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.1a

Chapter 6, Problem 6.Q.1a

[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.

Verified step by step guidance

Step 1: Understand the problem. The goal is to find the point estimate of the population mean, which is the sample mean. The sample mean is calculated by summing all the data points and dividing by the total number of data points.

Step 2: Identify the data provided. The table contains 20 winning times (in hours) for the Boston Marathon Women’s Open Division champions. These values are the sample data.

Step 3: Write the formula for the sample mean. The formula is:

x̄ = \frac{\sum_{i = 1}^{n} x_{i}}{n}

, where

x̄

is the sample mean,

n

is the number of data points, and

x_{i}

are the individual data points.

Step 4: Add all the values in the table. Carefully sum each of the 20 winning times provided in the table. This will give the total sum of the data points.

Step 5: Divide the total sum by the number of data points (20). This will yield the sample mean, which is the point estimate of the population mean.

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Key Concepts

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

Point Estimate

A point estimate is a single value used to estimate a population parameter. In this context, the point estimate of the population mean is calculated by taking the average of the winning times of the sampled Boston Marathon champions. It provides a quick summary of the data and serves as a best guess for the true population mean.

Population Mean

The population mean is the average of all values in a population, representing a central tendency. It is a key parameter in statistics, often denoted by the symbol μ. In this case, the population mean would reflect the average winning time of all Boston Marathon Women’s Open Division champions from 1980 to 2019, which we aim to estimate using the sample data.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It indicates how much individual data points differ from the mean. In this question, the population standard deviation is given as 0.068 hours, which helps assess the reliability of the point estimate and the spread of winning times among the sampled champions.

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Related Practice

Textbook Question

Finding Critical Values for χ2 In Exercises 3–8, find the critical values χR2 and χL2 for the level of confidence c and sample size n.

c = 0.98, n = 26

Textbook Question

Translating Statements In Exercises 29–34, translate the statement into a confidence interval. Approximate the level of confidence.

In a survey of 1052 parents of children ages 8–14, 68% say they are willing to get a second or part-time job to pay for their children’s college education, and 42% say they lose sleep worrying about college costs. The survey’s margin of error is ±3%. (Source: T. Rowe Price Group, Inc.)

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.

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)

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

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.

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?