Answer :
Sure! Let's walk through the solution step-by-step.
### Question
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.
### a. Hypothesis Testing
Step 1: Define the Hypotheses
We need to test the claim that the proportion of people over the age of 55 ([tex]\( p_1 \)[/tex]) who dream in black and white is greater than the proportion of people under the age of 25 ([tex]\( p_2 \)[/tex]).
- 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]
Among the given options:
- [tex]\( H_0: p_1 \leq p_2 \)[/tex]
- [tex]\( H_1: p_1 > p_2 \)[/tex]
This corresponds to option `A`.
### Step 2: Calculate the Sample Proportions
- For people over the age of 55: [tex]\( \hat{p_1} = \frac{69}{295} \approx 0.2339 \)[/tex]
- For people under the age of 25: [tex]\( \hat{p_2} = \frac{11}{290} \approx 0.0379 \)[/tex]
### Step 3: Calculate the Pooled Proportion
The pooled proportion ([tex]\( \hat{p} \)[/tex]) is given by:
[tex]\[ \hat{p} = \frac{69 + 11}{295 + 290} \approx 0.1368 \][/tex]
### Step 4: Calculate the Standard Error
The standard error ([tex]\( SE \)[/tex]) is calculated using the pooled proportion:
[tex]\[ SE = \sqrt{ \hat{p} \cdot (1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right) } \][/tex]
Plugging in the numbers:
[tex]\[ SE \approx \sqrt{ 0.1368 \cdot (1 - 0.1368) \left( \frac{1}{295} + \frac{1}{290} \right) } \approx 0.0284 \][/tex]
### Step 5: Calculate the Test Statistic (z-value)
The test statistic ([tex]\( z \)[/tex]) is given by:
[tex]\[ z = \frac{\hat{p_1} - \hat{p_2}}{SE} \][/tex]
[tex]\[ z \approx \frac{0.2339 - 0.0379}{0.0284} \approx 6.90 \][/tex]
### Step 6: Determine the Critical Value
For a one-tailed test with a significance level of 0.01, the critical z-value is:
[tex]\[ z_{critical} \approx 2.33 \][/tex]
### Step 7: Make a Decision
- Compare the test statistic to the critical value:
[tex]\[ z = 6.90 > z_{critical} = 2.33 \][/tex]
Since the test statistic exceeds the critical value, we reject the null hypothesis.
### Conclusion
There is sufficient evidence at the 0.01 significance level to support the claim that the proportion of people over the age of 55 who dream in black and white is greater than the proportion of people under the age of 25.
Summary of the Values:
- Sample Proportion for over 55: [tex]\( \hat{p_1} \approx 0.2339 \)[/tex]
- Sample Proportion for under 25: [tex]\( \hat{p_2} \approx 0.0379 \)[/tex]
- Pooled Proportion: [tex]\( \hat{p} \approx 0.1368 \)[/tex]
- Standard Error: [tex]\( SE \approx 0.0284 \)[/tex]
- Test Statistic (z-value): [tex]\( z \approx 6.90 \)[/tex]
- Critical Value: [tex]\( z_{critical} \approx 2.33 \)[/tex]
### Question
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.
### a. Hypothesis Testing
Step 1: Define the Hypotheses
We need to test the claim that the proportion of people over the age of 55 ([tex]\( p_1 \)[/tex]) who dream in black and white is greater than the proportion of people under the age of 25 ([tex]\( p_2 \)[/tex]).
- 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]
Among the given options:
- [tex]\( H_0: p_1 \leq p_2 \)[/tex]
- [tex]\( H_1: p_1 > p_2 \)[/tex]
This corresponds to option `A`.
### Step 2: Calculate the Sample Proportions
- For people over the age of 55: [tex]\( \hat{p_1} = \frac{69}{295} \approx 0.2339 \)[/tex]
- For people under the age of 25: [tex]\( \hat{p_2} = \frac{11}{290} \approx 0.0379 \)[/tex]
### Step 3: Calculate the Pooled Proportion
The pooled proportion ([tex]\( \hat{p} \)[/tex]) is given by:
[tex]\[ \hat{p} = \frac{69 + 11}{295 + 290} \approx 0.1368 \][/tex]
### Step 4: Calculate the Standard Error
The standard error ([tex]\( SE \)[/tex]) is calculated using the pooled proportion:
[tex]\[ SE = \sqrt{ \hat{p} \cdot (1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right) } \][/tex]
Plugging in the numbers:
[tex]\[ SE \approx \sqrt{ 0.1368 \cdot (1 - 0.1368) \left( \frac{1}{295} + \frac{1}{290} \right) } \approx 0.0284 \][/tex]
### Step 5: Calculate the Test Statistic (z-value)
The test statistic ([tex]\( z \)[/tex]) is given by:
[tex]\[ z = \frac{\hat{p_1} - \hat{p_2}}{SE} \][/tex]
[tex]\[ z \approx \frac{0.2339 - 0.0379}{0.0284} \approx 6.90 \][/tex]
### Step 6: Determine the Critical Value
For a one-tailed test with a significance level of 0.01, the critical z-value is:
[tex]\[ z_{critical} \approx 2.33 \][/tex]
### Step 7: Make a Decision
- Compare the test statistic to the critical value:
[tex]\[ z = 6.90 > z_{critical} = 2.33 \][/tex]
Since the test statistic exceeds the critical value, we reject the null hypothesis.
### Conclusion
There is sufficient evidence at the 0.01 significance level to support the claim that the proportion of people over the age of 55 who dream in black and white is greater than the proportion of people under the age of 25.
Summary of the Values:
- Sample Proportion for over 55: [tex]\( \hat{p_1} \approx 0.2339 \)[/tex]
- Sample Proportion for under 25: [tex]\( \hat{p_2} \approx 0.0379 \)[/tex]
- Pooled Proportion: [tex]\( \hat{p} \approx 0.1368 \)[/tex]
- Standard Error: [tex]\( SE \approx 0.0284 \)[/tex]
- Test Statistic (z-value): [tex]\( z \approx 6.90 \)[/tex]
- Critical Value: [tex]\( z_{critical} \approx 2.33 \)[/tex]
