© 2010 Pearson Prentice Hall. All rights reserved Chapter Estimating the Value of a Parameter Using Confidence Intervals 9

© 2010 Pearson Prentice Hall. All rights reserved Section The Logic in Constructing Confidence Intervals for a Population Mean When the Population Standard Deviation Is Known 9.1

© 2010 Pearson Prentice Hall. All rights reserved Compute a point estimate of the population mean 2.Construct and interpret a confidence interval for the population mean assuming that the population standard deviation is known 3.Understand the role of margin of error in constructing the confidence interval 4.Determine the sample size necessary for estimating the population mean within a specified margin of error Objectives

© 2010 Pearson Prentice Hall. All rights reserved 9-4 Objective 1 Compute a Point Estimate of the Population Mean

© 2010 Pearson Prentice Hall. All rights reserved 9-5 A point estimate is the value of a statistic that estimates the value of a parameter. For example, the sample mean,, is a point estimate of the population mean .

© 2010 Pearson Prentice Hall. All rights reserved 9-6 Pennies minted after 1982 are made from 97.5% zinc and 2.5% copper. The following data represent the weights (in grams) of 17 randomly selected pennies minted after Treat the data as a simple random sample. Estimate the population mean weight of pennies minted after Parallel Example 1: Computing a Point Estimate

© 2010 Pearson Prentice Hall. All rights reserved 9-7 The sample mean is The point estimate of  is grams. Solution

© 2010 Pearson Prentice Hall. All rights reserved 9-8 Objective 2 Construct and Interpret a Confidence Interval for the Population Mean

© 2010 Pearson Prentice Hall. All rights reserved 9-9 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-  )·100%.

© 2010 Pearson Prentice Hall. All rights reserved 9-10 For example, a 95% level of confidence (  =0.05) implies that if 100 different confidence intervals are constructed, each based on a different sample from the same population, we will expect 95 of the intervals to contain the parameter and 5 to not include the parameter.

© 2010 Pearson Prentice Hall. All rights reserved 9-11 Confidence interval estimates for the population mean are of the form Point estimate ± margin of error. The margin of error of a confidence interval estimate of a parameter is a measure of how accurate the point estimate is.

© 2010 Pearson Prentice Hall. All rights reserved 9-12 The margin of error depends on three factors: 1. Level of confidence: As the level of confidence increases, the margin of error also increases. 2. Sample size: As the size of the random sample increases, the margin of error decreases. 3. Standard deviation of the population: The more spread there is in the population, the wider our interval will be for a given level of confidence.

© 2010 Pearson Prentice Hall. All rights reserved 9-13 The shape of the distribution of all possible sample means will be normal, provided the population is normal or approximately normal, if the sample size is large (n≥30), with mean and standard deviation.

© 2010 Pearson Prentice Hall. All rights reserved 9-14 Because is normally distributed, we know 95% of all sample means lie within 1.96 standard deviations of the population mean,, and 2.5% of the sample means lie in each tail.

© 2010 Pearson Prentice Hall. All rights reserved 9-15

© 2010 Pearson Prentice Hall. All rights reserved % of all sample means are in the interval With a little algebraic manipulation, we can rewrite this inequality and obtain:

© 2010 Pearson Prentice Hall. All rights reserved 9-17 It is common to write the 95% confidence interval as so that it is of the form Point estimate ± margin of error..

© 2010 Pearson Prentice Hall. All rights reserved 9-18 We will use Minitab to simulate obtaining 30 simple random samples of size n=8 from a population that is normally distributed with  =50 and  =10. Construct a 95% confidence interval for each sample. How many of the samples result in intervals that contain  =50 ? Parallel Example 2: Using Simulation to Demonstrate the Idea of a Confidence Interval

© 2010 Pearson Prentice Hall. All rights reserved 9-19 Sample Mean 95.0% CI C ( 40.14, 54.00) C ( 42.40, 56.26) C ( 43.69, 57.54) C ( 40.98, 54.84) C ( 37.38, 51.24) C ( 44.57, 58.43) C ( 45.54, 59.40) C ( 36.56, 50.42) C ( 48.52, 62.38) C ( 43.15, 57.01) C ( 49.44, 63.30) C ( 42.12, 55.98) C ( 40.41, 54.27) C ( 43.40, 57.25) C ( 52.69, 66.54)

© 2010 Pearson Prentice Hall. All rights reserved 9-20 SAMPLE MEAN95% CI C ( 37.88, 51.74) C ( 44.12, 57.98) C ( 36.98, 50.84) C ( 39.57, 53.43) C ( 42.86, 56.72) C ( 41.82, 55.68) C ( 44.34, 58.20) C ( 40.87, 54.73) C ( 49.67, 63.52) C ( 40.77, 54.63) C ( 44.65, 58.51) C ( 40.44, 54.30) C ( 39.96, 53.82) C ( 44.99, 58.85) C ( 54.49, 68.35)

© 2010 Pearson Prentice Hall. All rights reserved 9-21 Note that 28 out of 30, or 93%, of the confidence intervals contain the population mean  =50. In general, for a 95% confidence interval, any sample mean that lies within 1.96 standard errors of the population mean will result in a confidence interval that contains . Whether a confidence interval contains  depends solely on the sample mean,.

