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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1:26 minutes

Problem 6.1.28

Textbook Question

In Exercises 25–28, use the confidence interval to find the margin of error and the sample mean.
(3.144, 3.176)

Verified step by step guidance

Identify the given confidence interval, which is (3.144, 3.176). The lower bound is 3.144, and the upper bound is 3.176.

To find the margin of error (E), use the formula:

E = \frac{\text{Upper Bound} - \text{Lower Bound}}{2}

. Substitute the values of the upper and lower bounds into this formula.

To find the sample mean (

\bar{x}

), use the formula:

\bar{x} = \frac{\text{Upper Bound} + \text{Lower Bound}}{2}

. Substitute the values of the upper and lower bounds into this formula.

Perform the subtraction and division in the margin of error formula to calculate the margin of error.

Perform the addition and division in the sample mean formula to calculate the sample mean.

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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 as an interval (e.g., (3.144, 3.176)) and is associated with a confidence level, indicating the probability that the interval contains the parameter. Understanding confidence intervals is crucial for estimating population parameters and assessing the precision of sample estimates.

Margin of Error

The margin of error quantifies the uncertainty associated with a sample estimate. It is calculated as half the width of the confidence interval, representing the maximum expected difference between the sample statistic and the population parameter. In the given interval (3.144, 3.176), the margin of error would be 0.016, indicating how much the sample mean could vary from the true mean.

Sample Mean

The sample mean is the average of a set of observations from a sample, serving as a point estimate of the population mean. It is calculated by summing all sample values and dividing by the number of observations. In the context of the confidence interval (3.144, 3.176), the sample mean can be found as the midpoint of the interval, which provides a central value around which the data is distributed.

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Textbook Question

Matching In Exercises 17–20, match the level of confidence c with the appropriate confidence interval. Assume each confidence interval is constructed for the same sample statistics.

c = 0.88

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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 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 = 0.95, 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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In Exercises 25–28, use the confidence interval to find the margin of error and the sample mean.(3.144, 3.176)

Key Concepts

Confidence Interval

Margin of Error

Sample Mean

Watch next

In Exercises 25–28, use the confidence interval to find the margin of error and the sample mean.
(3.144, 3.176)