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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2:04 minutes

Problem 6.1.48

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.

Verified step by step guidance

Identify the formula for determining the minimum sample size for estimating a population mean:

n = (z_{α/2) σ})^2

, where

z_{α/2}

is the critical z-value,

σ

is the population standard deviation, and

E

is the margin of error.

Determine the critical z-value for a 99% confidence level. Since the confidence level is 99%, the significance level

α

is 0.01, and

α_{2}

is 0.005. Use a z-table or statistical software to find the z-value corresponding to a cumulative probability of 0.995.

Substitute the given values into the formula. Here,

σ

= 1.4 and

E

= 2. The formula becomes:

n = (\frac{z_{α/2}}{\cdot} / 2)^2

Simplify the expression inside the parentheses by multiplying the critical z-value by the standard deviation and dividing by the margin of error.

Square the result from the previous step to calculate the minimum sample size. If the result is not a whole number, always round up to the nearest integer, as sample size must be an integer.

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Key Concepts

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

Sample Size Determination

Sample size determination is the process of calculating the number of observations needed in a sample to achieve a desired level of confidence and precision. It is crucial in statistics to ensure that the sample accurately reflects the population, allowing for valid inferences. The formula often used involves the desired confidence level, the population standard deviation, and the margin of error.

Confidence Level

The confidence level represents the probability that the sample mean will fall within a specified range of the population mean. In this case, a 99% confidence level indicates that if we were to take many samples, 99% of the time, the sample mean would be within two units of the true population mean. This concept is essential for understanding the reliability of the sample estimates.

Margin of Error

The margin of error is the range within which the true population parameter is expected to lie, given a certain level of confidence. In this scenario, a margin of error of two units means that the sample mean should be within two units of the population mean. This concept is vital for determining how precise the sample estimate needs to be in relation to the population parameter.

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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.

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

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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Textbook Question

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

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