2. Pooled Proportion Calculation:
The pooled proportion [tex]\( p_{pool} \)[/tex] is calculated as:
[tex]\[ p_{pool} = \frac{65 + 18}{296 + 292} \approx 0.1412 \][/tex]
3. Standard Error Calculation:
The standard error (SE) of the difference between two sample proportions is given by:
[tex]\[ SE = \sqrt{ p_{pool} \left(1 - p_{pool}\right) \left(\frac{1}{296} + \frac{1}{292}\right) } \approx 0.0287 \][/tex]
4. Test Statistic (z) Calculation:
The test statistic [tex]\( z \)[/tex] for the hypothesis test is calculated by the formula:
[tex]\[ z = \frac{(p_1 - p_2)}{SE} \approx \frac{0.2196 - 0.0616}{0.0287} \approx 5.5000 \][/tex]
Therefore, the test statistic is approximately 5.5000.
### c. Calculate the P-value
The next step is to calculate the p-value for the test statistic obtained. Since we're performing a one-tailed test, we look at the area to the right of the test statistic in the standard normal distribution.
Based on a standard normal distribution:
- The p-value corresponding to [tex]\( z = 5.5000 \)[/tex] is extremely small.
- Specifically, [tex]\( p-value \approx 1.8989 \times 10^{-8} \)[/tex].
### Conclusion
Since the p-value ([tex]\( \approx 1.8989 \times 10^{-8} \)[/tex]) is much smaller than our significance level ([tex]\( \alpha = 0.05 \)[/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 of those under 25.
To summarize:
Hypotheses:
[tex]\[ H_0: p_1 = p_2 \\ H_1: p_1 > p_2 \][/tex]
Test Statistic:
[tex]\[ z \approx 5.5000 \][/tex]
P-value:
[tex]\[ p-value \approx 1.8989 \times 10^{-8} \][/tex]
Decision:
Reject the null hypothesis.
Conclusion:
The proportion of people over 55 who dream in black and white is significantly greater than the proportion of people under 25 who do.
