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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1:46 minutes

Problem 6.1.34

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.

Verified step by step guidance

Step 1: Understand the concept of margin of error. The margin of error (E) is half the width of the confidence interval. It represents the maximum expected difference between the true population parameter and the sample estimate within the given confidence level.

Step 2: Calculate the width of the confidence interval by subtracting the lower bound from the upper bound. Use the formula:

Width = Upper - Lower

. In this case, the upper bound is 280.97 and the lower bound is 244.07.

Step 3: Divide the width of the confidence interval by 2 to find the margin of error. Use the formula:

E = \frac{Width}{2}

Step 4: Calculate the sample mean by finding the midpoint of the confidence interval. Use the formula:

Mean = \frac{Upper}{+}

. This gives the central value of the confidence interval.

Step 5: Summarize the results. The margin of error (E) and the sample mean are the two key values derived from the confidence interval. Ensure all calculations are accurate and clearly interpreted.

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Key Concepts

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

Margin of Error

The margin of error quantifies the uncertainty associated with a sample estimate. It represents the range within which the true population parameter is expected to lie, given a certain confidence level. In the context of a confidence interval, it is calculated as half the width of the interval, indicating how much the sample mean may differ from the actual population mean.

Confidence Interval

A confidence interval is a range of values, derived from a sample statistic, that is likely to contain the true population parameter with a specified level of confidence. For example, a 95% confidence interval suggests that if we were to take many samples, approximately 95% of the calculated intervals would contain the true mean. It provides a way to express the reliability of the estimate.

Sample Mean

The sample mean is the average value of a set of observations from a sample, calculated by summing all the sample values and dividing by the number of observations. It serves as a point estimate of the population mean and is central to inferential statistics, as it helps in making predictions about the population based on the sample data.

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Textbook Question

In Exercises 9–12, find the critical value tc for the level of confidence c and sample size n.

c = 0.98, n = 15

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Textbook Question

In Exercises 13–16, (a) find the margin of error for the values of c, s, and n, and (b) construct the confidence interval for using the t-distribution. Assume the population is normally distributed.

c = 0.90, s = 25.6, n = 16, xbar = 72.1

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Textbook Question

c = 0.98, s = 0.9, n = 12, xbar = 6.8

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Textbook Question

c = 0.99, s = 16.5, n = 20, xbar = 25.2

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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