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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2:05 minutes

Problem 6.1.43c

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

Verified step by step guidance

Understand the formula for the confidence interval: The width of a confidence interval is determined by the formula: $ \text{Width} = 2 \cdot Z \cdot \frac{\sigma}{\sqrt{n}} $, where $ Z $ is the critical value, $ \sigma $ is the population standard deviation, and $ n $ is the sample size.

Identify the role of the population standard deviation ($ \sigma $): The population standard deviation is in the numerator of the formula for the margin of error, which directly affects the width of the confidence interval.

Analyze the effect of increasing $ \sigma $: When $ \sigma $ increases, the value of $ \frac{\sigma}{\sqrt{n}} $ also increases, leading to a larger margin of error.

Relate the margin of error to the width: Since the width of the confidence interval is twice the margin of error, an increase in $ \sigma $ will result in a wider confidence interval.

Conclude the relationship: Therefore, an increase in the population standard deviation will increase the width of the confidence interval, assuming all other factors remain constant.

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Key Concepts

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

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It is expressed with a certain level of confidence, typically 95% or 99%. The width of the interval reflects the uncertainty associated with estimating the parameter; a wider interval indicates more uncertainty.

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. A larger standard deviation indicates greater variability in the data, which can lead to wider confidence intervals when estimating population parameters.

Effect of Standard Deviation on Confidence Interval Width

When the population standard deviation increases, the width of the confidence interval also increases, assuming all other factors remain constant. This is because a larger standard deviation indicates more variability in the data, necessitating a broader range to maintain the same level of confidence in capturing the true population parameter.

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In Exercise 35, would it be unusual for the population mean to be over $1500? Explain.

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In Exercise 37, does it seem likely that the population mean could be greater than $70? Explain.

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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.

a. Increase in the level of confidence

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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

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

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

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