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What鈥檚 wrong? 鈥淲ould you marry a person from a lower social class than your own?鈥� Researchers asked this question of a random sample of 385 black, never- married students at two historically black colleges in the South. Of the 149 men in the sample, 91 said 鈥淵es.鈥� Among the 236 women, 117 said 鈥淵es.鈥漒(^{14}\) Is there reason to think that different proportions of men and women in this student population would be willing to marry beneath their class? Holly carried out the significance test shown below to answer this question. Unfortunately, she made some mistakes along the way. Identify as many mistakes as you can, and tell how to correct each one. State: I want to perform a test of $$H_{0} : p_{1}=\rho_{2}$$ $$H_{a} : p_{1} \neq p_{2}$$ at the 95% confidence level. Plan: If conditions are met, I鈥檒l do a one-sample \(z\) test for comparing two proportions. \(\cdot\) Random The data came from a random sample of 385 black, never-married students. \(\cdot\) Normal One student's answer to the question should have no relationship to another student's answer. \(\cdot\) Independent The counts of successes and falures in the two groups - \(91,58,117,\) and \(119-\) are all at least 10 . Do: From the data, \(\hat{p}_{1}=\frac{91}{149}=0.61\) and \(\hat{p}_{2}=\frac{117}{236}=0.46\) \(\bullet\) Test statistic $$z=\frac{(0.61-0.46)-0}{\sqrt{\frac{0.61(0.39)}{149}+\frac{0.46(0.54)}{236}}}=2.91$$ \(\cdot P\) value From Table \(A, P(z \geq 2.91)=1-0.9982=\) 0.0018 . Conclude: The P-value, \(0.0018,\) is less than \(0.05,\) so I'll reject the null hypothesis. This proves that a higher proportion of men than women are willing to marry someone from a social class lower than their own.

Step by step solution

01

State the Hypotheses

The null hypothesis should be \( H_0 : p_1 = p_2 \), which states that there is no difference between the proportions of men \( p_1 \) and women \( p_2 \) willing to marry beneath their social class. The alternative hypothesis should be \( H_a : p_1 eq p_2 \), indicating a difference in proportions.

02

Planning Identification Error

The plan mentions using a 'one-sample \( z \)-test for comparing two proportions.' This is incorrect because a two-sample \( z \)-test should be used when comparing two independent sample proportions.

03

Check Conditions Correctly

The 'Normal' condition states that responses should be independent, but this condition actually refers to independence between the two groups: the sample of men and the sample of women. Check for independence within and between groups, not just individual responses.

04

Test Statistic Formula Error

The test statistic formula uses the difference \((0.61 - 0.46)\) directly. It should use the pooled sample proportion \( \hat{p} = \frac{91 + 117}{149+236} \) for calculating the standard error for the two-proportion \( z \)-test.

05

P-Value Calculation

The researcher calculated \( P(z \geq 2.91) \). However, because a two-tailed test for comparing two proportions should be used, calculate \( 2P(z > 2.91) \) to account for both tails.

06

Conclusion Clarification

The statement "proves that a higher proportion of men than women" is misleading since statistical tests do not prove hypotheses. The correct conclusion is that there is statistically significant evidence to suggest a difference in proportions, without implying one group is definitively higher based solely on this test.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Two-proportion Z-test

When comparing the proportions of two independent groups, such as the willingness of men and women to marry beneath their social class, we use a two-proportion Z-test. This test evaluates whether there is a statistically significant difference between the proportions of the two groups. In Holly's analysis, a mistake was noted where she referred to using a "one-sample Z-test." However, since we are comparing two distinct groups (men and women), the correct statistical method is the two-proportion Z-test.

To perform a two-proportion Z-test, we calculate the difference between the sample proportions and assess this difference's significance using a Z-statistic. It's essential to ensure the conditions of normality and independence are met before proceeding. If these conditions hold, we can calculate the Z-statistic and derive the p-value to help determine the statistical significance of the observed difference.

Statistical Significance

Statistical significance is a concept used to infer whether the results observed in a study, like the proportions of men and women willing to marry beneath their class, are likely due to random chance or reflect a real difference in the population.

In hypothesis testing, we use a predetermined significance level, typically 0.05, to assess whether to reject the null hypothesis. Holly's test resulted in a p-value of 0.0018. This p-value was compared against the 0.05 significance level. Because 0.0018 is less than 0.05, it suggests that the probability of observing such a difference (or more extreme) in the sample, assuming the null hypothesis is true, is very low. Thus, we consider the result statistically significant, supporting the possibility of a genuine difference in the population proportions.

Null and Alternative Hypotheses

In the context of Holly's test, she failed to clearly articulate the null and alternative hypotheses correctly. When testing if different proportions of men and women are willing to marry outside their social class, the null hypothesis \( H_0 \) should state that there is no difference in proportions: \( p_1 = p_2 \). This suggests that the men and women exhibit a similar willingness to marry down.

The alternative hypothesis \( H_a \) indicates a perceived difference, formulated as \( p_1 eq p_2 \), which implies a possible disparity in willingness between the two groups. Clear definition of these hypotheses is critical, as they guide the analysis and interpretation of statistical results.

Error Identification

Analyzing Holly's approach reveals several errors in the application of the hypothesis test. Firstly, her mention of "one-sample Z-test" was incorrect; a two-proportion Z-test was necessary. This distinction is essential for accurate data analysis.

She also miscalculated by not considering the "pooled proportion" in the test statistic formula. The correct approach involves using the combined proportion to determine the standard error correctly. Additionally, conditions for independence were not appropriately verified, which could affect the validity of results.

It's important to note that in a two-tailed test, both sides of the distribution should be considered. Holly only looked at one tail when calculating the p-value, an oversight corrected by considering both tails, something critical for correct hypothesis testing. Lastly, her conclusion overstated certainty. It is essential to communicate findings as supporting evidence rather than definitive proof, given the probabilistic nature of statistical inference.