Answer :
### Problem Statement
A study is conducted to determine the proportion of people who dream in black and white instead of color. Among 295 people over the age of 65, 69 dream in black and white. Among 290 people under the age of 25, 11 dream in black and white. You are to use a 0.01 significance level to test the claim that the proportion of people over 65 who dream in black and white is greater than the proportion for those under 25. Complete parts (a) through (c) below.
### Hypotheses
A. State the null and alternative hypotheses.
Let's consider the hypotheses:
- [tex]\( H_0 \)[/tex]: [tex]\( P_1 \leq P_2 \)[/tex] (The proportion of people over 65 who dream in black and white is less than or equal to the proportion of those under 25)
- [tex]\( H_1 \)[/tex]: [tex]\( P_1 > P_2 \)[/tex] (The proportion of people over 65 who dream in black and white is greater than the proportion of those under 25)
### Test Statistic
Determine the test statistic.
To identify the test statistic, we can use the z-score:
[tex]\[ z = 6.90 \][/tex]
(Rounded to two decimal places).
### P-value
Identify the P-value.
The P-value is:
[tex]\[ \text{P-value} = 0.000 \][/tex]
(Rounded to three decimal places).
### Conclusion
What is the conclusion based on the hypothesis test?
Given that the P-value is less than the significance level [tex]\(\alpha = 0.01\)[/tex], we reject the null hypothesis. Therefore, there is sufficient evidence to support the claim that the proportion of people over 65 who dream in black and white is greater than the proportion for those under 25.
### Confidence Interval
B. Test the claim by constructing an appropriate confidence interval.
To construct a 98% confidence interval for the difference in proportions ([tex]\(p_1 - p_2\)[/tex]):
- We have the sample proportions:
[tex]\( p_1 \)[/tex] = 0.2339 (proportion of people over 65 dreaming in black and white)
[tex]\( p_2 \)[/tex] = 0.0379 (proportion of people under 25 dreaming in black and white)
- The combined proportion:
[tex]\( p_{\text{combined}} \)[/tex] = 0.1368
- The standard error:
[tex]\( \text{SE} \)[/tex] = 0.0284
- The z-critical value for a 98% confidence level:
[tex]\( z_{\text{critical}} = 2.3263 \)[/tex]
- The margin of error:
[tex]\( \text{Margin of error} = 0.0661 \)[/tex]
The 98% confidence interval for the difference in proportions [tex]\( (p_1 - p_2) \)[/tex] is given by:
[tex]\[ (\text{lower bound}, \text{upper bound}) = (0.1299, 0.2621) \][/tex]
Thus, the 98% confidence interval is:
[tex]\[ (0.130, 0.262) \][/tex]
(Rounded to three decimal places as needed).
### Alternative Hypothesis
Other pairs of hypotheses such as [tex]\( H_0: P_1 = P_2 \)[/tex] and [tex]\( H_1: P_1 \neq P_2 \)[/tex] (two-tailed tests) or reversed hypotheses making [tex]\(H_0\)[/tex] indicate [tex]\( P_1 \geq P_2 \)[/tex] would be misaligned with the wording of the original one-tailed test claiming [tex]\( P_1 > P_2\)[/tex]. Therefore, the appropriate hypotheses are:
[tex]\[ H_0: P_1 \leq P_2 \\ H_1: P_1 > P_2 \][/tex]
Conclusively, based on statistical testing, we reject the null hypothesis and present a confidence interval proving a statistically significant difference between the proportions. This proves a higher proportion of older individuals over 65 dream in black and white compared to younger individuals under 25.
A study is conducted to determine the proportion of people who dream in black and white instead of color. Among 295 people over the age of 65, 69 dream in black and white. Among 290 people under the age of 25, 11 dream in black and white. You are to use a 0.01 significance level to test the claim that the proportion of people over 65 who dream in black and white is greater than the proportion for those under 25. Complete parts (a) through (c) below.
### Hypotheses
A. State the null and alternative hypotheses.
Let's consider the hypotheses:
- [tex]\( H_0 \)[/tex]: [tex]\( P_1 \leq P_2 \)[/tex] (The proportion of people over 65 who dream in black and white is less than or equal to the proportion of those under 25)
- [tex]\( H_1 \)[/tex]: [tex]\( P_1 > P_2 \)[/tex] (The proportion of people over 65 who dream in black and white is greater than the proportion of those under 25)
### Test Statistic
Determine the test statistic.
To identify the test statistic, we can use the z-score:
[tex]\[ z = 6.90 \][/tex]
(Rounded to two decimal places).
### P-value
Identify the P-value.
The P-value is:
[tex]\[ \text{P-value} = 0.000 \][/tex]
(Rounded to three decimal places).
### Conclusion
What is the conclusion based on the hypothesis test?
Given that the P-value is less than the significance level [tex]\(\alpha = 0.01\)[/tex], we reject the null hypothesis. Therefore, there is sufficient evidence to support the claim that the proportion of people over 65 who dream in black and white is greater than the proportion for those under 25.
### Confidence Interval
B. Test the claim by constructing an appropriate confidence interval.
To construct a 98% confidence interval for the difference in proportions ([tex]\(p_1 - p_2\)[/tex]):
- We have the sample proportions:
[tex]\( p_1 \)[/tex] = 0.2339 (proportion of people over 65 dreaming in black and white)
[tex]\( p_2 \)[/tex] = 0.0379 (proportion of people under 25 dreaming in black and white)
- The combined proportion:
[tex]\( p_{\text{combined}} \)[/tex] = 0.1368
- The standard error:
[tex]\( \text{SE} \)[/tex] = 0.0284
- The z-critical value for a 98% confidence level:
[tex]\( z_{\text{critical}} = 2.3263 \)[/tex]
- The margin of error:
[tex]\( \text{Margin of error} = 0.0661 \)[/tex]
The 98% confidence interval for the difference in proportions [tex]\( (p_1 - p_2) \)[/tex] is given by:
[tex]\[ (\text{lower bound}, \text{upper bound}) = (0.1299, 0.2621) \][/tex]
Thus, the 98% confidence interval is:
[tex]\[ (0.130, 0.262) \][/tex]
(Rounded to three decimal places as needed).
### Alternative Hypothesis
Other pairs of hypotheses such as [tex]\( H_0: P_1 = P_2 \)[/tex] and [tex]\( H_1: P_1 \neq P_2 \)[/tex] (two-tailed tests) or reversed hypotheses making [tex]\(H_0\)[/tex] indicate [tex]\( P_1 \geq P_2 \)[/tex] would be misaligned with the wording of the original one-tailed test claiming [tex]\( P_1 > P_2\)[/tex]. Therefore, the appropriate hypotheses are:
[tex]\[ H_0: P_1 \leq P_2 \\ H_1: P_1 > P_2 \][/tex]
Conclusively, based on statistical testing, we reject the null hypothesis and present a confidence interval proving a statistically significant difference between the proportions. This proves a higher proportion of older individuals over 65 dream in black and white compared to younger individuals under 25.
