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Ch. 6 - Confidence Intervals
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 6 - Confidence IntervalsProblem 6.3.9
Chapter 6, Problem 6.3.9

In Exercises 7–10, use the confidence interval to find the margin of error and the sample proportion.
(0.512, 0.596)

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Identify the given confidence interval, which is (0.512, 0.596). The lower bound is 0.512, and the upper bound is 0.596.
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 proportion (p̂), use the formula: \(\hat{p}\) = \(\frac{\text{Upper Bound}\) + \(\text{Lower Bound}\)}{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 E.
Perform the addition and division in the sample proportion formula to calculate p̂.

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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., (0.512, 0.596)) and is associated with a confidence level, typically 95% or 99%. This means that if we were to take many samples and construct confidence intervals for each, a certain percentage of those intervals would contain the true parameter.
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Introduction to Confidence Intervals

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 proportion and the true population proportion. In the given interval (0.512, 0.596), the margin of error can be found by subtracting the lower limit from the upper limit and dividing by two.
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Finding the Minimum Sample Size Needed for a Confidence Interval

Sample Proportion

The sample proportion is the ratio of the number of successes in a sample to the total number of observations in that sample. It is denoted as 'p̂' and provides an estimate of the true population proportion. In the context of the confidence interval (0.512, 0.596), the sample proportion can be calculated as the midpoint of the interval, which gives a point estimate of the population proportion.
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Sampling Distribution of Sample Proportion
Related Practice
Textbook Question

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

In a survey of 1052 parents of children ages 8–14, 68% say they are willing to get a second or part-time job to pay for their children’s college education, and 42% say they lose sleep worrying about college costs. The survey’s margin of error is ±3%. (Source: T. Rowe Price Group, Inc.)

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Textbook Question

"Finding p^ and q^ In Exercises 3–6, let p be the population proportion for the situation. Find point estimates of p and q.

Social Security In a survey of 351 retired Americans, 200 said that they rely on Social Security as major source of income. (Adapted from Gallup)"

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Textbook Question

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

In a survey of 1502 U.S. adults, 31% said that they use Pinterest. The survey’s margin of error is ±2.9%. (Source: Pew Research Center)

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Textbook Question

In Exercises 9–12, construct the indicated confidence intervals for (a) the population variance and (b) the population standard deviation . Assume the sample is from a normally distributed population.

c = 0.95, s^2 = 11.56, n = 30

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Textbook Question

[APPLET] The winning times (in hours) for a sample of 20 randomly selected Boston Marathon Women’s Open Division champions from 1980 to 2019 are shown in the table at the left. Assume the population standard deviation is 0.068 hour. (Source: Boston Athletic Association)

a. Find the point estimate of the population mean.

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Textbook Question

You wish to estimate the mean winning time for Boston Marathon Women’s Open Division champions. The estimate must be within 2 minutes of the population mean. Determine the minimum sample size required to construct a 99% confidence interval for the population mean. Use the population standard deviation from Exercise 1.

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