Math Problem Statement

A study was conducted to determine the proportion of people who dream in black and white instead of color. Among 306 people over the age of 55, 68 dream in black and white, and among 298 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 (p 1) is greater than the proportion of those under 25 (p 2). Question content area bottom Part 1 Find the p-value. Your answer should include 4 digits after the decimal point. p-value = enter your response here State the conclusion. A. Reject the null hypothesis. There is not enough evidence to support the claim that the proportion of people over 55 who dream in black and white is greater than that of people under 25. B. Fail to reject the null hypothesis. There is not enough evidence to support the claim that the proportion of people over 55 who dream in black and white is greater than that of people under 25. C. Fail to 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 that of people under 25. D. 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 that of people under 25.

Solution

from statsmodels.stats.proportion import proportions_ztest

Data

count = [68, 13] # Number of successes for each group nobs = [306, 298] # Total number of observations for each group

Test for the difference in proportions (one-tailed test)

stat, p_value = proportions_ztest(count, nobs, alternative='larger')

p_value5.982533052043423e-115.982533052043423e-11successThe $p$ -value is approximately $0.0000$ (rounded to four decimal places). This is significantly smaller than the significance level ( $\alpha = 0.01$ ).

Conclusion:

D. 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 that of people under 25.

Would you like a detailed explanation or additional insights? Here are five related questions:

How is the $z$ -statistic calculated in this hypothesis test?
What are the assumptions of the proportions $z$ -test used here?
Can you explain how the alternative hypothesis affects the interpretation of the $p$ -value?
How would the conclusion differ if the $p$ -value were larger than $0.01$ ?
What is the practical significance of the result in this context?

Tip: Always compare the $p$ -value with the significance level to make clear statistical decisions.

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Math Problem Analysis

Mathematical Concepts

Hypothesis Testing
Proportions
Z-Test for Proportions

Formulas

z = (p1 - p2) / sqrt( (p1(1-p1)/n1) + (p2(1-p2)/n2) )

Theorems

Z-test for differences in proportions

Suitable Grade Level

College/University

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