1. 1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University

2. 2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 8 Interval Estimation Population Mean:  Known Population Mean:  Known Population Mean:  Unknown Population Mean:  Unknown n Determining the Sample Size n Population Proportion

3. 3 3 Slide © 2008 Thomson South-Western. All Rights Reserved A point estimator cannot be expected to provide the A point estimator cannot be expected to provide the exact value of the population parameter. exact value of the population parameter. A point estimator cannot be expected to provide the A point estimator cannot be expected to provide the exact value of the population parameter. exact value of the population parameter. An interval estimate can be computed by adding and An interval estimate can be computed by adding and subtracting a margin of error to the point estimate. subtracting a margin of error to the point estimate. An interval estimate can be computed by adding and An interval estimate can be computed by adding and subtracting a margin of error to the point estimate. subtracting a margin of error to the point estimate. Point Estimate +/  Margin of Error The purpose of an interval estimate is to provide The purpose of an interval estimate is to provide information about how close the point estimate is to information about how close the point estimate is to the value of the parameter. the value of the parameter. The purpose of an interval estimate is to provide The purpose of an interval estimate is to provide information about how close the point estimate is to information about how close the point estimate is to the value of the parameter. the value of the parameter. Margin of Error and the Interval Estimate

4. 4 4 Slide © 2008 Thomson South-Western. All Rights Reserved The general form of an interval estimate of a The general form of an interval estimate of a population mean is population mean is The general form of an interval estimate of a The general form of an interval estimate of a population mean is population mean is Margin of Error and the Interval Estimate

5. 5 5 Slide © 2008 Thomson South-Western. All Rights Reserved Interval Estimation of a Population Mean:  Known n In order to develop an interval estimate of a population mean, the margin of error must be computed using either: the population standard deviation , or the population standard deviation , or the sample standard deviation s the sample standard deviation s  is rarely known exactly, but often a good estimate can be obtained based on historical data or other information.  is rarely known exactly, but often a good estimate can be obtained based on historical data or other information. We refer to such cases as the  known case. We refer to such cases as the  known case.

6. 6 6 Slide © 2008 Thomson South-Western. All Rights Reserved There is a 1   probability that the value of a There is a 1   probability that the value of a sample mean will provide a margin of error of or less.    /2 1 -  of all values 1 -  of all values Sampling distribution of Sampling distribution of Interval Estimation of a Population Mean:  Known

7. 7 7 Slide © 2008 Thomson South-Western. All Rights Reserved    /2 1 -  of all values 1 -  of all values Sampling distribution of Sampling distribution of [------------------------- -------------------------] interval does not include  interval includes  interval Interval Estimate of a Population Mean:  Known

8. 8 8 Slide © 2008 Thomson South-Western. All Rights Reserved Interval Estimate of  Interval Estimate of  Interval Estimate of a Population Mean:  Known where: is the sample mean 1 -  is the confidence coefficient 1 -  is the confidence coefficient z  /2 is the z value providing an area of z  /2 is the z value providing an area of  /2 in the upper tail of the standard  /2 in the upper tail of the standard normal probability distribution normal probability distribution  is the population standard deviation  is the population standard deviation n is the sample size n is the sample size

9. 9 9 Slide © 2008 Thomson South-Western. All Rights Reserved Interval Estimate of a Population Mean:  Known n Adequate Sample Size In most applications, a sample size of n = 30 is In most applications, a sample size of n = 30 is adequate. adequate. In most applications, a sample size of n = 30 is In most applications, a sample size of n = 30 is adequate. adequate. If the population distribution is highly skewed or If the population distribution is highly skewed or contains outliers, a sample size of 50 or more is contains outliers, a sample size of 50 or more is recommended. recommended. If the population distribution is highly skewed or If the population distribution is highly skewed or contains outliers, a sample size of 50 or more is contains outliers, a sample size of 50 or more is recommended. recommended. With n in the denominator, a larger sample size will With n in the denominator, a larger sample size will provide a smaller margin of error, a narrower interval, and greater precision. With n in the denominator, a larger sample size will With n in the denominator, a larger sample size will provide a smaller margin of error, a narrower interval, and greater precision.

