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

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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.

views

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

views

Textbook Question

Car Batteries The reserve capacities (in hours) of 18 randomly selected automotive batteries have a sample standard deviation of 0.25 hour. Use an 80% level of confidence.

views

Textbook Question

Juice Dispensing Machine A beverage company uses a machine to fill half-gallon bottles with fruit juice (see figure). The company wants to estimate the mean volume of water the machine is putting in the bottles within 0.25 fluid ounce.

[IMAGE]

a. Determine the minimum sample size required to construct a 95% confidence interval for the population mean. Assume the population standard deviation is 1 fluid ounce.

views

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.

views

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)

views