Question

A study was conducted to determine the proportion of people who dream in black and white instead of color. Among319 people over the age of​ 55, 67 dream in black and​ white, and among 307 people under the age of​ 25, 16 dream in black and white. Use a 0.01 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?

Identify the test statistic.

z=

​(Round to two decimal places as​ needed.)

Identify the​ P-value.

​P-value=

​(Round to three decimal places as​ needed.)

What is the conclusion based on the hypothesis​ test?

The​ P-value is less than or greater than the significance level of alpha=0.01​, so, reject or fail to reject

the null hypothesis. There is sufficient or insufficient 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 98​% confidence interval is ___<( p 1-p2) ?< __.

​(Round to three decimal places as​ needed.)

What is the conclusion based on the confidence​ interval?

Because the confidence interval limits include or do not include ​0, it appears that the two proportions are equal. or not equal. Because the confidence interval limits include positive and negative or only negative or only positive

​values, it appears that the proportion of people over 55 who dream in black and white is not significantly different from or lesser than or 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. 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.

C.Yes. The results can be used to verify the given explanation because the difference in proportions is statistically significant.

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 cannot be used to verify the cause of such a difference.

Answer 1

(A) We have 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

Using TI 84 calculator

STAT>TESTS>2-PropZtest

x1 = 67, n1 = 319

x2 = 16, n2 = 307

p1 > p2

ENTER

z = 5.82

p value = 0.000

The​ P-value is less than the significance level of alpha=0.01​, 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)

Using TI 84 calculator

STAT>TESTS>2-PropZInt

x1 = 67, n1 = 319

x2 = 16, n2 = 307

c-level = 0.98

ENTER

confidence interval does not include 0, so we can write

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 significantly greater than the proportion for those under 25.

(C) since p value is less than significance level and the confidence interval also supported the hypothesis test

option C