10. 10 Slide © 2008 Thomson South-Western. All Rights Reserved Interval Estimate of a Population Mean:  Known n Adequate Sample Size (continued) If the population is believed to be at least If the population is believed to be at least approximately normal, a sample size of less than 15 approximately normal, a sample size of less than 15 can be used. can be used. If the population is believed to be at least If the population is believed to be at least approximately normal, a sample size of less than 15 approximately normal, a sample size of less than 15 can be used. can be used. If the population is not normally distributed but is If the population is not normally distributed but is roughly symmetric, a sample size as small as 15 roughly symmetric, a sample size as small as 15 will suffice. will suffice. If the population is not normally distributed but is If the population is not normally distributed but is roughly symmetric, a sample size as small as 15 roughly symmetric, a sample size as small as 15 will suffice. will suffice. If the population is normally distributed, repeated generation of 95% confidence intervals will produce confidence intervals containing the mean exactly 95% of the time. If the population is normally distributed, repeated generation of 95% confidence intervals will produce confidence intervals containing the mean exactly 95% of the time.

11. 11 Slide © 2008 Thomson South-Western. All Rights Reserved S D Interval Estimate of Population Mean:  Known n Example: Discount Sounds Discount Sounds has 260 retail outlets throughout the United States. The firm is evaluating a potential location for a new outlet, based in part, on the mean annual income of the individuals in the marketing area of the new location. A sample of size n = 36 was taken; the sample mean income is $31,100. The population is not believed to be highly skewed. The population standard deviation is estimated to be $4,500, and the confidence coefficient to be used in the interval estimate is.95.

12. 12 Slide © 2008 Thomson South-Western. All Rights Reserved 95% of the sample means that can be observed are within + 1.96 of the population mean . The margin of error is: Thus, at 95% confidence, the margin of error Thus, at 95% confidence, the margin of error is $1,470. is $1,470. S D Interval Estimate of Population Mean:  Known

13. 13 Slide © 2008 Thomson South-Western. All Rights Reserved Interval Estimate of Population Mean:  Known Values of z a/2 for the most commonly used confidence levels: Confidence Levelaa/2z a/2 90%.10.051.645 95%.05.0251.960 99%.01.0052.576 Values of z a/2 for the most commonly used confidence levels: Confidence Levelaa/2z a/2 90%.10.051.645 95%.05.0251.960 99%.01.0052.576 Comparing the results for the 90%, 95%, and 99% confidence levels, we see that in order to have a higher degree of confidence, the margin of error and thus the width of the confidence interval must be larger. Comparing the results for the 90%, 95%, and 99% confidence levels, we see that in order to have a higher degree of confidence, the margin of error and thus the width of the confidence interval must be larger.

14. 14 Slide © 2008 Thomson South-Western. All Rights Reserved Interval estimate of  is: Interval Estimate of Population Mean:  Known S D We are 95% confident that the interval contains the population mean. We say that this interval has been established at the 95% confidence level. The value 95% is referred to as the confidence coefficient, and the interval $29,630 to $32,570 is called the 95% confidence interval. $31,100 + $1,470 or $29,630 to $32,570

15. 15 Slide © 2008 Thomson South-Western. All Rights Reserved Interval Estimation of a Population Mean:  Unknown If an estimate of the population standard deviation  cannot be developed prior to sampling, we use the sample standard deviation s to estimate . If an estimate of the population standard deviation  cannot be developed prior to sampling, we use the sample standard deviation s to estimate . This is the  unknown case. This is the  unknown case. In this case, the interval estimate for  is based on the t distribution. In this case, the interval estimate for  is based on the t distribution. n (We’ll assume for now that the population is normally distributed.)

