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Chapter 9: Problem 23

A school social worker wishes to estimate the mean amount of time each week that high school students spend with friends. She obtains a random sample of 1175 high school students and finds that the mean weekly time spent with friends is 9.2 hours. Assuming the population standard deviation amount of time spent with friends is 6.7 hours, construct and interpret a \(90 \%\) confidence interval for the mean time spent with friends each week.

Short Answer

Expert verified
The 90% confidence interval is (8.88, 9.52) hours.

Step by step solution

01

- Identify the Given Information

Given: Sample size (\(n\))= 1175, Sample mean (\(\bar{x}\)) = 9.2 hours, Population standard deviation (\(\sigma\)) = 6.7 hours. We need to construct a 90% confidence interval for the mean weekly time spent with friends.
02

- Determine the Z-value for 90% Confidence Interval

For a 90% confidence interval, the Z-value is = 1.645. This value is obtained from the Z-table.
03

- Calculate the Standard Error

The standard error (SE) is calculated using the formula \(SE = \frac{\sigma}{\sqrt{n}}\). Substitute \(\sigma = 6.7\) and \(n = 1175\): \(SE = \frac{6.7}{\sqrt{1175}} \approx 0.195\)
04

- Calculate the Margin of Error

The margin of error (ME) is calculated using the Z-value and standard error: \(ME = Z \times SE\). Substitute \(Z = 1.645\) and \(SE = 0.195\): \(ME = 1.645 \times 0.195 \approx 0.32\)
05

- Construct the Confidence Interval

The confidence interval is given by the formula: \(\bar{x} \pm ME\). Substituting \(\bar{x} = 9.2\) and \(ME = 0.32\): \(CI = 9.2 \pm 0.32\) Therefore, the 90% confidence interval is (8.88, 9.52).
06

- Interpret the Confidence Interval

We are 90% confident that the mean time high school students spend with friends each week lies between 8.88 and 9.52 hours.

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Key Concepts

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

Sample Mean
A sample mean, often denoted by \(\bar{x}\), is the average value found from a sample of a population. It is calculated by summing all the observed values and then dividing by the number of values. In our example, the school social worker surveyed 1175 high school students and found that, on average, they spend 9.2 hours per week with friends. This sample mean is used as an estimate to make inferences about the entire population of high school students. The sample mean is crucial because it serves as the central point of our confidence interval estimate.
Population Standard Deviation
The population standard deviation, denoted by \(\sigma\), measures the spread of all observations in the entire population. It tells us how much the individual data points deviate from the population mean. Unlike the sample standard deviation, which is calculated from a sample, the population standard deviation is a fixed value. In the exercise, the population standard deviation is assumed to be 6.7 hours, which means that the amount of time high school students spend with friends varies around 6.7 hours from the mean in the entire population. This parameter is important for calculating the standard error.
Margin of Error
The margin of error (ME) quantifies the range of values above and below the sample mean in a confidence interval. It helps to express the degree of uncertainty or variability in the sample estimate. The margin of error is calculated by multiplying the critical value (Z-value) by the standard error. In our scenario, the Z-value for a 90% confidence interval is 1.645. Thus, the margin of error is computed as \(ME = Z \times SE = 1.645 \times 0.195 = 0.32\). This means that our sample mean of 9.2 hours can vary by 卤0.32 hours.
Standard Error
The standard error (SE) measures the variability of the sample mean from the population mean. It is calculated using the formula \(SE = \frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size. For the high school social worker鈥檚 data, the standard error is calculated as \(SE = \frac{6.7}{\sqrt{1175}} 鈮� 0.195\). This small standard error indicates that the sample mean (9.2 hours) is a fairly precise estimate of the population mean.
Z-value
The Z-value, also known as the critical value or Z-score, corresponds to the desired confidence level and helps in determining the margin of error. It is derived from the standard normal distribution table. For a 90% confidence interval, the Z-value is 1.645. This Z-value tells us how many standard deviations the sample mean is from the population mean in a standard normal distribution. In the exercise, using the Z-value of 1.645 helps in constructing the confidence interval around the sample mean, ensuring that the interval estimates the true population mean with 90% confidence.

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Most popular questions from this chapter

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