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Chapter 9: Problem 5

Construct the appropriate confidence interval. A simple random sample of size \(n=300\) individuals who are currently employed is asked if they work at home at least once per week. Of the 300 employed individuals surveyed, 35 responded that they did work at home at least once per week. Construct a \(99 \%\) confidence interval about the population proportion of employed individuals who work at home at least once per week.

Short Answer

Expert verified
The \(99\%\) confidence interval is \(0.0697\ to \(0.1637\).

Step by step solution

01

Identify the sample proportion

First, identify the sample size (=300) and the number of individuals who work at home at least once per week (=35). Calculate the sample proportion \(=\frac{35}{300}\).
02

Calculate the sample proportion

The sample proportion \( \hat{p} \) is calculated as follows:\[ \hat{p} = \frac{35}{300} = 0.1167 \]
03

Find the critical value

For a \(99\%\) confidence interval, the critical value \(Z_{\alpha/2}\) is found from the standard normal distribution. The critical value is \(Z_{\alpha/2} = 2.576\).
04

Calculate the standard error

The standard error (SE) of the sample proportion is calculated using:\[ SE = \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} } = \sqrt{ \frac{0.1167 \times 0.8833}{300} } = 0.01826 \]
05

Calculate the margin of error

The margin of error (MOE) is calculated using the formula:\[ MOE = Z_{\alpha/2} \times SE = 2.576 \times 0.01826 \approx 0.0470 \]
06

Calculate the confidence interval

The confidence interval for the population proportion is calculated using the formula:\[\hat{p} \pm MOE\]Substitute the values for \( \hat{p} \) and the calculated MOE:\[ 0.1167 \pm 0.0470 \]This gives the interval:\[ 0.0697 \ to 0.1637 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Proportion Calculation
The sample proportion is the ratio of individuals who exhibit a certain characteristic to the total sample size. In the provided exercise, 35 out of 300 employed individuals work from home at least once per week. Calculate the sample proportion (\text p\text caret \text) using the formula:

\[ \text \text \text caret \text \text= \frac{35}{300} \approx 0.1167 \] \text helpful since it allows you to estimate the proportion of a population with specific features. In this case, the proportion indicates that roughly 11.67% of employed individuals work from home at least once a week.

Remember:
  • Sample Size (n) = 300
  • Number of Individuals Working at Home (x) = 35
  • Sample Proportion (\text p \text caret \text = 0.1167
Critical Value
The critical value is a factor used to calculate the confidence interval. It corresponds to the desired level of confidence and comes from the standard normal distribution (z-distribution). For a 99% confidence interval, you find the critical value (\text Z \text \text alpha/2 \text) using statistical tables or software, which yields:

\[ Z \text \text alpha/2 \text= 2.576 \] This value is significant because a higher confidence level (99%) requires a larger critical value, ensuring a wider confidence interval, thus more reliability in the estimate.
Standard Error
The standard error (SE) measures the accuracy of the sample proportion as an estimate of the population proportion. It considers the sample size and variance, calculated as:

\[ SE = \sqrt{\frac{\text p \text caret \text \times (1 - \text p \text caret \text)}{n}} \ = \sqrt{\frac{0.1167 \times 0.8833}{300}} \approx 0.01826 \]

This small value indicates a relatively precise estimate. A larger sample size helps reduce the SE, enhancing the precision of our population proportion estimate.
Margin of Error
The margin of error (MOE) quantifies the range within which the true population proportion likely falls. It is influenced by both the standard error and the critical value, and is calculated as follows:

\[ MOE = Z \text \text alpha/2 \times SE \ = 2.576 \times 0.01826 \approx 0.0470 \]

The MOE signifies that if you surveyed all individuals, the proportion of employed individuals working from home at least once per week would be within the range of \text p\text \text caret卤\text0.0470.
99% Confidence Interval
A confidence interval shows the range in which the population proportion is likely to be. In our case, the 99% confidence interval is calculated by adding and subtracting the margin of error from the sample proportion: \[\text p \text \text caret \text 卤 MOE\text= 0.1167 卤 0.0470\]\resulting in: \= [0.0697, 0.1637]\Thus, we are 99% confident that the true proportion of employed individuals who work from home at least once per week lies between 6.97% and 16.37%.

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Most popular questions from this chapter

Construct the appropriate confidence interval. A simple random sample of size \(n=40\) is drawn from a population. The sample mean is found to be \(\bar{x}=120.5\) and the sample standard deviation is found to be \(s=12.9 .\) Construct a \(99 \%\) confidence interval about the population mean.

A researcher wishes to estimate the mean number of miles on 4 -year-old Saturn SCIs. (a) How many cars should be in a sample to estimate the mean number of miles within 1000 miles with \(90 \%\) confidence, assuming that \(\sigma=19,700 ?\) (b) How many cars should be in a sample to estimate the mean number of miles within 500 miles with \(90 \%\) confidence, assuming that \(\sigma=19,700 ?\) (c) What effect does doubling the required accuracy have on the sample size? Why is this the expected result?

A simple random sample of size \(n\) is drawn from a population whose population standard deviation, \(\sigma,\) is known to be \(5.3 .\) The sample mean, \(\bar{x},\) is determined to be 34.2 (a) Compute the \(95 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is 35 (b) Compute the \(95 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is \(50 .\) How does increasing the sample size affect the margin of error, \(E ?\) (c) Compute the \(99 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is \(35 .\) Compare the results to those obtained in part (a). How does increasing the level of confidence affect the size of the margin of error, \(E ?\) (d) Can we compute a confidence interval about \(\mu\) based on the information given if the sample size is \(n=15 ?\) Why? If the sample size is \(n=15,\) what must be true regarding the population from which the sample was drawn?

Suppose the arrival of cars at Burger King's drive-through follows a Poisson process with \(\mu=4\) cars every 10 minutes. (a) Simulate obtaining 30 samples of size \(n=40\) from this population. (b) Construct \(90 \%\) confidence intervals for each of the 30 samples. [Note: \(\sigma=\sqrt{\mu}\) in a Poisson process.] (c) How many of the intervals do you expect to include the population mean? How many actually contain the population mean?

Own a Gun? In a Harris Poll conducted in May 2000 , \(39 \%\) of the people polled answered yes to the following question: "Do you happen to have in your home or garage any guns or revolvers?" The margin of error in the poll was \(\pm 3 \%=\pm 0.03,\) and the estimate was made with \(95 \%\) confidence. How many people were surveyed?
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