Step 1 :Define the null hypothesis (H0) as the proportion of people over 55 who dream in black and white is equal to the proportion of those under 25. The alternative hypothesis (H1) is that the proportion of people over 55 who dream in black and white is greater than the proportion of those under 25.
Step 2 :Given the P-value is 0, which is less than the significance level of 0.05, we reject the null hypothesis. This suggests that 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 3 :Calculate the sample proportions and their difference: For people over 55: \(p1 = \frac{79}{304} = 0.260\), For people under 25: \(p2 = \frac{12}{299} = 0.040\), Difference: \(p1 - p2 = 0.260 - 0.040 = 0.220\)
Step 4 :Calculate the standard error (SE) of the difference: \(SE = \sqrt{\left(\frac{p1*(1-p1)}{304}\right) + \left(\frac{p2*(1-p2)}{299}\right)} = \sqrt{\left(\frac{0.260*0.740}{304}\right) + \left(\frac{0.040*0.960}{299}\right)} = 0.027\)
Step 5 :Construct a 90% confidence interval for the difference in proportions: \(p1 - p2 \pm Z*SE\), where Z is the Z-score for a 90% confidence interval, which is 1.645. Lower limit: \(0.220 - 1.645*0.027 = 0.176\), Upper limit: \(0.220 + 1.645*0.027 = 0.264\)
Step 6 :\(\boxed{\text{Therefore, the 90% confidence interval for the difference in proportions is } 0.176 < (p1 - p2) < 0.264. \text{ This interval does not contain 0, which further supports the conclusion that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25.}}\)