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.1d

Chapter 6, Problem 6.Q.1d

[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.

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 (n = 20). Use the formula x̄ = (Σx) / n.

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

Step 3: Compute the standard error of the mean (SE) using the formula SE = σ / √n, where n is the sample size.

Step 4: Determine the z-score for the population mean of 2.52 hours using the formula z = (x̄ - μ) / SE, where μ is the hypothesized population mean (2.52 hours).

Step 5: Compare the calculated z-score to the critical z-value for a chosen significance level (e.g., α = 0.05). If the z-score falls within the critical region, it suggests that the population mean is unlikely to be greater than 2.52 hours. Otherwise, it may be plausible.

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Key Concepts

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

Population Mean

The population mean is the average of all values in a population. It is a key parameter in statistics that represents the central tendency of a dataset. In this context, it refers to the average winning time of all Boston Marathon Women’s Open Division champions from 1980 to 2019. Understanding the population mean helps in assessing whether the sample mean is a good estimate of the overall average.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. It involves formulating a null hypothesis (e.g., the population mean is less than or equal to 2.52 hours) and an alternative hypothesis (e.g., the population mean is greater than 2.52 hours). By analyzing the sample data, one can determine whether to reject or fail to reject the null hypothesis, providing insights into the population mean.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In this case, the population standard deviation is given as 0.068 hours, indicating how much the winning times deviate from the population mean. A smaller standard deviation suggests that the winning times are closely clustered around the mean, while a larger standard deviation indicates more variability, which is crucial for understanding the reliability of the sample mean in estimating the population mean.

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Calculating Standard Deviation

Related Practice

Textbook Question

The data set represents the amounts of time (in minutes) spent checking email for a random sample of employees at a company.

c. Repeat part (b), assuming σ = 3.5 minutes. Compare the results.

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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)

a. Find the point estimate of the population mean.

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Textbook Question

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

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Textbook Question

You wish to estimate the mean winning time for Boston Marathon Women’s Open Division champions. The estimate must be within 2 minutes of the population mean. Determine the minimum sample size required to construct a 99% confidence interval for the population mean. Use the population standard deviation from Exercise 1.

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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

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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