Out Of A Sample Of 760 People, 367 Own Their Homes. Construct A $95\%$ Confidence Interval For The Population Proportion Of People Who Own Their Homes.A. $CI = (43.62\%, 52.96\%)$B. $CI = (44.74\%, 51.84\%)$C. $CI

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Introduction

In statistics, a confidence interval is a range of values within which a population parameter is likely to lie. When dealing with proportions, it's essential to construct a confidence interval to estimate the population proportion. In this article, we'll discuss how to construct a $95%$ confidence interval for the population proportion of people who own their homes, given a sample of 760 people.

Given Information

  • Sample size ($n$): 760
  • Number of people who own their homes ($x$): 367
  • Confidence level: $95%$

Calculating the Sample Proportion

The sample proportion ($\hat{p}$) is calculated by dividing the number of people who own their homes by the sample size.

p^=xn=367760=0.4832\hat{p} = \frac{x}{n} = \frac{367}{760} = 0.4832

Calculating the Standard Error

The standard error ($SE$) is calculated using the formula:

SE=p^(1−p^)nSE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

Substituting the values, we get:

SE=0.4832(1−0.4832)760=0.0153SE = \sqrt{\frac{0.4832(1-0.4832)}{760}} = 0.0153

Calculating the Margin of Error

The margin of error ($ME$) is calculated using the formula:

ME=zα/2×SEME = z_{\alpha/2} \times SE

where $z_{\alpha/2}$ is the critical value from the standard normal distribution for a $95%$ confidence level, which is approximately 1.96.

ME=1.96×0.0153=0.0300ME = 1.96 \times 0.0153 = 0.0300

Constructing the Confidence Interval

The confidence interval ($CI$) is constructed by adding and subtracting the margin of error from the sample proportion.

CI=p^±ME=0.4832±0.0300CI = \hat{p} \pm ME = 0.4832 \pm 0.0300

Therefore, the $95%$ confidence interval for the population proportion of people who own their homes is:

CI=(0.4532,0.5132)CI = (0.4532, 0.5132)

Converting the Confidence Interval to a Percentage

To express the confidence interval as a percentage, we multiply the lower and upper bounds by 100.

CI=(45.32%,51.32%)CI = (45.32\%, 51.32\%)

Comparison with the Given Options

Comparing the calculated confidence interval with the given options, we can see that:

  • Option A: $CI = (43.62%, 52.96%)$
  • Option B: $CI = (44.74%, 51.84%)$
  • Option C: Not provided

Our calculated confidence interval, $CI = (45.32%, 51.32%)$, is closest to Option A.

Conclusion

Q: What is a confidence interval?

A: A confidence interval is a range of values within which a population parameter is likely to lie. It's a way to estimate the population parameter based on a sample of data.

Q: Why is it important to construct a confidence interval?

A: Constructing a confidence interval is important because it allows us to make inferences about the population parameter based on a sample of data. It also helps us to understand the uncertainty associated with our estimates.

Q: What is the difference between a point estimate and a confidence interval?

A: A point estimate is a single value that represents the population parameter, while a confidence interval is a range of values within which the population parameter is likely to lie.

Q: How do I choose the confidence level?

A: The confidence level is typically chosen based on the desired level of precision and the amount of uncertainty associated with the estimate. Common confidence levels include 90%, 95%, and 99%.

Q: What is the critical value (z-score) used in constructing a confidence interval?

A: The critical value (z-score) is a value from the standard normal distribution that corresponds to the desired confidence level. For example, a 95% confidence level corresponds to a z-score of approximately 1.96.

Q: How do I calculate the margin of error?

A: The margin of error is calculated by multiplying the standard error by the critical value (z-score). The formula is:

ME=zα/2×SEME = z_{\alpha/2} \times SE

Q: What is the standard error (SE)?

A: The standard error is a measure of the variability of the sample proportion. It's calculated using the formula:

SE=p^(1−p^)nSE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

Q: Can I use a confidence interval to make inferences about the population?

A: Yes, a confidence interval can be used to make inferences about the population. If the confidence interval includes a value of interest, it suggests that the value is plausible based on the sample data.

Q: What are some common mistakes to avoid when constructing a confidence interval?

A: Some common mistakes to avoid include:

  • Not checking the assumptions of the confidence interval (e.g., normality of the data)
  • Not using the correct formula for the margin of error
  • Not considering the effect of sample size on the confidence interval
  • Not interpreting the confidence interval correctly

Q: Can I use a confidence interval to compare two or more groups?

A: Yes, a confidence interval can be used to compare two or more groups. This is known as a confidence interval for the difference between two proportions.

Q: What are some real-world applications of confidence intervals?

A: Confidence intervals have many real-world applications, including:

  • Estimating the proportion of people who own a particular product
  • Evaluating the effectiveness of a new treatment
  • Comparing the performance of two or more groups
  • Estimating the population mean or proportion based on a sample of data