https://file.aiquickdraw.com/mathbot/file/172741074646572okoabt.jpg

To solve the problem in the screenshot, let's break it down.

### Part 1: Identifying the Correct Statistical Test

The scenario asks to construct a 95% confidence interval for the population proportion of households with kids in Thousand Oaks. You are given the following dataTo solve the problem in the screenshot, let's break it down.

### Part 1: Identifying the Correct Statistical Test

The scenario asks to construct a 95% confidence interval for the population proportion of households with kids in Thousand Oaks. You are given the following 
- 300 households were sampled.
- 184 households have kids.

To calculate a confidence interval for a population proportion, you need to use the **Confidence Interval for a Population Proportion** formula.

### Part 2: Confidence Interval Calculation

The formula for the confidence interval for a population proportion is:

\[
\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
\]

Where:
- \(\hat{p}\) is the sample proportion.
- \(Z_{\alpha/2}\) is the Z-score for the desired confidence level (for a 95% confidence interval, \(Z = 1.96\)).
- \(n\) is the sample size.

#### Step 1: Calculate \(\hat{p}\)
\[
\hat{p} = \frac{184}{300} = 0.6133
\]

#### Step 2: Margin of Error
\[
\text{Margin of error} = Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = 1.96 \times \sqrt{\frac{0.6133 \times (1 - 0.6133)}{300}}
\]
\[
\text{Margin of error} = 1.96 \times \sqrt{\frac{0.6133 \times 0.3867}{300}} = 1.96 \times \sqrt{\frac{0.2371}{300}} = 1.96 \times \sqrt{0.0007903}
\]
\[
\text{Margin of error} \approx 1.96 \times 0.0281 = 0.0551
\]

#### Step 3: Confidence Interval
The confidence interval is:
\[
\hat{p} \pm \text{Margin of Error} = 0.6133 \pm 0.0551
\]

\[
\text{Confidence interval} = (0.558, 0.668)
\]

### Part 3: Expressing the Answers

#### a) Open interval:
\[
(0.558, 0.668)
\]

#### b) Tri-linear inequality:
\[
0.558 < p < 0.668
\]

#### c) Point estimate and margin of error:
\[
p = 0.613 \pm 0.055
\]

Would you like further clarification or details on any step?

---

Here are 5 related questions to extend the topic:
1. How do you determine the appropriate Z-score for a different confidence level, such as 99%?
2. What changes in the formula when the sample size increases?
3. How is a hypothesis test for a population proportion conducted?
4. How do you calculate the sample size needed for a specific margin of error?
5. Can confidence intervals overlap, and what does it imply when they do?

**Tip**: Always double-check the margin of error calculation to ensure accuracy in confidence interval estimations.

A toy store would like to open a new store in the city of Thousand Oaks, and they would like to know what proportion of households have kids. Out of 300 households sampled, 184 had kids. Based on this, to construct a 95% confidence interval for the true population proportion of Thousand Oaks households with kids, you need to use which one of the following calculators?

Learn how to calculate the 95% confidence interval for the population proportion of households with kids in Thousand Oaks. Out of a sample of 300 households, 184 had kids. This problem involves calculating confidence intervals and understanding margin of error in population proportions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation