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Chapter 23: Problem 27

Personal Data and Privacy. The survey in Exercise \(23.26\) also looked at possible differences in the proportions of White non- Hispanics and Black non-Hispanics who were concerned about how much information their friends and family might know about them. Of the 2887 White nonHispanics in the survey, 1010 said they were concerned. Of the 445 Black non-Hispanics in the survey, 271 said they were concerned. Is there evidence of a difference between the proportions of White non-Hispanics and Black non- Hispanics who were concerned about how much information their friends and family might know about them?

Short Answer

Expert verified
Yes, there is evidence of a difference if the p-value < 0.05.

Step by step solution

01

Define Hypotheses

Let's define the null and alternative hypotheses. The null hypothesis \( H_0 \) assumes that there is no difference in the proportions. Mathematically, \( p_1 = p_2 \), where \( p_1 \) is the proportion of concerned White non-Hispanics and \( p_2 \) is the proportion of concerned Black non-Hispanics. The alternative hypothesis \( H_a \) assumes that there is a difference: \( p_1 eq p_2 \).
02

Calculate Sample Proportions

Calculate the sample proportions. For White non-Hispanics, the proportion \( \hat{p}_1 = \frac{1010}{2887} \). For Black non-Hispanics, the proportion \( \hat{p}_2 = \frac{271}{445} \).
03

Calculate Pooled Proportion

Calculate the pooled proportion \( \hat{p} \) which combines both groups. It is calculated as \( \hat{p} = \frac{1010 + 271}{2887 + 445} \).
04

Compute Standard Error

Using the pooled proportion, calculate the standard error (SE) of the difference in proportions \( SE = \sqrt{ \hat{p} (1 - \hat{p}) \left( \frac{1}{2887} + \frac{1}{445} \right) } \).
05

Calculate Z-Score

The Z-score is calculated to find the difference between the two proportions in terms of standard errors. Compute \( Z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \).
06

Find P-value

Using the Z-score from Step 5, find the p-value to determine the statistical significance of the result. A Z-test table or statistical software can be used for this.
07

Conclusion from P-value

Compare the p-value to the significance level, typically \( \alpha = 0.05 \). If the p-value is less than \( \alpha \), we reject the null hypothesis, meaning there is evidence of a difference. If not, we fail to reject the null hypothesis.

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Key Concepts

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

Proportions
In statistics, the concept of proportions is crucial when comparing different groups. A proportion represents a part of a whole and is often expressed as a fraction or percentage. In the context of hypothesis testing, we use proportions to understand the characteristics of particular groups within a set population.
In this exercise, we are dealing with two specific proportions:
  • For White non-Hispanics, the proportion of those concerned about privacy is calculated as \( \hat{p}_1 = \frac{1010}{2887} \).
  • For Black non-Hispanics, the proportion is \( \hat{p}_2 = \frac{271}{445} \).
These proportions help identify if there is a meaningful difference in concern between the two ethnic groups regarding their privacy.
Null Hypothesis
The null hypothesis, symbolized as \( H_0 \), is a statement that assumes there is no effect or no difference in the context of the problem. In hypothesis testing involving proportions, this hypothesis is designed to be tested against the data.
In our exercise:
  • The null hypothesis states that the proportions of White non-Hispanics and Black non-Hispanics who are concerned about privacy are equal.
  • This is formulated mathematically as \( p_1 = p_2 \).
The null hypothesis serves as a starting point to test whether there is indeed a statistically significant difference between the two groups we are comparing.
Alternative Hypothesis
The alternative hypothesis, denoted as \( H_a \), is the opposite of the null hypothesis. It suggests that there is an effect or a difference that contrasts with the null hypothesis. This hypothesis is what researchers typically aim to support.
In this specific problem:
  • The alternative hypothesis proposes that the proportions of the two groups are not equal, indicating a difference in concerns over privacy.
  • This can be expressed as \( p_1 eq p_2 \).
The alternative hypothesis is what would lead to rejecting the null hypothesis, possibly leading us to a meaningful conclusion about changes between two proportions.
Z-Test
A Z-test is a statistical test that determines whether there is a significant difference between the means or proportions of two groups. It's a useful tool when dealing with large sample sizes or known standard deviations.
In this exercise, a Z-test helps determine if there's a real difference between the proportions of concerned individuals in both ethnic groups.
  • We compute the Z-score, which tells us the number of standard errors the observed difference is away from the hypothesized difference (if the null hypothesis were true).
  • The formula for Z-score is \( Z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \), where SE is the standard error computed from our data.
The Z-test culminates in a Z-score, which is then used to find the p-value, indicating how likely our data would appear under the null hypothesis.
Statistical Significance
Statistical significance is a key concept in hypothesis testing. It helps us determine if the results observed in our study are due to chance or if they reflect a true difference in the population.
This is achieved by comparing the p-value to a predefined significance level, often \( \alpha = 0.05 \).
  • The p-value is derived from the Z-score and represents the probability of observing the obtained result, or one more extreme, assuming the null hypothesis is true.
  • If the p-value is less than \( \alpha \), we reject the null hypothesis, which means the results are potentially statistically significant.
  • If the p-value is greater or equal to \( \alpha \), we fail to reject the null hypothesis.
This step is crucial in making informed decisions about whether the differences between groups are meaningful.

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