1. 8-1 Confidence Intervals Chapter Contents Confidence Interval for a Mean (μ) with Known σ Confidence Interval for a Mean (μ) with Unknown σ Confidence Interval for a Proportion (π) Estimating from Finite Populations Sample Size Determination for a Mean Sample Size Determination for a Proportion Chapter 8

2. 8-2 Chapter Learning Objectives (LO’s) Construct a 90, 95, or 99 percent confidence interval for μ. Know when to use Student’s t instead of z to estimate μ. Construct a 90, 95, or 99 percent confidence interval for π. Construct confidence intervals for finite populations. Calculate sample size to estimate a mean or proportion. Construct a confidence interval for a variance (optional). Chapter 8 Confidence Intervals

3. What is a Confidence Interval? What is a Confidence Interval? Chapter 8 Confidence Interval for a Mean (  ) with known (  ) 8-3

4. What is a Confidence Interval? What is a Confidence Interval? The confidence interval for  with known  is:The confidence interval for  with known  is: Chapter 8 Confidence Interval for a Mean (  ) with known (  ) 8-4

5. A higher confidence level leads to a wider confidence interval.A higher confidence level leads to a wider confidence interval. Choosing a Confidence Level Choosing a Confidence Level Greater confidence implies loss of precision (i.e. greater margin of error).Greater confidence implies loss of precision (i.e. greater margin of error). 95% confidence is most often used.95% confidence is most often used. Chapter 8 Confidence Intervals for Example 8.2 8-5

6. A confidence interval either does or does not contain .A confidence interval either does or does not contain . The confidence level quantifies the risk.The confidence level quantifies the risk. Out of 100 confidence intervals, approximately 95% may contain , while approximately 5% might not contain  when constructing 95% confidence intervals.Out of 100 confidence intervals, approximately 95% may contain , while approximately 5% might not contain  when constructing 95% confidence intervals. Interpretation Interpretation Chapter 8 When Can We Assume Normality? When Can We Assume Normality? If  is known and the population is normal, then we can safely use the formula to compute the confidence interval. If  is known and we do not know whether the population is normal, a common rule of thumb is that n  30 is sufficient to use the formula as long as the distribution Is approximately symmetric with no outliers. Larger n may be needed to assume normality if you are sampling from a strongly skewed population or one with outliers. 8-6

7. Use the Student’s t distribution instead of the normal distribution when the population is normal but the standard deviation  is unknown and the sample size is small.Use the Student’s t distribution instead of the normal distribution when the population is normal but the standard deviation  is unknown and the sample size is small. Student’s t Distribution Student’s t Distribution Chapter 8 Confidence Interval for a Mean (  ) with Unknown (  ) 8-7

8. Student’s t Distribution Student’s t Distribution Chapter 8 8-8

9. 8-9 Student’s t Distribution Student’s t Distribution t distributions are symmetric and shaped like the standard normal distribution.t distributions are symmetric and shaped like the standard normal distribution. The t distribution is dependent on the size of the sample.The t distribution is dependent on the size of the sample. Figure 8.11 Chapter 8 Comparison of Normal and Student’s t

10. 8-10 Degrees of Freedom Degrees of Freedom Degrees of Freedom (d.f.) is a parameter based on the sample size that is used to determine the value of the t statistic.Degrees of Freedom (d.f.) is a parameter based on the sample size that is used to determine the value of the t statistic. Degrees of freedom tell how many observations are used to calculate , less the number of intermediate estimates used in the calculation. The d.f for the t distribution in this case, is given by d.f. = n -1.Degrees of freedom tell how many observations are used to calculate , less the number of intermediate estimates used in the calculation. The d.f for the t distribution in this case, is given by d.f. = n -1. Chapter 8 As n increases, the t distribution approaches the shape of the normal distribution.As n increases, the t distribution approaches the shape of the normal distribution. For a given confidence level, t is always larger than z, so a confidence interval based on t is always wider than if z were used.For a given confidence level, t is always larger than z, so a confidence interval based on t is always wider than if z were used.

11. 8-11 Comparison of z and t Comparison of z and t For very small samples, t-values differ substantially from the normal. As degrees of freedom increase, the t-values approach the normal z-values. For example, for n = 31, the degrees of freedom, d.f. = 31 – 1 = 30. Chapter 8 So for a 90 percent confidence interval, we would use t = 1.697, which is only slightly larger than z = 1.645.

