C HAPTER 9 C ONFIDENCE I NTERVALS 9.1 The Logic of Constructing Confidence Intervals Obj: Construct confidence intervals for means

E STIMATE THE VALUE OF A MEAN The high school had a vending machine with only milk. Estimate the average milk (plain or flavored) consumption per capita in the U.S. in Make a reasonable guess. On what do you base your answer? How confident are you? Give an interval centered about your guess in which you are 50% confident. Give an interval centered about your guess in which you are 90% confident. Give an interval centered about your guess in which you are 99% confident !

D EFINITIONS A point estimate is the value of a statistic that estimates the value of a parameter. A confidence interval for an unknown parameter consists of an interval of numbers. The level of confidence represents the expected proportion of intervals that will contain the parameter if a large number of different samples is obtained. The level of confidence is denoted (1 – α)100%.

C ONFIDENCE I NTERVALS Confidence interval estimated for the population mean are of the form Point estimate + margin of error The margin of error depends on the level of confidence, sample size, and standard deviation.

G RAPH Common critical values: If level of confidence is:α/2 is:critical value is: 90% % %

C ONSTRUCTING A C ONFIDENCE I NTERVAL If a sample size of n is taken from a population with an unknown mean, μ, and a known standard deviation, σ, then a (1 – α)100% confidence interval for μ is: Lower Bound = x – z α/2 Upper Bound = x + z α/2 We must have a sample size n > 30 or the population must be normally distributed.

M ARGIN OF E RROR The margin of error, E, in a confidence interval in which σ is known is given by E = z α/2

E XAMPLE We know that scores of the Stanford-Binet IQ test are normally distributed with μ = 100 and σ = 16. Use the following 20 sample means from samples of size n = 15 to construct 95% confidence intervals for the population mean μ Margin of Error = z 0.05/2 = 8.10

C ONFIDENCE INTERVALS How many of the confidence intervals actually contain the population mean?

E XAMPLE A simple random sample of size n is drawn from a population whose population standard deviation is known to be 3.8. The sample mean is determined to be Compute the 90% confidence interval about μ if the sample size n is 45. Compute the 90% confidence interval about μ if the sample size is 55. Compute the 98% confidence interval about μ if the sample size is 45.

E XAMPLE For a billing process, the number of days for customers to pay their bill from the date of invoice is approximately normally distributed, with mean μ = 47 days and σ = 11 days. A random sample of size 10 bills from the billing process during the month of June results in the following data: Use the data to compute a point estimate for the data. Construct a 95% confidence interval for the mean.

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