Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

Statistics

7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

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1:37 minutes

Problem 6.Q.1a

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.

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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Textbook Question

When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.

b. Increase in the sample size

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Textbook Question

When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.

c. Increase in the population standard deviation

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Textbook Question

Determining a Minimum Sample Size Determine the minimum sample size required when you want to be 99% confident that the sample mean is within two units of the population mean and σ = 1.4. Assume the population is normally distributed.

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Textbook Question

Paint Can Volumes A paint manufacturer uses a machine to fill gallon cans with paint (see figure). The manufacturer wants to estimate the mean volume of paint the machine is putting in the cans within 0.5 ounce. Assume the population of volumes is normally distributed.

a. Determine the minimum sample size required to construct a 90% confidence interval for the population mean. Assume the population standard deviation is 0.75 ounce.

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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)

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

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Textbook Question

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

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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

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

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