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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1:57 minutes

Problem 6.1.29

Textbook Question

In Exercises 29–32, determine the minimum sample size n needed to estimate for the values of c, σ, and E.
c = 0.90, σ = 6.8, E = 1.

Verified step by step guidance

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

n = (z

_c⋅σE)2, where z_c is the critical z-value, σ is the population standard deviation, and E is the margin of error.

Determine the critical z-value (z_c) for the given confidence level c = 0.90. Use a z-table or statistical software to find the z-value corresponding to the middle 90% of the standard normal distribution.

Substitute the given values into the formula: σ = 6.8 and E = 1. Replace z_c with the critical z-value obtained in the previous step.

Simplify the fraction

z

_c⋅σE by multiplying z_c and σ, then dividing by E.

Square the result of the fraction to compute the minimum sample size n. If n 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 or replicates needed in a statistical study to ensure that the results are reliable and valid. It is crucial for achieving a desired level of confidence and margin of error in estimates. The formula often used involves the population standard deviation, the desired confidence level, and the margin of error.

Confidence Level (c)

The confidence level represents the probability that the confidence interval will contain the true population parameter. A confidence level of 0.90 indicates that there is a 90% chance that the interval estimate will capture the true value. This level influences the width of the confidence interval and, consequently, the required sample size.

Margin of Error (E)

The margin of error is the range within which the true population parameter is expected to lie, given a certain confidence level. It quantifies the uncertainty associated with sample estimates. A smaller margin of error requires a larger sample size, as it indicates a desire for more precise estimates of the population parameter.

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Textbook Question

In Exercises 21–24, construct the indicated confidence interval for the population mean μ.

c = 0.95, xbar = 31.39, σ = 0.80, n = 82.

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Textbook Question

In Exercises 21–24, construct the indicated confidence interval for the population mean μ.

c = 0.80, xbar = 20.6, σ = 4.7, n = 100.

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Textbook Question

In Exercises 25–28, use the confidence interval to find the margin of error and the sample mean.

(21.61, 30.15)

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Textbook Question

In Exercises 25–28, use the confidence interval to find the margin of error and the sample mean.

(3.144, 3.176)

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Textbook Question

In Exercises 29–32, determine the minimum sample size n needed to estimate for the values of c, σ, and E.

c = 0.95, σ = 2.5, E = 1.

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Textbook Question

In Exercises 29–32, determine the minimum sample size n needed to estimate for the values of c, σ, and E.

c = 0.80, σ = 4.1, E = 2.

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Textbook Question

Use technology to find the standard deviation of the set of 36 sample means. How does it compare with the standard deviation of the ages found in Exercise 5? Does this agree with the result predicted by the Central Limit Theorem?

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Textbook Question

The initial pressures for bicycle tires when first filled are normally distributed, with a mean of 70 pounds per square inch (psi) and a standard deviation of 1.2 psi.

b. A random sample of 15 tires is drawn from this population. What is the probability that the mean tire pressure of the sample is less than 69 psi?

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In Exercises 29–32, determine the minimum sample size n needed to estimate for the values of c, σ, and E.c = 0.90, σ = 6.8, E = 1.

Key Concepts

Sample Size Determination

Confidence Level (c)

Margin of Error (E)

Watch next

In Exercises 29–32, determine the minimum sample size n needed to estimate for the values of c, σ, and E.
c = 0.90, σ = 6.8, E = 1.