Understanding Confidence Intervals

Questions and Answers

What is the primary purpose of a confidence interval?

To make a probability statement about the sample statistic

To estimate a population parameter with a certain level of uncertainty (correct)

To compare the means of two populations

To determine the sample size required for a study

What is the relationship between the confidence level and the margin of error?

The margin of error is always half of the confidence level

As the confidence level decreases, the margin of error increases (correct)

As the confidence level increases, the margin of error decreases

The confidence level and margin of error are independent of each other

What is the formula for a one-sample confidence interval?

CI = sample statistic ± (critical value × margin of error)

CI = sample statistic ± (critical value × standard deviation)

CI = sample statistic ± (critical value × standard error) (correct)

CI = sample statistic ± (critical value × confidence level)

What is the main difference between a one-sample and two-sample confidence interval?

The formula used to calculate the interval Signup and view all the answers

What is the correct interpretation of a 95% confidence interval?

There is a 95% probability that the interval contains the true population parameter Signup and view all the answers

What is a common misconception about confidence intervals?

The true population parameter has a certain probability of lying within the interval Signup and view all the answers

What calculator function would you use to calculate a confidence interval for a population proportion?

1-PropZInt Signup and view all the answers

What is the effect of increasing the confidence level on the interval width?

The interval width increases Signup and view all the answers

What is the purpose of the critical value in a confidence interval calculation?

To calculate the margin of error Signup and view all the answers

What is a key assumption underlying confidence interval calculations?

The population is normally distributed Signup and view all the answers

Study Notes

Confidence Intervals

Definition

A confidence interval is a range of values within which a population parameter is likely to lie
It is a measure of the amount of uncertainty associated with a sample statistic

Key Concepts

Confidence Level: The probability that the interval contains the true population parameter (e.g. 95%)
Margin of Error: The maximum amount by which the sample statistic may differ from the true population parameter
Interval Width: The difference between the upper and lower bounds of the interval

Types of Confidence Intervals

One-Sample Confidence Interval
- Used to estimate a population mean (μ) or proportion (p)
- Formula: CI = sample statistic ± (critical value × standard error)
Two-Sample Confidence Interval
- Used to compare the means or proportions of two populations
- Formula: CI = (sample statistic 1 - sample statistic 2) ± (critical value × standard error)

Interpretation

95% Confidence Interval: There is a 95% probability that the true population parameter lies within the interval
Do not interpret the confidence interval as a probability statement about the sample statistic

Common Misconceptions

Do not say that the true population parameter has a 95% probability of lying within the interval
Do not assume that the interval is a range of values within which the sample statistic will lie

Calculator Functions

TI-83/84 Calculator: Use the 1-PropZInt or TInterval functions to calculate confidence intervals
Other Calculators: Check your calculator's manual for equivalent functions

Confidence Intervals

A range of values within which a population parameter is likely to lie, measuring the uncertainty associated with a sample statistic

Key Concepts

Confidence Level: The probability that the interval contains the true population parameter, e.g., 95%
Margin of Error: The maximum amount by which the sample statistic may differ from the true population parameter
Interval Width: The difference between the upper and lower bounds of the interval

Types of Confidence Intervals

One-Sample Confidence Interval: Estimates a population mean (μ) or proportion (p) using the formula: CI = sample statistic ± (critical value × standard error)
Two-Sample Confidence Interval: Compares the means or proportions of two populations using the formula: CI = (sample statistic 1 - sample statistic 2) ± (critical value × standard error)

Interpretation

A 95% Confidence Interval means there is a 95% probability that the true population parameter lies within the interval
Do not interpret the confidence interval as a probability statement about the sample statistic

Important Notes

Common Misconceptions: Do not say the true population parameter has a 95% probability of lying within the interval, or assume the interval is a range of values within which the sample statistic will lie

Calculator Functions

TI-83/84 Calculator: Use the 1-PropZInt or TInterval functions to calculate confidence intervals
Other Calculators: Check your calculator's manual for equivalent functions