Transcript Ch7-3
Section 7.3 Estimating a Population mean µ (σ known) Objective Find the confidence interval for a population mean µ when σ is known Determine the sample size needed to estimate a population mean µ when σ is known Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 1 Best Point Estimation The best point estimate for a population mean µ (σ known) is the sample mean x Best point estimate : x Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 2 Notation = population mean = population standard deviation x = sample mean n = number of sample values E = margin of error z/2 = z-score separating an area of α/2 in the right tail of the standard normal distribution Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 3 Requirements (1) The population standard deviation σ is known (2) One or both of the following: The population is normally distributed or n > 30 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 4 Margin of Error Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 5 Confidence Interval ( x – E, x + E ) where Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 6 Definition The two values x – E and x + E are called confidence interval limits. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7 Round-Off Rules for Confidence Intervals Used to Estimate µ 1. When using the original set of data, round the confidence interval limits to one more decimal place than used in original set of data. 2. When the original set of data is unknown and only the summary statistics (n, x, s) are used, round the confidence interval limits to the same number of decimal places used for the sample mean. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 8 Example Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of size 42 has a mean of 38.4 Direct Computation Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 9 Example Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of size 42 has a mean of 38.4 Using StatCrunch Stat → Z statistics → One Sample → with Summary Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10 Example Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of size 42 has a mean of 38.4 Using StatCrunch Enter Parameters Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 11 Example Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of size 42 has a mean of 38.4 Using StatCrunch Click Next Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 12 Example Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of size 42 has a mean of 38.4 Using StatCrunch Select ‘Confidence Interval’ Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 13 Example Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of size 42 has a mean of 38.4 Using StatCrunch Enter Confidence Level, then click ‘Calculate’ Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 14 Example Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of size 42 has a mean of 38.4 Using StatCrunch Standard Error Lower Limit Upper Limit From the output, we find the Confidence interval is CI = (35.862, 40.938) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 15 Sample Size for Estimating a Population Mean = population mean σ = population standard deviation x = sample mean E = desired margin of error zα/2 = z score separating an area of /2 in the right tail of the standard normal distribution n= (z/2) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 2 E 16 Round-Off Rule for Determining Sample Size If the computed sample size n is not a whole number, round the value of n up to the next larger whole number. Examples: n = 310.67 n = 295.23 n = 113.01 round up to 311 round up to 296 round up to 114 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 17 Example We want to estimate the mean IQ score for the population of statistics students. How many statistics students must be randomly selected for IQ tests if we want 95% confidence that the sample mean is within 3 IQ points of the population mean? What we know: /2 = 0.025 = 0.05 n = 1.96 • 15 = 15 2 = 96.04 = 97 3 z / 2 = 1.96 (using StatCrunch) E=3 With a simple random sample of only 97 statistics students, we will be 95% confident that the sample mean is within 3 IQ points of the true population mean . Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 18 Summary Confidence Interval of a Mean µ (σ known) σ = population standard deviation x = sample mean n = number sample values 1 – α = Confidence Level ( x – E, x + E ) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 19 Summary Sample Size for Estimating a Mean µ (σ known) E = desired margin of error σ = population standard deviation x = sample mean 1 – α = Confidence Level n= (z/2) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 2 E 20