Question

identify the test statistic
z = 6.72
(round to two decimal places as needed.)
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 0, it appears that the two proportions are
because the confidence interval limits include values, it appears that the proportion of people over 55 who dream in black and white is
the proportion for those under 25.

Explanation:

Step1: Analyze Confidence Interval

The 90% confidence interval for \((p_1 - p_2)\) is \(0.141 < (p_1 - p_2) < 0.230\). We check if the interval includes 0 (for equal proportions) or shows a direction.

Step2: Interpret Confidence Interval

Since the confidence interval limits (\(0.141\) and \(0.230\)) are both positive, it means \(p_1 - p_2>0\) (so \(p_1 > p_2\)). Also, the interval does not include 0, so the proportions are not equal. The proportion for people over 55 (let's say \(p_1\)) and under 25 (\(p_2\)): the interval for \(p_1 - p_2\) is positive, so \(p_1>p_2\). So, "Because the confidence interval limits are \(0.141\) and \(0.230\), it appears that the two proportions are different. 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."

Answer:

Because the confidence interval limits are \(0.141\) and \(0.230\), it appears that the two proportions are different. 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. (Filling the blanks: different; greater than)