Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.3.15

Chapter 6, Problem 6.3.15

LGBT Identification In a survey of 15,349 U.S. adults, 860 identify as lesbian, gay, bisexual, or transgender. Construct a 95% confidence interval for the population proportion of U.S. adults who identify as lesbian, gay, bisexual, or transgender. (Adapted from Gallup)

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Step 1: Identify the sample proportion (p̂). The sample proportion is calculated as the number of individuals identifying as LGBT divided by the total sample size. Use the formula:

p̂ = \frac{x}{n}

, where x = 860 and n = 15,349.

Step 2: Determine the standard error (SE) of the sample proportion. The formula for the standard error is:

SE = \sqrt{\frac{p̂}{(}}

. Substitute the value of p̂ from Step 1 and n = 15,349 into this formula.

Step 3: Identify the critical value (z*) for a 95% confidence level. For a 95% confidence interval, the critical value z* is approximately 1.96. This value corresponds to the z-score that captures the middle 95% of the standard normal distribution.

Step 4: Calculate the margin of error (ME). The formula for the margin of error is:

ME = z* × SE

. Use the z* value from Step 3 and the SE value from Step 2 to compute the margin of error.

Step 5: Construct the confidence interval. The confidence interval is given by:

[p̂ - ME, p̂ + ME]

. Substitute the values of p̂ from Step 1 and ME from Step 4 to find the lower and upper bounds of the interval.

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Key Concepts

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

Population Proportion

The population proportion refers to the fraction of a population that exhibits a certain characteristic, in this case, the proportion of U.S. adults identifying as lesbian, gay, bisexual, or transgender. It is denoted as 'p' and is calculated by dividing the number of individuals with the characteristic by the total number of individuals in the population.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence, such as 95%. It provides an estimate of uncertainty around the sample proportion and is calculated using the sample proportion, the standard error, and a critical value from the normal distribution.

Standard Error

The standard error measures the variability of the sample proportion and is crucial for constructing confidence intervals. It is calculated using the formula SE = sqrt[(p(1-p)/n)], where 'p' is the sample proportion and 'n' is the sample size. A smaller standard error indicates that the sample proportion is a more accurate estimate of the population proportion.

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Related Practice

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

101

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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)

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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

When estimating the population mean, why not construct a 99% confidence interval every time?

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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

In Exercise 31, the population mean salary is \$67,319. Does the t-value fall between -t0.98 and t0.98? (Source: Salary.com)

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