Textbook Question
In Exercises 29–32, determine the minimum sample size n needed to estimate for the values of c, σ, and E.
c = 0.95, σ = 2.5, E = 1.
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7. Sampling Distributions & Confidence Intervals: Mean
Confidence Intervals for Population Mean
3:56 minutesProblem 6.2.17
Textbook Question
Constructing a Confidence Interval In Exercises 17–20, you are given the sample mean and the sample standard deviation. Assume the population is normally distributed and use the t-distribution to find the margin of error and construct a 95% confidence interval for the population mean. Interpret the results.
Commute Time In a random sample of eight people, the mean commute time to work was 35.5 minutes and the standard deviation was 7.2 minute
Verified step by step guidance1
Step 1: Identify the given values from the problem. The sample mean (\( \bar{x} \)) is 35.5 minutes, the sample standard deviation (\( s \)) is 7.2 minutes, the sample size (\( n \)) is 8, and the confidence level is 95%.
Step 2: Determine the critical value (\( t^* \)) for a 95% confidence level using the t-distribution. The degrees of freedom (\( df \)) are calculated as \( n - 1 \), which is \( 8 - 1 = 7 \). Use a t-table or statistical software to find \( t^* \) for \( df = 7 \) and a two-tailed test with \( \alpha = 0.05 \).
Step 3: Calculate the standard error of the mean (\( SE \)) using the formula \( SE = \frac{s}{\sqrt{n}} \). Substitute \( s = 7.2 \) and \( n = 8 \) into the formula.
Step 4: Compute the margin of error (\( ME \)) using the formula \( ME = t^* \cdot SE \). Multiply the critical value (\( t^* \)) by the standard error (\( SE \)) calculated in the previous step.
Step 5: Construct the confidence interval for the population mean using the formula \( \bar{x} \pm ME \). Add and subtract the margin of error (\( ME \)) from the sample mean (\( \bar{x} \)) to find the lower and upper bounds of the confidence interval. Interpret the results by stating that you are 95% confident the true population mean commute time lies within this interval.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Confidence Interval
A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence, typically expressed as a percentage. For example, a 95% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 95% of those intervals would contain the true population mean.
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t-Distribution
The t-distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution, but has heavier tails. It is used instead of the normal distribution when the sample size is small (typically less than 30) and the population standard deviation is unknown. The t-distribution accounts for the additional uncertainty introduced by estimating the population standard deviation from the sample.
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Margin of Error
The margin of error is a statistic that expresses the amount of random sampling error in a survey's results. It quantifies the uncertainty associated with the sample estimate and is calculated as the product of the critical value from the t-distribution and the standard error of the sample mean. A smaller margin of error indicates a more precise estimate of the population parameter.
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