16. 16 Slide © 2008 Thomson South-Western. All Rights Reserved The t distribution is a family of similar probability The t distribution is a family of similar probability distributions. distributions. The t distribution is a family of similar probability The t distribution is a family of similar probability distributions. distributions. t Distribution A specific t distribution depends on a parameter A specific t distribution depends on a parameter known as the degrees of freedom. known as the degrees of freedom. A specific t distribution depends on a parameter A specific t distribution depends on a parameter known as the degrees of freedom. known as the degrees of freedom. Degrees of freedom refer to the number of Degrees of freedom refer to the number of independent pieces of information that go into the independent pieces of information that go into the computation of s. Only n-1 of the values are computation of s. Only n-1 of the values are independent because the remaining value can be independent because the remaining value can be determined exactly by subtracting all other values determined exactly by subtracting all other values from the sample mean cumulatively and keeping a from the sample mean cumulatively and keeping a running sum. The final sum is the dependent value. running sum. The final sum is the dependent value. Degrees of freedom refer to the number of Degrees of freedom refer to the number of independent pieces of information that go into the independent pieces of information that go into the computation of s. Only n-1 of the values are computation of s. Only n-1 of the values are independent because the remaining value can be independent because the remaining value can be determined exactly by subtracting all other values determined exactly by subtracting all other values from the sample mean cumulatively and keeping a from the sample mean cumulatively and keeping a running sum. The final sum is the dependent value. running sum. The final sum is the dependent value.

17. 17 Slide © 2008 Thomson South-Western. All Rights Reserved t Distribution A t distribution with more degrees of freedom has A t distribution with more degrees of freedom has less dispersion, variability, and more closely less dispersion, variability, and more closely resembles the normal distribution. resembles the normal distribution. A t distribution with more degrees of freedom has A t distribution with more degrees of freedom has less dispersion, variability, and more closely less dispersion, variability, and more closely resembles the normal distribution. resembles the normal distribution. As the number of degrees of freedom increases, the As the number of degrees of freedom increases, the difference between the t distribution and the difference between the t distribution and the standard normal probability distribution becomes standard normal probability distribution becomes smaller and smaller. smaller and smaller. As the number of degrees of freedom increases, the As the number of degrees of freedom increases, the difference between the t distribution and the difference between the t distribution and the standard normal probability distribution becomes standard normal probability distribution becomes smaller and smaller. smaller and smaller.

18. 18 Slide © 2008 Thomson South-Western. All Rights Reserved t Distribution Standardnormaldistribution t distribution (20 degrees of freedom) t distribution (10 degrees of freedom) of freedom) 0 z, t

19. 19 Slide © 2008 Thomson South-Western. All Rights Reserved For more than 100 degrees of freedom, the standard For more than 100 degrees of freedom, the standard normal z value provides a good approximation to normal z value provides a good approximation to the t value. the t value. For more than 100 degrees of freedom, the standard For more than 100 degrees of freedom, the standard normal z value provides a good approximation to normal z value provides a good approximation to the t value. the t value. t Distribution The standard normal z values can be found in the The standard normal z values can be found in the infinite degrees (  ) row of the t distribution table. infinite degrees (  ) row of the t distribution table. The standard normal z values can be found in the The standard normal z values can be found in the infinite degrees (  ) row of the t distribution table. infinite degrees (  ) row of the t distribution table.

20. 20 Slide © 2008 Thomson South-Western. All Rights Reserved t Distribution Standard normal z values

21. 21 Slide © 2008 Thomson South-Western. All Rights Reserved n Interval Estimate where: 1 -  = the confidence coefficient t  /2 = the t value providing an area of  /2 t  /2 = the t value providing an area of  /2 in the upper tail of a t distribution in the upper tail of a t distribution with n - 1 degrees of freedom with n - 1 degrees of freedom s = the sample standard deviation s = the sample standard deviation Interval Estimation of a Population Mean:  Unknown

22. 22 Slide © 2008 Thomson South-Western. All Rights Reserved A reporter for a student newspaper is writing an article on the cost of off-campus housing. A sample of 16 efficiency apartments within a half-mile of campus resulted in a sample mean of $650 per month and a sample standard deviation of $55. Interval Estimation of a Population Mean:  Unknown n Example: Apartment Rents

23. 23 Slide © 2008 Thomson South-Western. All Rights Reserved Let us provide a 95% confidence interval estimate of the mean rent per month for the population of efficiency apartments within a half-mile of campus. We will assume this population to be normally distributed. Interval Estimation of a Population Mean:  Unknown n Example: Apartment Rents

