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, 11 dream in black and white. use a 0.05 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. complete parts (a) through (c) below.

a. test the claim using a hypothesis test

consider the first sample to be the sample of people over the age of 55 and the second sample to be the sample of people under the age of 25. what are the null and alternative hypotheses for the hypothesis test?

a. ( h_0: p_1
eq p_2 ) ( h_1: p_1 = p_2 )

b. ( h_0: p_1 = p_2 ) ( h_1: p_1
eq p_2 )

c. ( h_0: p_1 leq p_2 ) ( h_1: p_1
eq p_2 )

d. ( h_0: p_1 geq p_2 ) ( h_1: p_1
eq p_2 )

e. ( h_0: p_1 = p_2 ) ( h_1: p_1 > p_2 )

f. ( h_0: p_1 = p_2 ) ( h_1: p_1 < p_2 )

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, 11 dream in black and white. use a 0.05 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. complete parts (a) through (c) below.

a. test the claim using a hypothesis test

consider the first sample to be the sample of people over the age of 55 and the second sample to be the sample of people under the age of 25. what are the null and alternative hypotheses for the hypothesis test?

a. ( h_0: p_1
eq p_2 ) ( h_1: p_1 = p_2 )

b. ( h_0: p_1 = p_2 ) ( h_1: p_1
eq p_2 )

c. ( h_0: p_1 leq p_2 ) ( h_1: p_1
eq p_2 )

d. ( h_0: p_1 geq p_2 ) ( h_1: p_1
eq p_2 )

e. ( h_0: p_1 = p_2 ) ( h_1: p_1 > p_2 )

f. ( h_0: p_1 = p_2 ) ( h_1: p_1 < p_2 )

