The 2010 Census showed that 31% of King Country residents lived within Seattle city limits. Seattle city officials wondered if the proportion of county residents currently living in Seattle is higher than the 2010 level. a. Define your parameter, and write the null hypothesis, alternative hypothesis for hypothesis testing. b. Show that the necessary conditions (Randomization Condition, 10% Condition, Success/Failure Condition) are satisfied if they randomly select a sample of 1000 King County residents. Briefly explain how each condition is satisfied. c. Suppose that out of a recent sample of 1000 King County residents, 345 lived in Seattle city limits. Perform the hypothesis test and compute the P-value. (To show your work: Write down which calculator/program you are using and what values you are entering into the calculator.) d. Does this provide strong evidence that the proportion of King County residents living in Seattle has increased from the 2010 level? Briefly explain.
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Answer only C and D
![### Investigating the Proportion of King County Residents Living in Seattle
In this exercise, we will analyze data from the 2010 Census and subsequent sample data to determine if the proportion of King County residents living in Seattle has increased since 2010.
#### Given Data
- **2010 Census Result:** 31% of King County residents lived within Seattle city limits.
- **Current Investigation:** Seattle city officials want to determine if this percentage has increased.
#### Tasks
**a. Parameter Definition and Hypotheses**
1. **Parameter Definition:**
- Let \( p \) represent the current proportion of King County residents living within Seattle city limits.
2. **Hypotheses:**
- **Null Hypothesis (H0):** \( p = 0.31 \) (The proportion has not changed from the 2010 level.)
- **Alternative Hypothesis (H1):** \( p > 0.31 \) (The proportion has increased since 2010.)
**b. Necessary Conditions for Hypothesis Testing**
To proceed with hypothesis testing, the following conditions must be satisfied:
1. **Randomization Condition:** The sample of 1000 King County residents must be randomly selected.
2. **10% Condition:** The sample size is less than 10% of the population of King County residents.
3. **Success/Failure Condition:** The sample size \( n \) should be large enough to expect at least 10 successes and 10 failures:
- \( np_0 = 1000 \times 0.31 = 310 \) (expected successes)
- \( n(1 - p_0) = 1000 \times 0.69 = 690 \) (expected failures)
Since both values are greater than 10, the Success/Failure Condition is satisfied.
**c. Hypothesis Test and P-value Calculation**
- **Sample Data:** Out of a recent sample of 1000 King County residents, 345 lived in Seattle city limits.
1. **Proportion of sample living in Seattle:** \( \hat{p} = \frac{345}{1000} = 0.345 \)
2. **Standard deviation of the sampling distribution:**
\[
\sigma = \sqrt{\frac{p_0 (1 - p_0)}{n}} = \sqrt{\frac{0.31 \times 0.69}{1000](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85a16b31-7619-4bbb-b006-067a19fea2d2%2F2e7a4866-4f45-4520-ad17-833da23b5279%2Ft1dhnd.png&w=3840&q=75)
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