© 2010 Pearson Prentice Hall. All rights reserved 9-22 Interpretation of a Confidence Interval A (1-  )·100% confidence interval indicates that, if we obtained many simple random samples of size n from the population whose mean, , is unknown, then approximately (1-  )·100% of the intervals will contain . For example, if we constructed a 99% confidence interval with a lower bound of 52 and an upper bound of 71, we would interpret the interval as follows: “We are 99% confident that the population mean, , is between 52 and 71.”

© 2010 Pearson Prentice Hall. All rights reserved 9-23 Constructing a (1-  )·100% Confidence Interval for ,  Known Suppose that a simple random sample of size n is taken from a population with unknown mean, , and known standard deviation . A (1-  )·100% confidence interval for  is given by where is the critical Z-value. Note: The sample size must be large (n≥30) or the population must be normally distributed. LowerUpper Bound:Bound:

© 2010 Pearson Prentice Hall. All rights reserved 9-24 Construct a 99% confidence interval about the population mean weight (in grams) of pennies minted after Assume  =0.02 grams Parallel Example 3: Constructing a Confidence Interval

© 2010 Pearson Prentice Hall. All rights reserved 9-25

© 2010 Pearson Prentice Hall. All rights reserved 9-26 Weight (in grams) of Pennies

© 2010 Pearson Prentice Hall. All rights reserved 9-27 Lower bound: = = = Upper bound: = = = We are 99% confident that the mean weight of pennies minted after 1982 is between and grams.

© 2010 Pearson Prentice Hall. All rights reserved 9-28 Objective 3 Understand the Role of the Margin of Error in Constructing a Confidence Interval

© 2010 Pearson Prentice Hall. All rights reserved 9-29 The margin of error, E, in a (1-  )·100% confidence interval in which  is known is given by where n is the sample size. Note: We require that the population from which the sample was drawn be normally distributed or the samples size n be greater than or equal to 30.

© 2010 Pearson Prentice Hall. All rights reserved 9-30 Parallel Example 5: Role of the Level of Confidence in the Margin of Error Construct a 90% confidence interval for the mean weight of pennies minted after Comment on the effect that decreasing the level of confidence has on the margin of error.

© 2010 Pearson Prentice Hall. All rights reserved 9-31 Lower bound: = = = Upper bound: = = = We are 90% confident that the mean weight of pennies minted after 1982 is between and grams.

© 2010 Pearson Prentice Hall. All rights reserved 9-32 Notice that the margin of error decreased from to when the level of confidence decreased from 99% to 90%. The interval is therefore wider for the higher level of confidence. Confidence Level Margin of Error Confidence Interval 90%0.008(2.456, 2.472) 99%0.012(2.452, 2.476)

© 2010 Pearson Prentice Hall. All rights reserved 9-33 Parallel Example 6: Role of Sample Size in the Margin of Error Suppose that we obtained a simple random sample of pennies minted after Construct a 99% confidence interval with n=35. Assume the larger sample size results in the same sample mean, The standard deviation is still  =0.02. Comment on the effect increasing sample size has on the width of the interval.

© 2010 Pearson Prentice Hall. All rights reserved 9-34 Lower bound: = = = Upper bound: = = = We are 99% confident that the mean weight of pennies minted after 1982 is between and grams.

© 2010 Pearson Prentice Hall. All rights reserved 9-35 Notice that the margin of error decreased from to when the sample size increased from 17 to 35. The interval is therefore narrower for the larger sample size. Sample Size Margin of Error Confidence Interval (2.452, 2.476) (2.455, 2.473)

© 2010 Pearson Prentice Hall. All rights reserved 9-36 Objective 4 Determine the Sample Size Necessary for Estimating the Population Mean within a Specified Margin of Error

© 2010 Pearson Prentice Hall. All rights reserved 9-37 Determining the Sample Size n The sample size required to estimate the population mean, , with a level of confidence (1-  )·100% with a specified margin of error, E, is given by where n is rounded up to the nearest whole number.

© 2010 Pearson Prentice Hall. All rights reserved 9-38 Back to the pennies. How large a sample would be required to estimate the mean weight of a penny manufactured after 1982 within grams with 99% confidence? Assume  =0.02. Parallel Example 7: Determining the Sample Size

© 2010 Pearson Prentice Hall. All rights reserved 9-39  =0.02 E=0.005 Rounding up, we find n=107.