1. 1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 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

2. 2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Best Point Estimation The best point estimate for a population mean µ (σ known) is the sample mean x Best point estimate : x

3. 3 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.  = population mean  = population standard deviation = 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 Notation

4. 4 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (1) The population standard deviation σ is known (2) One or both of the following: The population is normally distributed or n > 30 Requirements

5. 5 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Margin of Error

6. 6 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Confidence Interval ( x – E, x + E ) where

7. 7 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Definition The two values x – E and x + E are called confidence interval limits.

8. 8 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 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. Round-Off Rules for Confidence Intervals Used to Estimate µ

9. 9 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 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 Example

10. 10 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using StatCrunch Stat → Z statistics → One Sample → with Summary 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 Example

11. 11 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using StatCrunch Enter Parameters 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 Example

12. 12 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using StatCrunch Click Next 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 Example

13. 13 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using StatCrunch Select ‘Confidence Interval’ 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 Example

14. 14 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using StatCrunch Enter Confidence Level, then click ‘Calculate’ 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 Example

15. 15 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using StatCrunch From the output, we find the Confidence interval is CI = (35.862, 40.938) Lower Limit Upper Limit Standard Error 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 Example

16. 16 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Sample Size for Estimating a Population Mean ( z  /2 )   n = E 2  = population mean σ = population standard deviation = sample mean E = desired margin of error z α/2 = z score separating an area of  /2 in the right tail of the standard normal distribution

17. 17 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 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 round up to 311 n = 295.23 round up to 296 n = 113.01 round up to 114

18. 18 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.  / 2 = 0.025 z  / 2 = 1.96 (using StatCrunch) 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? Example n = 1.96 15 = 96.04 = 97 3 2 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 . What we know:  = 0.05 E = 3  = 15

19. 19 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Summary Confidence Interval of a Mean µ (σ known) ( x – E, x + E ) σ = population standard deviation x = sample mean n = number sample values 1 – α = Confidence Level

20. 20 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. ( z  /2 )   n = E 2 E = desired margin of error σ = population standard deviation x = sample mean 1 – α = Confidence Level Summary Sample Size for Estimating a Mean µ (σ known)