Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.3.18a

Chapter 6, Problem 6.3.18a

Senate Filibuster You wish to estimate, with 99% confidence, the population proportion of U.S. adults who disapprove of the U.S Senate’s use of the filibuster. Your estimate must be accurate within 2% of the population proportion.
a. No preliminary estimate is available. Find the minimum sample size needed.

Verified step by step guidance

Step 1: Identify the formula for determining the minimum sample size for estimating a population proportion. The formula is:

n >= \frac{z^{2}}{} E^{2}

, where

z

is the z-score corresponding to the confidence level,

p

is the estimated population proportion, and

E

is the margin of error.

Step 2: Since no preliminary estimate of the population proportion is available, use

p = \frac{1}{2}

(i.e., 0.5) to maximize the required sample size. This is a conservative approach that ensures the sample size is large enough.

Step 3: Determine the z-score for a 99% confidence level. For a 99% confidence level, the z-score is approximately

2.576

. This value corresponds to the critical value in a standard normal distribution.

Step 4: Substitute the values into the formula. Use

z = 2.576

p = 0.5

, and

E = 0.02

(2% margin of error). The formula becomes:

n >= \frac{{2.576}^{2}}{} {0.02}^{2}

Step 5: Simplify the expression to calculate the minimum sample size. First, calculate the numerator

{2.576}^{2} * 0.5 * 0.5

, then divide by the square of the margin of error

{0.02}^{2}

. Round up the result to the nearest whole number, as sample size must be an integer.

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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, U.S. adults who disapprove of the Senate's use of the filibuster. Understanding this concept is crucial for estimating how widespread an opinion is within a larger group, which is essential for statistical analysis and survey design.

Sample Size Determination

Sample size determination is the process of calculating the number of observations or replicates needed to ensure that the sample accurately reflects the population. In this scenario, it involves using the desired confidence level and margin of error to find the minimum sample size required to estimate the population proportion with a specified accuracy.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. In this case, a 99% confidence level indicates that if the same sampling method were repeated multiple times, 99% of the calculated intervals would contain the true population proportion, providing a measure of reliability for the estimate.

Recommended video:

06:33

Introduction to Confidence Intervals

Related Practice

Textbook Question

Constructing Confidence Intervals In Exercises 13–24, assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance σ^2. Interpret the results.

Drive-Thru Times The times (in seconds) spent by a random sample of 28 customers in the drive-thru of a fast-food restaurant have a sample standard deviation of 56.1. Use a 98% level of confidence.

Textbook Question

Drug Concentration The times (in minutes) for the drug concentration to peak when the drug epinephrine is injected into 15 randomly selected patients are listed. Use a 90% level of confidence.

Textbook Question

When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.

a. Increase in the level of confidence

Textbook Question

Fast Food You wish to estimate, with 90% confidence, the population proportion of U.S. families who eat fast food at least once per week. Your estimate must be accurate within 3% of the population proportion.

a. No preliminary estimate is available. Find the minimum sample size needed.

Textbook Question

Cholesterol Contents of Cheese A cheese processing company wants to estimate the mean cholesterol content of all one-ounce servings of a type of cheese. The estimate must be within 0.75 milligram of the population mean.

a. Determine the minimum sample size required to construct a 95% confidence interval for the population mean. Assume the population standard deviation is 3.10 milligrams.

Textbook Question

Annual Precipitation The annual precipitation amounts (in inches) of a random sample of 61 years for Chicago, Illinois, have a sample standard deviation of 6.46. Use a 98% level of confidence. (Source: National Oceanic and Atmospheric Administration)