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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1:36 minutes

Problem 6.Q.1b

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.

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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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)

a. Find the point estimate of the population mean.

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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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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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