Question

a study was conducted to determine the proportion of people who dream in black and white instead of color. among 288 people over the age of 55, 64 dream in black and white, and among 299 people under the age of 25, complete parts (a) through (c) below. identify the p - value. p - value = 0.000 (round to three decimal places as needed) what is the conclusion based on the hypothesis test? the p - value is less than the significance level of α = 0.05, so 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. b. test the claim by constructing an appropriate confidence interval. the 90% confidence interval is 0.141 < (p₁ - p₂) < 0.230 (round to three decimal places as needed) what is the conclusion based on the confidence interval? because the confidence interval limits do not include 0, it appears that the two proportions are not equal. because the confidence interval limits include only positive values, it appears that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25. c. an explanation for the results is that those over the age of 55 grew up exposed to media that was displayed in black and white. can these results be used to verify that explanation? a. yes. the results can be used to verify the given explanation because the difference in proportions is practically significant. b. yes. the results can be used to verify the given explanation because the difference in proportions is statistically significant. c. no. the results speak to a possible difference between the proportions of people over 55 and under 25 who dream in black and white, but the results cannot be used to verify the cause of such a difference. d. no. the results speak to a possible difference between the proportions of people over 55 and under 25 who dream in black and white, but the results are not statistically significant enough to verify the cause of such a difference.

Explanation:

Part (a)

Step 1: Recall P - value interpretation

The P - value is given as \(0.000\). For a hypothesis test with significance level \(\alpha = 0.05\), we compare the P - value with \(\alpha\). If \(P - value<\alpha\), we reject the null hypothesis. Here, \(0.000 < 0.05\).

Step 2: Conclusion from hypothesis test

Since the P - value (\(0.000\)) is less than the significance level \(\alpha=0.05\), we reject the null hypothesis. This means 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.

Step 1: Recall confidence interval interpretation

The 90% confidence interval for \((p_1 - p_2)\) is \(0.141<(p_1 - p_2)<0.230\). We check if the interval contains \(0\). If the confidence interval for the difference of proportions does not contain \(0\), it suggests that the two proportions are not equal. Also, since all values in the interval are positive, it appears that the proportion for those over 55 ( \(p_1\)) is greater than the proportion for those under 25 ( \(p_2\)).

Step 2: Conclusion from confidence interval

Because the confidence interval limits do not include \(0\), it appears that the two proportions are not equal. Because the confidence interval limits include only positive values, it appears that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25.

Step 1: Understand the nature of statistical results

Statistical hypothesis tests and confidence intervals can tell us about the existence of a difference between two groups (in this case, the proportions of people over 55 and under 25 who dream in black and white). However, they cannot establish causation. Just because we find a difference in proportions does not mean we can say that the exposure to black - and - white media (the proposed explanation) is the cause of the difference in dreaming in black and white.

Step 2: Evaluate the options

• Option A: Says the results can verify the explanation. But we can't verify causation from this study, so A is wrong.
• Option B: Says the results can verify the explanation. We can't verify causation, so B is wrong.
• Option C: Says the results speak to a possible difference but can't verify the cause. This is correct because we have a difference (from parts a and b) but we can't say the media exposure is the cause.
• Option D: Says the results are not statistically significant enough, but from parts a and b we have significant results, so D is wrong.

Answer:

The P - value is \(0.000\). The conclusion is: The P - value is less than the significance level of \(\alpha = 0.05\), so 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.

Part (b)