Question
In a survey of 400 people, 122 said they own a house. What is the margin of error for a 95% confidence interval of the sample proportion of people who own a house? Determine the margin of error for this study rounded to the nearest hundredth and interpret its meaning in context. (2 points) The margin of error r=□ %. With 95% confidence, the sample proportion of people who own a house is between □ % and □ %
Answers
GPT-4o mini
To calculate the margin of error for the sample proportion at a 95% confidence level, we can follow these steps:
1. **Calculate the sample proportion (\( \hat{p} \)):**
\[
\hat{p} = \frac{x}{n} = \frac{122}{400} = 0.305
\]
2. **Calculate the standard error (SE) of the sample proportion:**
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.305 \times (1 - 0.305)}{400}}
\]
\[
SE = \sqrt{\frac{0.305 \times 0.695}{400}} = \sqrt{\frac{0.212475}{400}} \approx \sqrt{0.0005311875} \approx 0.02305
\]
3. **Find the critical value (z) for a 95% confidence level:**
For a 95% confidence interval, the critical value (\( z \)) is approximately 1.96.
4. **Calculate the margin of error (ME):**
\[
ME = z \cdot SE = 1.96 \cdot 0.02305 \approx 0.0452
\]
Converting to a percentage:
\[
ME \approx 0.0452 \times 100 \approx 4.52\%
\]
Now, we can summarize the results:
- The margin of error \( r \) is approximately **4.52%**.
- Therefore, the confidence interval for the sample proportion of people who own a house is:
\[
\hat{p} \pm ME = 30.5\% \pm 4.52\%
\]
Calculating the bounds:
- Lower bound: \( 30.5\% - 4.52\% \approx 25.98\% \)
- Upper bound: \( 30.5\% + 4.52\% \approx 35.02\% \)
Thus, the interval is approximately **between 25.98% and 35.02%**.
### Final answer:
The margin of error \( r \) is **4.52%**. With 95% confidence, the sample proportion of people who own a house is between **25.98%** and **35.02%**.
### Interpretation:
This means that we are 95% confident that the true proportion of people who own a house in the population from which the sample was drawn falls between 25.98% and 35.02%. This range gives us an indication of the variability in our estimation based on the surveyed sample.
1. **Calculate the sample proportion (\( \hat{p} \)):**
\[
\hat{p} = \frac{x}{n} = \frac{122}{400} = 0.305
\]
2. **Calculate the standard error (SE) of the sample proportion:**
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.305 \times (1 - 0.305)}{400}}
\]
\[
SE = \sqrt{\frac{0.305 \times 0.695}{400}} = \sqrt{\frac{0.212475}{400}} \approx \sqrt{0.0005311875} \approx 0.02305
\]
3. **Find the critical value (z) for a 95% confidence level:**
For a 95% confidence interval, the critical value (\( z \)) is approximately 1.96.
4. **Calculate the margin of error (ME):**
\[
ME = z \cdot SE = 1.96 \cdot 0.02305 \approx 0.0452
\]
Converting to a percentage:
\[
ME \approx 0.0452 \times 100 \approx 4.52\%
\]
Now, we can summarize the results:
- The margin of error \( r \) is approximately **4.52%**.
- Therefore, the confidence interval for the sample proportion of people who own a house is:
\[
\hat{p} \pm ME = 30.5\% \pm 4.52\%
\]
Calculating the bounds:
- Lower bound: \( 30.5\% - 4.52\% \approx 25.98\% \)
- Upper bound: \( 30.5\% + 4.52\% \approx 35.02\% \)
Thus, the interval is approximately **between 25.98% and 35.02%**.
### Final answer:
The margin of error \( r \) is **4.52%**. With 95% confidence, the sample proportion of people who own a house is between **25.98%** and **35.02%**.
### Interpretation:
This means that we are 95% confident that the true proportion of people who own a house in the population from which the sample was drawn falls between 25.98% and 35.02%. This range gives us an indication of the variability in our estimation based on the surveyed sample.