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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

"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)"

Textbook Question

True or False? In Exercises 1 and 2, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

The point estimate for the population proportion of failures is 1-p^

Textbook Question

In Exercises 13–16, find the margin of error for the values of c, σ and n.

c = 0.975, σ = 4.6, n = 100

Textbook Question

Bisexual Idenfitication In a survey of 692 lesbian, gay, bisexual, or transgender U.S adults, 378 said that they consider themselves bisexual. Construct a 90% confidence interval for the population proportion of lesbian, gay, bisexual, or transgender U.S. adults who consider themselves bisexual. (Adapted from Gallup)

Textbook Question

For the same sample statistics, which level of confidence would produce the widest confidence interval? Explain your reasoning.

a. 90%

b. 95%

c. 98%

d. 99%

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.