Dallas, Texas, and Stafford, Texas, are numbered 57 and 58 on the top 100 most diverse cities in the U.S.  A random sample of 300 residents from each city were surveyed.  The results are listed below.



**Dallas (X1)**

201 of 300 surveyed were non-white

**Stafford (X2)**

180 of 300 surveyed were non-white


At the 0.01 level of significance, is there enough evidence to conclude that Dallas has a higher proportion of non-white residents than Stafford?

a) Which are the null and alternative hypotheses?


b) What is the test statistic?  Round to 2 decimal places.

Incorrect

c) What is the p-value?  Round to 4 decimal places.


To determine if there is enough evidence to conclude that Dallas has a higher proportion of non-white residents than Stafford, we will conduct a hypothesis test for the difference in proportions.

### a) Null and Alternative Hypotheses

The hypotheses for this test are:

- **Null Hypothesis (\(H_0\))**: The proportion of non-white residents in Dallas is less than or equal to the proportion of non-white residents in Stafford. \( p_1 \leq p_2 \)
- **Alternative Hypothesis (\(H_a\))**: The proportion of non-white residents in Dallas is greater than the proportion of non-white residents in Stafford. \( p_1 > p_2 \)

### b) Test Statistic

First, we need to calculate the sample proportions:

\[ \hat{p}_1 = \frac{X_1}{n_1} = \frac{201}{300} = 0.67 \]
\[ \hat{p}_2 = \frac{X_2}{n_2} = \frac{180}{300} = 0.60 \]

The combined proportion \( \hat{p} \) is calculated as:

\[ \hat{p} = \frac{X_1 + X_2}{n_1 + n_2} = \frac{201 + 180}{300 + 300} = \frac{381}{600} = 0.635 \]

Next, we calculate the standard error (SE) for the difference in proportions:

\[ SE = \sqrt{\hat{p} (1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \]

\[ SE = \sqrt{0.635 (1 - 0.635) \left( \frac{1}{300} + \frac{1}{300} \right)} \]

\[ SE = \sqrt{0.635 \times 0.365 \times \frac{2}{300}} \]

\[ SE = \sqrt{0.635 \times 0.365 \times 0.00667} \]

\[ SE = \sqrt{0.001550} \]

\[ SE \approx 0.0394 \]

Now, we calculate the test statistic \( z \):

\[ z = \frac{\hat{p}_1 - \hat{p}_2}{SE} = \frac{0.67 - 0.60}{0.0394} = \frac{0.07}{0.0394} \approx 1.78 \]

### c) p-value

The p-value for a one-tailed test with \( z = 1.78 \) can be found using standard normal distribution tables or a statistical calculator. 

For \( z = 1.78 \), the p-value is approximately \( 0.0375 \).

### Decision

Given the significance level \( \alpha = 0.01 \), we compare the p-value with \( \alpha \):

Since \( 0.0375 > 0.01 \), we **fail to reject the null hypothesis**.

### Conclusion

There is not enough evidence at the 0.01 level of significance to conclude that Dallas has a higher proportion of non-white residents than Stafford.

To summarize:

1. **a) Null and Alternative Hypotheses:**
   - \( H_0: p_1 \leq p_2 \)
   - \( H_a: p_1 > p_2 \)

2. **b) Test Statistic:**
   - \( z = 1.78 \)

3. **c) p-value:**
   - \( p \approx 0.0375 \)

4. **Decision:**
   - Fail to reject the null hypothesis.

5. **Conclusion:**
   - There is not enough evidence to conclude that Dallas has a higher proportion of non-white residents than Stafford at the 0.01 level of significance.

Do you want details or have any questions?

### Additional Questions:

1. What assumptions are made in performing this test for the difference in proportions?
2. How would the conclusions change if the significance level was \( \alpha = 0.05 \)?
3. What would be the effect size for this test?
4. How do you interpret the standard error in this context?
5. What other statistical tests could be used to analyze this data?
6. How would you construct a confidence interval for the difference in proportions?
7. What could be some reasons for the observed proportions of non-white residents in these cities?
8. How might the results differ with a larger sample size?

**Tip:** When conducting hypothesis tests, always verify that the sample size is sufficiently large to justify the use of normal approximation.

Explore a hypothesis test comparing the proportions of non-white residents in Dallas and Stafford, Texas, using statistical methods. Determine if there is sufficient evidence to conclude differences at the 0.01 significance level.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation