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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1:48 minutes

Problem 6.1.30

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.

Verified step by step guidance

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

n = (\frac{z_{c} \cdot σ}{E}) 2

, where zc is the critical z-value, σ is the population standard deviation, and E is the margin of error.

Determine the critical z-value (zc) for the given confidence level c = 0.95. Use a z-table or statistical software to find the z-value corresponding to a cumulative probability of 0.975 (since the confidence level is two-tailed).

Substitute the given values into the formula:

σ = 2.5

(population standard deviation) and

E = 1

(margin of error).

Simplify the numerator of the fraction by multiplying the critical z-value (zc) by the population standard deviation (σ). Then divide the result by the margin of error (E).

Square the result from the previous step to calculate the minimum sample size n. If n is not a whole number, always round up to the nearest whole number, 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 precision in estimating population parameters, such as means or proportions, while controlling for the margin of error.

Confidence Level (c)

The confidence level, denoted as c, represents the probability that the confidence interval will contain the true population parameter. A confidence level of 0.95 indicates that if the same population were sampled multiple times, approximately 95% of the calculated confidence intervals would capture the true parameter, reflecting a high degree of certainty in the estimate.

Margin of Error (E)

The margin of error, denoted as E, quantifies the range within which the true population parameter is expected to lie, given a certain confidence level. It is a critical component in survey results and statistical estimates, as it indicates the potential deviation from the observed sample statistic, thus influencing the reliability of the conclusions drawn from the data.

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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 = 0.90, σ = 6.8, 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

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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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Textbook Question

Constructing a Confidence Interval In Exercises 17–20, you are given the sample mean and the sample standard deviation. Assume the population is normally distributed and use the t-distribution to find the margin of error and construct a 95% confidence interval for the population mean. Interpret the results.

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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 = 0.95, σ = 2.5, 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 = 0.95, σ = 2.5, E = 1.