Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo

Larson 8th Edition

Ch. 6 - Confidence Intervals

Problema 6.3.16

Capitolo 6, Problema 6.3.16

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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Step 1: Identify the sample proportion (p̂) and the sample size (n). The sample proportion is calculated as p̂ = x / n, where x is the number of individuals who consider themselves bisexual (378), and n is the total sample size (692).

Step 2: Determine the critical value (z*) for a 90% confidence level. For a 90% confidence interval, the critical value corresponds to the z-score that leaves 5% in each tail of the standard normal distribution. Use a z-table or statistical software to find this value.

Step 3: Calculate the standard error (SE) of the sample proportion using the formula: SE = sqrt((p̂ * (1 - p̂)) / n). This measures the variability of the sample proportion.

Step 4: Compute the margin of error (ME) using the formula: ME = z* × SE. This represents the range of error around the sample proportion.

Step 5: Construct the confidence interval for the population proportion using the formula: Confidence Interval = p̂ ± ME. This provides the range of plausible values for the true population proportion at the 90% confidence level.

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Population Proportion

The population proportion refers to the fraction of a population that exhibits a certain characteristic. In this context, it represents the proportion of lesbian, gay, bisexual, or transgender U.S. adults who identify as bisexual. Understanding this concept is crucial for interpreting survey results and making inferences about the larger population based on sample data.

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05:45

Constructing Confidence Intervals for Proportions

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. In this case, a 90% confidence interval means that if we were to take many samples and construct intervals, approximately 90% of those intervals would contain the true proportion of bisexual individuals in the population. This concept helps quantify the uncertainty associated with sample estimates.

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06:33

Introduction to Confidence Intervals

Sample Size and Its Impact

Sample size refers to the number of observations or data points collected in a survey or study. A larger sample size generally leads to more reliable estimates of population parameters and narrower confidence intervals. In this scenario, the sample size of 692 provides a basis for estimating the population proportion of bisexual individuals, influencing the precision of the confidence interval constructed.

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Sampling Distribution of Sample Proportion

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Domanda del libro di testo

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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Domanda del libro di testo

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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Domanda del libro di testo

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

In a survey of 2094 U.S. adults who have used an online dating app, 57% said their personal experience with online dating was positive. The survey’s margin of error is ±3.6%. (Source: Pew Research Center)

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Domanda del libro di testo

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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Domanda del libro di testo

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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Matching In Exercises 17–20, match the level of confidence c with the appropriate confidence interval. Assume each confidence interval is constructed for the same sample statistics.

c = 0.98

100

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