Confidence Intervals

Definitions

Confidence Interval Estimate: a range of values so constructed that a specified proportion of the intervals costructed that way would contain the true value of a parameter.

Critical Value: a point on the test distribution that establishes the width of the confidence interval for a given level of confidence.

Degrees of Freedom: the number of independent observations in a sample minus the number of population parameters that must be estimated from sample data.

Interval Estimate: defined by two numbers, between which a population parameter is said to lie.

Level of Confidence: the percentage of all possible samples that can be expected to include the true population parameter.

Margin of Error: the range of values above and below the sample statistic.

Point Estimate: a single value given as an estimate of a parameter of a population.

Student's t-Distribution: a (family of) probability distribution that is used to estimate population parameters when the population variance is unknown. It depends on a single parameter, the degrees of freedom.

These are the illustrations I have used in class:

Understanding the t-distribution and its normal approximation from Kristoffer Magnusson's blog.

Clinton maintains lead after claiming nomination - CBS News poll

• review the last paragraph
• can you explain the size of the sample in this survey?

Notes

The general formula for the interval estimate is Point Estimate ± (Critical Value)×(Standard Error).

The choice of a point estimate is the easiest:

• to estimate population mean μ, use the sample mean x̄
• to estimate population proportion π, use the sample proportion p.

Choosing critical value depends on 2 things, desired confidence level and knowledge of the population standard deviation.

• desired confidence level gives you α
• if you know the population standard deviation σ, use normal distribution (for the mean)
• if you do not know the population standard deviation σ, use Student's t distribution (for the mean) with degrees of freedom (n-1)
• for the proportion, use normal distribution

The standard error is the sampling distribution's parameter.

• population standard deviation σ divided by square root of the sample size, if you know the population standard deviation.
• sample standard deviation s divided by square root of the sample size, if you do not know the population standard deviation.

Read These

Chapter 8. Confidence Interval Estimation in the textbook:

8.1 Confidence Interval Estimate for the Mean (σ Known) (pp. 273-278)

8.2 Confidence Interval Estimate for the Mean (σ Unknown) (pp. 279-285)

8.3 Confidence Interval Estimate for the Proportion (pp. 287-289)

8.4 Determining Sample Size (pp. 290-294)

You may omit sections 8.5 Confidence Interval Estimation and Ethical Issues, 8.6 Application of Confidence Interval Estimation in Auditing, 8.7 Estimation and Sample Size Estimation for Finite Populations, and 8.8 Bootstrapping (pp. 295-296)

Watch This

Confidence Intervals for a Population Mean This is the list of 10 videos discussing the concepts and and the practical advice to build confidence intervals. You may want to review all of them or just those that you feel you have difficulties with.

Answer These

Do problem 8.63 (p. 300 in the textbook).

Do problem 8.65 (p. 300 in the textbook).