12. 8-12 Example GMAT Scores Again Example GMAT Scores Again Figure 8.13 Chapter 8

13. 8-13 Example GMAT Scores Again Example GMAT Scores Again Construct a 90% confidence interval for the mean GMAT score of all MBA applicants.Construct a 90% confidence interval for the mean GMAT score of all MBA applicants. x = 510 s = 73.77 Since  is unknown, use the Student’s t for the confidence interval with d.f. = 20 – 1 = 19.Since  is unknown, use the Student’s t for the confidence interval with d.f. = 20 – 1 = 19. First find t  /2 = t.05 = 1.729 from T-table.First find t  /2 = t.05 = 1.729 from T-table. Chapter 8

14. 8-14 For a 90% confidence interval, use Appendix D to find t 0.05 = 1.729 with d.f. = 19.For a 90% confidence interval, use Appendix D to find t 0.05 = 1.729 with d.f. = 19. Note: One can use Excel, Minitab, etc. to obtain these values as well as to construct confidence Intervals. Chapter 8 We are 90 percent confident that the true mean GMAT score might be within the interval [481.48, 538.52]

15. 8-15 Confidence Interval Width Confidence Interval Width Confidence interval width reflects - the sample size, - the confidence level and - the standard deviation. To obtain a narrower interval and more precision - increase the sample size or - lower the confidence level (e.g., from 90% to 80% confidence ). Chapter 8

16. 8-16 Using T-Table Using T-Table Beyond d.f. = 50, Appendix D shows d.f. in steps of 5 or 10.Beyond d.f. = 50, Appendix D shows d.f. in steps of 5 or 10. If the table does not give the exact degrees of freedom, use the t-value for the next lower degrees of freedom.If the table does not give the exact degrees of freedom, use the t-value for the next lower degrees of freedom. This is a conservative procedure since it causes the interval to be slightly wider.This is a conservative procedure since it causes the interval to be slightly wider. A conservative statistician may use the t distribution for confidence intervals when σ is unknown because using z would underestimate the margin of error. Chapter 8

17. A proportion is a mean of data whose only values are 0 or 1.A proportion is a mean of data whose only values are 0 or 1. Chapter 8 Confidence Interval for a Proportion (  ) 8-17

18. The distribution of a sample proportion p = x/n is symmetric if  =.50 and regardless of , approaches symmetry as n increases.The distribution of a sample proportion p = x/n is symmetric if  =.50 and regardless of , approaches symmetry as n increases. Applying the CLT Applying the CLT Chapter 8 8-18

19. Rule of Thumb: The sample proportion p = x/n may be assumed to be normal if both n   10 and n(1-  )  10.Rule of Thumb: The sample proportion p = x/n may be assumed to be normal if both n   10 and n(1-  )  10. When is it Safe to Assume Normality of p? When is it Safe to Assume Normality of p? Sample size to assume normality: Table 8.9 Chapter 8 8-19

20. Confidence Interval for  Confidence Interval for  Since  is unknown, the confidence interval for p = x/n (assuming a large sample) isSince  is unknown, the confidence interval for p = x/n (assuming a large sample) is Chapter 8 8-20

21. Example Auditing Example Auditing Chapter 8 8-21

22. 8-22 Chapter 8 N = population size; n = sample size

23. 8-23 To estimate a population mean with a precision of + E (allowable error), you would need a sample of size. Now,To estimate a population mean with a precision of + E (allowable error), you would need a sample of size. Now, Sample Size to Estimate  Sample Size to Estimate  Chapter 8 Sample Size determination for a Mean

24. 8-24 Method 1: Take a Preliminary Sample Take a small preliminary sample and use the sample s in place of  in the sample size formula.Method 1: Take a Preliminary Sample Take a small preliminary sample and use the sample s in place of  in the sample size formula. Method 2: Assume Uniform Population Estimate rough upper and lower limits a and b and set  = [(b-a)/12] ½.Method 2: Assume Uniform Population Estimate rough upper and lower limits a and b and set  = [(b-a)/12] ½. How to Estimate  ? How to Estimate  ? Chapter 8 Method 3: Assume Normal Population Estimate rough upper and lower limits a and b and set  = (b-a)/4. This assumes normality with most of the data with  ± 2  so the range is 4 .Method 3: Assume Normal Population Estimate rough upper and lower limits a and b and set  = (b-a)/4. This assumes normality with most of the data with  ± 2  so the range is 4 . Method 4: Poisson Arrivals In the special case when  is a Poisson arrival rate, then  = Method 4: Poisson Arrivals In the special case when  is a Poisson arrival rate, then  = 

25. 8-25 To estimate a population proportion with a precision of ± E (allowable error), you would need a sample of sizeTo estimate a population proportion with a precision of ± E (allowable error), you would need a sample of size Since  is a number between 0 and 1, the allowable error E is also between 0 and 1.Since  is a number between 0 and 1, the allowable error E is also between 0 and 1. Chapter 8

26. 8-26 Method 1: Assume that  =.50 This conservative method ensures the desired precision. However, the sample may end up being larger than necessary.Method 1: Assume that  =.50 This conservative method ensures the desired precision. However, the sample may end up being larger than necessary. Method 2: Take a Preliminary Sample Take a small preliminary sample and use the sample p in place of  in the sample size formula.Method 2: Take a Preliminary Sample Take a small preliminary sample and use the sample p in place of  in the sample size formula. Method 3: Use a Prior Sample or Historical Data How often are such samples available? Unfortunately,  might be different enough to make it a questionable assumption.Method 3: Use a Prior Sample or Historical Data How often are such samples available? Unfortunately,  might be different enough to make it a questionable assumption. How to Estimate  ? How to Estimate  ? Chapter 8