CONFIDENCE INTERVALS of Means AP STATISTICS, CHAPTER 19 Mrs. Watkins

INFERENCE: Allows us to estimate population parameters based on results of one sample We “infer” that something is true about the population.

Purpose of a Confidence Interval Using a sample to estimate the true value of a population parameter, like the mean μ.

Estimators: A.Point estimates: as an estimate of μ B.Interval estimates: a range of values over which the true parameter will be located EX: __________________ ___________________

Confidence Interval (ESTIMATE) + MARGIN OF ERROR

A 95% confidence interval is We are 95% confident that the true population mean lies between _________ and ________. If we wanted to be more confident, then the interval will be _______________.

“the net” Wider intervals give more CONFIDENCE. Narrower intervals give more PRECISION.

Confidence Levels: 90% means 10% left out, so 5% on each side. Invnorm(.05) = -1.65, so this is what we call the Critical Value 95% --what would critical value be? 99% --what would critical value be?

FORMULA for Z Confidence Interval for Means

EXAMPLE #1 Estimate the mean size of a chocolate chip cookie using a 95% confidence interval. 1. Formula—what numbers do we need? 2. Meaning—what does the final answer indicate?

Meaning of the Confidence Interval Meaning of the Confidence Interval: We are 95% confident that the true population parameter (mean or proportion) is between _____________ and _____________.

EXAMPLE #2 A psychologist is doing a study on teen and sleep patterns. Prior studies show that teens should get 8 hours of sleep per night. A recent sample of 15 students indicates that the mean hours of sleep is 6.8 with a standard deviation of 1.25 hours. Calculate a 90% confidence intervals for the mean hours of sleep. Then do a 99% confidence interval.

Important questions about Confidence Intervals 1.How sure are we of that result? 2.What is our margin of error? 3.What effect does sample size have on our confidence in our prediction?

Meaning of the Confidence Level Meaning of the Confidence Level: In all possible samples of size n, the true population mean will be contained in 95% of the intervals generated.

Assumptions for Z conf. int. 1.Random Sample—usually stated 2. Population Standard Deviation KNOWN 3. Approximately normally distributed **May be stated or you may have to check box plot, normal prob plot or mean and median

t confidence intervals t distribution We use a t distribution when we do NOT know the population standard deviation. We RARELY know the population standard deviation, so most confidence intervals t intervals for the MEAN will be t intervals.

What is difference between z and t? Critical Values are different because t is actually a series of symmetric, mound shaped curves. Values depend on DEGREES of FREEDOM, df = n – 1 where n is sample size.

T distribution

Formula for t confidence interval

Different steps? Not much. 1. Different critical value 2. Same interpretation 3. Assumptions for t interval—very similar a. Random Sample? Usually stated b. Approximately normal? Usually have to check c. Population st. deviation unknown

Find the critical values 1.Z int, 95%, n = 12 2.t int, 95%, n = 12 3.t int, 98%, n = 22 4.Z int, 92%, n = 15 5.t int, 90%, n = 62 6.Z int, 94%, n = 30