A random sample of households in Orange County showed that 18 out of 209 of them were burglarized in the past 5 years.  And in a random sample of households in Cook County, 24 out of 199 of them were burglarized in that same time period.  We will test the claim that the proportion of Orange County households that were burglarized (p1) is greater than the proportion of Cook County households that were burglarized (p2).
For each part below, enter only a numeric value in the answer box. For example, do not type "z =" or "t =" before your answers. Round each of your answers to 3 places after the decimal point.
(a) Calculate the value of  the pooled sample proportion, that is used in this hypothesis test.
    
P = 
(b) Calculate the value of the test statistic used in this test.
     Test statistic's value = 
(c) Use your calculator to find the P-value of this test.
     P-value = 
(d) Use your calculator to find the critical value(s) used to test this claim at the 0.07 significance level. If there are two critical values, then list them both with a comma between them.
     Critical value(s) = 

Let's solve this step by step.

### (a) Pooled Sample Proportion (P)

The pooled sample proportion (\( P \)) is calculated as the total number of burglarized households divided by the total number of households across both counties.

\[
P = \frac{x_1 + x_2}{n_1 + n_2}
\]

Where:
- \( x_1 = 18 \) (burglarized households in Orange County)
- \( n_1 = 209 \) (total households in Orange County)
- \( x_2 = 24 \) (burglarized households in Cook County)
- \( n_2 = 199 \) (total households in Cook County)

\[
P = \frac{18 + 24}{209 + 199} = \frac{42}{408}
\]

Let's compute the pooled proportion.

\[
P = 0.103
\]

### (b) Test Statistic

The test statistic is calculated using the formula for the difference between two proportions:

\[
z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{P(1 - P) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}}
\]

Where:
- \( \hat{p}_1 = \frac{18}{209} \) (sample proportion in Orange County)
- \( \hat{p}_2 = \frac{24}{199} \) (sample proportion in Cook County)
- \( P \) is the pooled sample proportion calculated in part (a).

Let’s calculate this now.

### (c) P-value

Once the z-value is calculated in part (b), we use the standard normal distribution to find the p-value associated with that z-value.

### (d) Critical Value(s)

For a one-tailed test (since we are testing if \( p_1 > p_2 \)), at the 0.07 significance level, we need to find the critical z-value from the standard normal distribution corresponding to a significance level of 0.07.

Let me calculate these values.Here are the results:

### (a) Pooled sample proportion:
\[
P = 0.103
\]

### (b) Test statistic:
\[
\text{Test statistic's value} = -1.146
\]

### (c) P-value:
\[
\text{P-value} = 0.874
\]

### (d) Critical value at the 0.07 significance level (one-tailed test):
\[
\text{Critical value} = 1.476
\]

Let me know if you have any questions or need further clarification!

Here are 5 follow-up questions:
1. How do we interpret the p-value in this context?
2. Why do we use a pooled proportion in hypothesis tests comparing two proportions?
3. What does the negative test statistic indicate about the relationship between \( p_1 \) and \( p_2 \)?
4. How would the results change if we used a two-tailed test instead of a one-tailed test?
5. What assumptions are required for this hypothesis test to be valid?

**Tip**: Always check whether the sample sizes and success counts meet the conditions for a valid hypothesis test on proportions.

A random sample of households in Orange County showed that 18 out of 209 of them were burglarized in the past 5 years. And in a random sample of households in Cook County, 24 out of 199 of them were burglarized in that same time period. We will test the claim that the proportion of Orange County households that were burglarized (p1) is greater than the proportion of Cook County households that were burglarized (p2). (a) Calculate the value of the pooled sample proportion. (b) Calculate the value of the test statistic. (c) Find the P-value of this test. (d) Find the critical value(s) used to test this claim at the 0.07 significance level.

Solve a hypothesis test comparing burglary rates in Orange County and Cook County. This statistical problem covers key concepts like the pooled sample proportion, test statistic, and critical values, with a significance level of 0.07. Suitable for advanced high school or college-level statistics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation