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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1:59 minutes

Problem 6.1.43b

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

Verified step by step guidance

Understand the relationship between sample size and the width of a confidence interval: The width of a confidence interval is inversely related to the square root of the sample size. This means that as the sample size increases, the width of the confidence interval decreases.

Recall the formula for the margin of error in a confidence interval: $ \text{Margin of Error} = z^* \cdot \frac{\sigma}{\sqrt{n}} $, where $ z^* $ is the critical value, $ \sigma $ is the population standard deviation, and $ n $ is the sample size.

Analyze the effect of increasing the sample size $ n $: Since $ n $ appears in the denominator under a square root, increasing $ n $ will decrease the value of $ \frac{\sigma}{\sqrt{n}} $, which in turn reduces the margin of error.

Conclude how this impacts the confidence interval: A smaller margin of error results in a narrower confidence interval, meaning the range of values within which the population parameter is estimated becomes more precise.

Summarize the effect: Increasing the sample size reduces the width of the confidence interval, improving the precision of the estimate while keeping all other factors 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%, indicating the degree of certainty about the estimate. The width of the interval reflects the precision of the estimate; narrower intervals suggest more precise estimates.

Sample Size

Sample size refers to the number of observations or data points collected in a study. A larger sample size generally leads to more reliable estimates of population parameters, as it reduces the impact of random variability. In the context of confidence intervals, increasing the sample size typically results in a narrower interval, indicating greater precision in estimating the population parameter.

Margin of Error

The margin of error is the amount of error that can be tolerated in the estimate of a population parameter. It is influenced by the sample size and the variability of the data. A larger sample size decreases the margin of error, which in turn reduces the width of the confidence interval, allowing for a more accurate representation of the population parameter.

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Textbook Question

Finding the Margin of Error In Exercises 33 and 34, use the confidence interval to find the estimated margin of error. Then find the sample mean. Book Prices A store manager reports a confidence interval of (244.07, 280.97) when estimating the mean price (in dollars) for the population of textbooks.

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Textbook Question

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

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Textbook Question

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.

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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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

Key Concepts

Confidence Interval

Sample Size

Margin of Error

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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