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

2. 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 2012. 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. 20.04!

3. 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%.

4. 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.

5. G RAPH Common critical values: If level of confidence is:α/2 is:critical value is: 90% 0.051.645 95% 0.0251.96 99% 0.0052.575

6. 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.

7. 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

8. 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. 104.3293.97108.73104.11100.67 96.8799.74100.25101.32 94.24 102.2394.32 97.66101.44 98.19 107.15100.38 95.89104.43102.28 Margin of Error = z 0.05/2 = 8.10

9. C ONFIDENCE INTERVALS 104.32 + 8.10 93.97 + 8.10 108.73 + 8.10 104.11 + 8.10 100.67 + 8.10 96.87 + 8.10 99.74 + 8.10... How many of the confidence intervals actually contain the population mean?

10. 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 59.2. 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.

11. 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: 55 45 45 42 65 5835 36 3460 Use the data to compute a point estimate for the data. Construct a 95% confidence interval for the mean.

12. A SSIGNMENT Page 4587 – 19 odd, 22 – 28 even