Question

A study was conducted to determine the proportion of people who dream in black and white instead of color. Among 309 people over the age of​ 55, 63 dream in black and​ white, and among 315 people under the age of​ 25,13 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)

(1). Identify the test statistics.

(2) What is the P value

(b). Test the claim by constructing an appropriate confidence interval.

c. An explanation given for the results is that those over the age of 55 grew up exposed to media that was mostly displayed in black and white. Can these results be used to verify that​ explanation?

Answer 1

(a) Basic data:

= Sample proportion of men over the age of 55 who see dreams in Black and White = 63/309 = 0.2039

= Sample proportion of men below the age of 55 who see dreams in Black and White =13/315 = 0.0413

= Overall Proportion = (63 + 13) / (309 + 315) = 0.1218

1 - = 0.8782

= 0.01

The Hypothesis:

H0: p1 = p2

Ha: p1 > p2

The Test Statistic:

The p Value: The p value (Right Tail) for Z = 6.15, is; p value = 0.000

The Conclusion: The p value is lesser than the significance level of = 0.01, so reject the null hypothesis. There is sufficient evidence to conclude that the sample proportion of men over the age of 55 who see dreams in Black and White is greater than the sample proportion of men below the age of 25 who see dreams in Black and White.

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(b) For the 99% Confidence interval

= 0.2039 and 1 - = 0.7961

= 0.0413 and 1 - = 0.9587

The Zcritical (2 tail) for = 0.01, is 2.576

The Confidence Interval is given by (- ) ME, where

(- ) = 0.2039 – 0.0413 = 0.1626

The Lower Limit = 0.1626 - 0.0658 = 0.0968

The Upper Limit = 0.1626 + 0.0658 = 0.2284

The 99% Confidence Interval is 0.0968 < p1 - p2 < 0.2284

Since both value of the CI are positive, we can conclude that p1 is greater than p2.

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(c) No, these results cannot be used to give explanations for causations. It only gives the fact that we are 99% confident that the true population difference in proportions lie between the limits of 0.0968 and 0.2284.

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