Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.3.31

Chapter 6, Problem 6.3.31

Translating Statements In Exercises 29–34, translate the statement into a confidence interval. Approximate the level of confidence.
In a survey of 1000 U.S. adults, 71% think teaching is one of the most important jobs in our country today. The survey’s margin of error is ±3%. (Source: Rasmussen Reports)

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Identify the sample proportion (p̂) from the problem. Here, 71% of the surveyed adults think teaching is one of the most important jobs. Convert this percentage to a decimal: p̂ = 0.71.

Determine the margin of error (ME) provided in the problem. The margin of error is ±3%, which can be written as ME = 0.03 in decimal form.

Construct the confidence interval using the formula: Confidence Interval = p̂ ± ME. This means the lower bound of the interval is p̂ - ME, and the upper bound is p̂ + ME.

Substitute the values into the formula. The lower bound is 0.71 - 0.03, and the upper bound is 0.71 + 0.03. This gives the range of the confidence interval.

Approximate the level of confidence. The margin of error is typically associated with a 95% confidence level unless otherwise stated. Therefore, the confidence level for this interval is approximately 95%.

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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 estimate, typically calculated as the sample proportion plus or minus the margin of error. For example, if 71% of respondents support a statement and the margin of error is ±3%, the confidence interval would be 68% to 74%.

Margin of Error

The margin of error quantifies the uncertainty associated with a sample estimate. It indicates the range within which the true population parameter is expected to fall, based on the sample data. In this case, a margin of error of ±3% means that the true percentage of U.S. adults who think teaching is important could be 3% higher or lower than the reported 71%.

Level of Confidence

The level of confidence reflects the degree of certainty that the true population parameter lies within the confidence interval. Common levels of confidence are 90%, 95%, and 99%. The level of confidence can be approximated based on the sample size and margin of error; for a sample of 1000 with a margin of error of ±3%, a 95% confidence level is often assumed.

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Introduction to Confidence Intervals

Related Practice

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.

Textbook Question

Translating Statements In Exercises 29–34, translate the statement into a confidence interval. Approximate the level of confidence.

In a survey of 220 U.S. adults ages 18–29, 65% said that they use Snapchat. The survey’s margin of error is ±7.9%. (Source: Pew Research Center)

Textbook Question

In Exercises 35–40, use the standard normal distribution or the t-distribution to construct a 95% confidence interval for the population mean. Justify your decision. If neither distribution can be used, explain why. Interpret the results.

In a random sample of 18 months from January 2011 through December 2020, the mean interest rate for 30-year fixed rate home mortgages was 3.95% and the standard deviation was 0.49%. Assume the interest rates are normally distributed. (Source: Freddie Mac)

Textbook Question

Finding Critical Values for χ2 In Exercises 3–8, find the critical values χR2 and χL2 for the level of confidence c and sample size n.

c = 0.99, n = 15

Textbook Question

In Exercise 28, the population mean weekly time spent on homework by students is 7.8 hours. Does the t-value fall between -t0.99 and t0.99?

Textbook Question

Why Check It? Why is it necessary to check that np^ ≥ 5 and nq^ ≥ 5?