Accepted Answer

# Brief Explanations: The claim is that the proportion of people over 55 ( $ p_1 $) who dream in black and white is greater than the proportion for those under 25 ( $ p_2 $). The null hypothesis ($ H_0 $) is a statement of equality or no difference, and the alternative hypothesis ($ H_1 $) reflects the claim. So $ H_0: p_1 \leq p_2 $ and $ H_1: p_1 > p_2 $ doesn't match options, wait no—wait the first sample is over 55 ($ p_1 $), second under 25 ($ p_2 $). The claim is $ p_1 > p_2 $? Wait no, the problem says "the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25". Wait no, wait the options: Let's re-express. The null hypothesis is the opposite of the claim or a statement of no difference. The claim is $ p_1 > p_2 $ (over 55 proportion > under 25). So null hypothesis $ H_0: p_1 \leq p_2 $, alternative $ H_1: p_1 > p_2 $? Wait no, looking at options: Option C is $ H_0: p_1 \leq p_2 $, $ H_1: p_1 eq p_2 $? No. Wait the problem says "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". Wait maybe I misread. Wait the first sample: over 55, 288 people, 64 dream in B&W. Second sample: under 25, 299 people, 11 dream? Wait no, the text: "Among 288 people over the age of 55, 64 dream in black and white, and among 299 people under the age of 25, 11..." Wait no, maybe the claim is $ p_1 > p_2 $, where $ p_1 $ is over 55, $ p_2 $ under 25. So null hypothesis $ H_0: p_1 \leq p_2 $, alternative $ H_1: p_1 > p_2 $. But looking at options, Option C is $ H_0: p_1 \leq p_2 $, $ H_1: p_1 eq p_2 $? No. Wait no, maybe the claim is that the proportion for over 55 is greater, so the alternative is $ p_1 > p_2 $, null is $ p_1 \leq p_2 $. Wait the options: Let's list options: A. $ H_0: p_1 eq p_2 $, $ H_1: p_1 = p_2 $ → no, null and alternative reversed. B. $ H_0: p_1 = p_2 $, $ H_1: p_1 eq p_2 $ → two-tailed, not matching claim. C. $ H_0: p_1 \leq p_2 $, $ H_1: p_1 eq p_2 $ → no, alternative should be $ > $. D. $ H_0: p_1 \geq p_2 $, $ H_1: p_1 eq p_2 $ → no. E. $ H_0: p_1 = p_2 $, $ H_1: p_1 > p_2 $ → no, null is equality, but claim is $ p_1 > p_2 $, so null should be $ p_1 \leq p_2 $. Wait maybe I made a mistake. Wait the problem says "the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25". So $ p_1 > p_2 $, where $ p_1 $ is over 55, $ p_2 $ under 25. So null hypothesis is $ H_0: p_1 \leq p_2 $, alternative $ H_1: p_1 > p_2 $. But none of the options have that? Wait no, looking at the options again: Wait the options: A. $ H_0: p_1 eq p_2 $, $ H_1: p_1 = p_2 $ B. $ H_0: p_1 = p_2 $, $ H_1: p_1 eq p_2 $ C. $ H_0: p_1 \leq p_2 $, $ H_1: p_1 eq p_2 $ D. $ H_0: p_1 \geq p_2 $, $ H_1: p_1 eq p_2 $ E. $ H_0: p_1 = p_2 $, $ H_1: p_1 > p_2 $ F. $ H_0: p_1 = p_2 $, $ H_1: p_1 < p_2 $ Wait no, the original options as per image: A. $ H_0: p_1 eq p_2 $, $ H_1: p_1 = p_2 $ B. $ H_0: p_1 = p_2 $, $ H_1: p_1 eq p_2 $ C. $ H_0: p_1 \leq p_2 $, $ H_1: p_1 eq p_2 $ D. $ H_0: p_1 \geq p_2 $, $ H_1: p_1 eq p_2 $ E. $ H_0: p_1 = p_2 $, $ H_1: p_1 > p_2 $ F. $ H_0: p_1 = p_2 $, $ H_1: p_1 < p_2 $ Wait no, the user's image: Let's re-express the options from the image: A. $ H_0: p_1 eq p_2 $, $ H_1: p_1 = p_2 $ B. $ H_0: p_1 = p_2 $, $ H_1: p_1 eq p_2 $ C. $ H_0: p_1 \leq p_2 $, $ H_1: p_1 eq p_2 $ D. $ H_0: p_1 \geq p_2 $, $ H_1: p_1 eq p_2 $ E. $ H_0: p_1 = p_2 $, $ H_1: p_1 > p_2 $ F. $ H_0: p_1 = p_2 $, $ H_1: p_1 < p_2 $ Wait the claim is "the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25". So $ p_1 > p_2 $ (where $ p_1 $ is over 55, $ p_2 $ under 25). So the null hypothesis is the opposite or a statement of equality? Wait, in hypothesis testing, the null hypothesis is a statement of no difference or the opposite of the claim, and the alternative is the claim. So if the claim is $ p_1 > p_2 $, then $ H_0: p_1 \leq p_2 $, $ H_1: p_1 > p_2 $. But none of the options have that. Wait maybe I misread the problem. Wait the problem says "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". Wait maybe the first sample is under 25 and second over 55? No, the problem says "the first sample to be the sample of people over the age of 55 and the second sample to be the sample of people under the age of 25". So $ p_1 $ is over 55, $ p_2 $ under 25. The claim is $ p_1 > p_2 $. So the alternative hypothesis is $ H_1: p_1 > p_2 $, and the null hypothesis is $ H_0: p_1 \leq p_2 $. But looking at the options, Option E is $ H_0: p_1 = p_2 $, $ H_1: p_1 > p_2 $. Wait, maybe the problem's claim is misinterpreted. Wait, maybe the claim is that the proportion for over 55 is greater, so the null is $ p_1 = p_2 $ (a common null for proportion comparison when testing a difference), and alternative is $ p_1 > p_2 $. So Option E: $ H_0: p_1 = p_2 $, $ H_1: p_1 > p_2 $. Wait no, the null hypothesis should be the opposite of the claim or a statement of equality. Wait, in two-proportion z-test, when testing if $ p_1 > p_2 $, the null is $ p_1 = p_2 $ (or $ p_1 \leq p_2 $), and alternative is $ p_1 > p_2 $. But the options: Option E is $ H_0: p_1 = p_2 $, $ H_1: p_1 > p_2 $. That matches the claim (since if $ H_0 $ is $ p_1 = p_2 $, and $ H_1 $ is $ p_1 > p_2 $, that's a right-tailed test for the claim that $ p_1 > p_2 $). Wait, maybe the problem's null is $ p_1 = p_2 $ (a common null hypothesis for comparing two proportions, assuming equality under null). So the correct option is E? Wait no, let's check again. Wait the options: Option E: $ H_0: p_1 = p_2 $, $ H_1: p_1 > p_2 $ Yes, because the claim is that $ p_1 $ (over 55) is greater than $ p_2 $ (under 25), so alternative is $ p_1 > p_2 $, null is $ p_1 = p_2 $ (or $ p_1 \leq p_2 $, but in options, E is $ H_0: p_1 = p_2 $, $ H_1: p_1 > p_2 $). Wait, maybe the problem's null is set as equality. So the correct option is E? Wait no, looking at the options again, maybe I made a mistake. Wait the original problem's options: A. $ H_0: p_1 eq p_2 $, $ H_1: p_1 = p_2 $ → reversed, invalid. B. $ H_0: p_1 = p_2 $, $ H_1: p_1 eq p_2 $ → two-tailed, not matching claim. C. $ H_0: p_1 \leq p_2 $, $ H_1: p_1 eq p_2 $ → two-tailed, null is $ p_1 \leq p_2 $, alternative two-tailed, no. D. $ H_0: p_1 \geq p_2 $, $ H_1: p_1 eq p_2 $ → two-tailed, null $ p_1 \geq p_2 $, no. E. $ H_0: p_1 = p_2 $, $ H_1: p_1 > p_2 $ → one-tailed, alternative is claim, null is equality. F. $ H_0: p_1 = p_2 $, $ H_1: p_1 < p_2 $ → opposite of claim. So the correct option is E? Wait no, the claim is that over 55 proportion is greater than under 25, so $ p_1 > p_2 $, so alternative is $ p_1 > p_2 $, null is $ p_1 = p_2 $ (or $ p_1 \leq p_2 $). So Option E is correct. # Answer: E. $ H_0: p_1 = p_2 $, $ H_1: p_1 > p_2 $

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