Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.Q.1b

Chapter 6, Problem 6.Q.1b

[APPLET] The winning times (in hours) for a sample of 20 randomly selected Boston Marathon Women’s Open Division champions from 1980 to 2019 are shown in the table at the left. Assume the population standard deviation is 0.068 hour. (Source: Boston Athletic Association)

b. Find the margin of error for a 95% confidence level.

Verified step by step guidance

Step 1: Calculate the sample mean (x̄) of the winning times. Add all the values in the table and divide by the total number of observations (20). Use the formula: x̄ = (Σx) / n.

Step 2: Identify the population standard deviation (σ), which is given as 0.068 hours.

Step 3: Determine the sample size (n), which is the number of observations in the table. In this case, n = 20.

Step 4: Find the critical value (z*) for a 95% confidence level. For a normal distribution, the z* value corresponding to a 95% confidence level is approximately 1.96.

Step 5: Calculate the margin of error (E) using the formula: E = z* × (σ / √n). Substitute the values for z*, σ, and n into the formula to compute the margin of error.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Margin of Error

The margin of error quantifies the uncertainty in a sample estimate. It indicates the range within which the true population parameter is expected to lie, given a certain confidence level. For a 95% confidence level, the margin of error is typically calculated using the formula: ME = z * (σ/√n), where z is the z-score corresponding to the confidence level, σ is the population standard deviation, and n is the sample size.

Confidence Level

The confidence level represents the degree of certainty that the population parameter lies within the margin of error. A 95% confidence level means that if we were to take many samples and construct confidence intervals for each, approximately 95% of those intervals would contain the true population parameter. This level is commonly used in statistics to balance precision and reliability.

Population Standard Deviation

The population standard deviation is a measure of the dispersion or spread of a set of values in a population. It quantifies how much individual data points deviate from the population mean. In this case, the given standard deviation of 0.068 hours is crucial for calculating the margin of error, as it reflects the variability of winning times among the Boston Marathon champions.

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Related Practice

Textbook Question

In Exercises 9–12, find the critical value tc for the level of confidence c and sample size n.

c = 0.98, n = 15

111

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Textbook Question

[APPLET] The waking times (in minutes past 5:00 A.M.) of 40 people who start work at 8:00 A.M. are shown in the table at the left. Assume the population standard deviation is 45 minutes. Find (a) the point estimate of the population mean μ and (b) the margin of error for a 90% confidence interval.

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Textbook Question

In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.

c = 0.90, n = 16

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Textbook Question

In a random sample of 12 senior-level civil engineers, the mean annual earnings were \$133,326 and the standard deviation was \$36,729. Assume the annual earnings are normally distributed and construct a 95% confidence interval for the population mean annual earnings for senior-level civil engineers. Interpret the results. (Adapted from Salary.com)

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Textbook Question

[APPLET] The winning times (in hours) for a sample of 20 randomly selected Boston Marathon Women’s Open Division champions from 1980 to 2019 are shown in the table at the left. Assume the population standard deviation is 0.068 hour. (Source: Boston Athletic Association)

d. Does it seem likely that the population mean could be greater than 2.52 hours? Explain.

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Textbook Question

You research the salaries of senior-level civil engineers and find that the population mean is \$131,935. In Exercise 4, does the t-value fall between -t0.95 and t0.95?

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