The number of AIDS cases reported for Santa Clara County, California is broken down by race in the table below. Source: "HIV/AIDS Epidemiology Santa Clara County", Santa Clara County Public Health Department, May 2011. Cases 2267 1182 464 Asian/Pacific Islander 227 41.40 Race White Hispanic Black Total

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### Distribution of AIDS Cases in Santa Clara County

The number of AIDS cases reported for Santa Clara County, California is broken down by race in the table below. **Source:** "HIV/AIDS Epidemiology Santa Clara County," Santa Clara County Public Health Department, May 2011.

| Race                      | Cases |
|---------------------------|-------|
| White                     | 2267  |
| Hispanic                  | 1182  |
| Black                     | 464   |
| Asian/Pacific Islander    | 227   |
| **Total**                 | **4140** |

---

**Directions:** Conduct a chi-square test for goodness-of-fit to determine whether or not the occurrence of AIDS cases is consistent with the race distribution of Santa Clara County.

1. **Choose the correct null and alternative hypotheses.**

   - \( H_0 \): The distribution of AIDS cases is different from the race distribution in Santa Clara County.
   - \( H_a \): The distribution of AIDS cases is consistent with the race distribution in Santa Clara County.

   - \( H_0 \): The distribution of AIDS cases is consistent with the race distribution in Santa Clara County.
   - \( H_a \): The distribution of AIDS cases is different from the race distribution in Santa Clara County.

2. **Compute the test statistic.**

   The population distribution of Santa Clara County by race is provided in the table below. Use these percentages to compute the expected number of cases for each racial group. Round each of the expected counts to 2 decimal places.

   | Race                     | Proportion | Expected cases |
   |--------------------------|------------|----------------|
   | White                    | 0.455      |                |
   | Hispanic                 | 0.252      |                |
   | Black                    | 0.025      |                |
   | Asian/Pacific Islander   | 0.268      |                |
   | **Total**                | **1**      |                |

### Instructions for Analysis:
- Use the given proportions to calculate the expected cases for each racial group.
- Fill in the "Expected cases" column by multiplying the total number of cases (4140) by each race's proportion.
- Compare the observed and expected cases to conduct the chi-square test.

This data provides insight into the distribution of AIDS cases across different racial groups in Santa Clara County, which can be critical for public health planning and interventions.
Transcribed Image Text:### Distribution of AIDS Cases in Santa Clara County The number of AIDS cases reported for Santa Clara County, California is broken down by race in the table below. **Source:** "HIV/AIDS Epidemiology Santa Clara County," Santa Clara County Public Health Department, May 2011. | Race | Cases | |---------------------------|-------| | White | 2267 | | Hispanic | 1182 | | Black | 464 | | Asian/Pacific Islander | 227 | | **Total** | **4140** | --- **Directions:** Conduct a chi-square test for goodness-of-fit to determine whether or not the occurrence of AIDS cases is consistent with the race distribution of Santa Clara County. 1. **Choose the correct null and alternative hypotheses.** - \( H_0 \): The distribution of AIDS cases is different from the race distribution in Santa Clara County. - \( H_a \): The distribution of AIDS cases is consistent with the race distribution in Santa Clara County. - \( H_0 \): The distribution of AIDS cases is consistent with the race distribution in Santa Clara County. - \( H_a \): The distribution of AIDS cases is different from the race distribution in Santa Clara County. 2. **Compute the test statistic.** The population distribution of Santa Clara County by race is provided in the table below. Use these percentages to compute the expected number of cases for each racial group. Round each of the expected counts to 2 decimal places. | Race | Proportion | Expected cases | |--------------------------|------------|----------------| | White | 0.455 | | | Hispanic | 0.252 | | | Black | 0.025 | | | Asian/Pacific Islander | 0.268 | | | **Total** | **1** | | ### Instructions for Analysis: - Use the given proportions to calculate the expected cases for each racial group. - Fill in the "Expected cases" column by multiplying the total number of cases (4140) by each race's proportion. - Compare the observed and expected cases to conduct the chi-square test. This data provides insight into the distribution of AIDS cases across different racial groups in Santa Clara County, which can be critical for public health planning and interventions.
### Chi-Square Test for Proportions

#### 2. Compute the Test Statistic
The population distribution of Santa Clara County by race is provided in the table below. Use these percentages to compute the expected number of cases for each racial group. Round each of the expected counts to 2 decimal places.

| Race                 | Proportion | Expected Cases |
|----------------------|------------|----------------|
| White                | 0.455      |                |
| Hispanic             | 0.252      |                |
| Black                | 0.025      |                |
| Asian/Pacific Islander | 0.268      |                |
| **Total** | **1**        |                |

Determine the value of the test statistic. Round your answer to 1 decimal place.

\[
\chi^2 = \_\_\_\_\_
\]

#### 3. Compute the p-value. Round your answer to 4 decimal places.

\[
p \text{-value} = \_\_\_\_\_
\]

#### 4. Interpret the Results of the Significance Test
- **Option 1:** The differences between the distribution of AIDS cases and the distribution of the general population in Santa Clara County are not statistically significant. From a practical perspective, the differences are minor.

- **Option 2:** The differences between the distribution of AIDS cases and the distribution of the general population in Santa Clara County is statistically significant. The differences are also important from a practical perspective. For example, the number of blacks with AIDS is times more than expected and the number of Asian/Pacific Islanders with AIDS is times less than expected.
Transcribed Image Text:### Chi-Square Test for Proportions #### 2. Compute the Test Statistic The population distribution of Santa Clara County by race is provided in the table below. Use these percentages to compute the expected number of cases for each racial group. Round each of the expected counts to 2 decimal places. | Race | Proportion | Expected Cases | |----------------------|------------|----------------| | White | 0.455 | | | Hispanic | 0.252 | | | Black | 0.025 | | | Asian/Pacific Islander | 0.268 | | | **Total** | **1** | | Determine the value of the test statistic. Round your answer to 1 decimal place. \[ \chi^2 = \_\_\_\_\_ \] #### 3. Compute the p-value. Round your answer to 4 decimal places. \[ p \text{-value} = \_\_\_\_\_ \] #### 4. Interpret the Results of the Significance Test - **Option 1:** The differences between the distribution of AIDS cases and the distribution of the general population in Santa Clara County are not statistically significant. From a practical perspective, the differences are minor. - **Option 2:** The differences between the distribution of AIDS cases and the distribution of the general population in Santa Clara County is statistically significant. The differences are also important from a practical perspective. For example, the number of blacks with AIDS is times more than expected and the number of Asian/Pacific Islanders with AIDS is times less than expected.
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