Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.3.16

Chapter 6, Problem 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)

Verified step by step guidance

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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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 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.

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.

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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Related Practice

Textbook Question

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

(0.512, 0.596)

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

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

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 661 non-retired Americans, 218 said that they expect to rely on Social Security as major source of income when they retire. (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.