24. 24 Slide © 2008 Thomson South-Western. All Rights Reserved At 95% confidence,  =.05, and  /2 =.025. In the t distribution table we see that t.025 = 2.131. t.025 is based on n  1 = 16  1 = 15 degrees of freedom. Interval Estimation of a Population Mean:  Unknown

25. 25 Slide © 2008 Thomson South-Western. All Rights Reserved We are 95% confident that the mean rent per month We are 95% confident that the mean rent per month for the population of efficiency apartments within a half-mile of campus is between $620.70 and $679.30. n Interval Estimate Interval Estimation of a Population Mean:  Unknown Margin of Error

26. 26 Slide © 2008 Thomson South-Western. All Rights Reserved Summary of Interval Estimation Procedures for a Population Mean Can the population standard deviation  be assumed deviation  be assumed known ? known ? Use the sample standard deviation s to estimate  Use YesNo Use  Known Case  Unknown Case

27. 27 Slide © 2008 Thomson South-Western. All Rights Reserved Let E = the desired margin of error. E is the amount Let E = the desired margin of error. E is the amount added to and subtracted from the point estimate to added to and subtracted from the point estimate to obtain an interval estimate. obtain an interval estimate. Let E = the desired margin of error. E is the amount Let E = the desired margin of error. E is the amount added to and subtracted from the point estimate to added to and subtracted from the point estimate to obtain an interval estimate. obtain an interval estimate. Sample Size for an Interval Estimate of a Population Mean z a/2 is the confidence level to be used in developing z a/2 is the confidence level to be used in developing the interval estimate. the interval estimate. z a/2 is the confidence level to be used in developing z a/2 is the confidence level to be used in developing the interval estimate. the interval estimate. If σ unknown you can: 1.Use an estimate of the population standard deviation, σ, from prior analyses. deviation, σ, from prior analyses. 2.Use a pilot study to identify an estimate of the standard deviation, σ. standard deviation, σ. 3. Use judgment or a best guess for the value of σ. If σ unknown you can: 1.Use an estimate of the population standard deviation, σ, from prior analyses. deviation, σ, from prior analyses. 2.Use a pilot study to identify an estimate of the standard deviation, σ. standard deviation, σ. 3. Use judgment or a best guess for the value of σ.

28. 28 Slide © 2008 Thomson South-Western. All Rights Reserved Sample Size for an Interval Estimate of a Population Mean n Margin of Error n Necessary Sample Size

29. 29 Slide © 2008 Thomson South-Western. All Rights Reserved Recall that Discount Sounds is evaluating a potential location for a new retail outlet, based in part, on the mean annual income of the individuals in Recall that Discount Sounds is evaluating a potential location for a new retail outlet, based in part, on the mean annual income of the individuals in the marketing area of the new location. Suppose that Discount Sounds’ management team wants an estimate of the population mean such that there is a.95 probability that the sampling error is $500 or less. How large a sample size is needed to meet the required precision? S D Sample Size for an Interval Estimate of a Population Mean

30. 30 Slide © 2008 Thomson South-Western. All Rights Reserved At 95% confidence, z.025 = 1.96. Recall that  = 4,500. Sample Size for an Interval Estimate of a Population Mean S D A sample of size 312 is needed to reach a desired A sample of size 312 is needed to reach a desired precision of + $500 at 95% confidence. precision of + $500 at 95% confidence.

31. 31 Slide © 2008 Thomson South-Western. All Rights Reserved The general form of an interval estimate of a The general form of an interval estimate of a population proportion is population proportion is The general form of an interval estimate of a The general form of an interval estimate of a population proportion is population proportion is Interval Estimation of a Population Proportion

32. 32 Slide © 2008 Thomson South-Western. All Rights Reserved Interval Estimation of a Population Proportion The sampling distribution of plays a key role in The sampling distribution of plays a key role in computing the margin of error for this interval computing the margin of error for this interval estimate. estimate. The sampling distribution of plays a key role in The sampling distribution of plays a key role in computing the margin of error for this interval computing the margin of error for this interval estimate. estimate. The sampling distribution of can be approximated The sampling distribution of can be approximated by a normal distribution whenever np > 5 and by a normal distribution whenever np > 5 and n (1 – p ) > 5. n (1 – p ) > 5. The sampling distribution of can be approximated The sampling distribution of can be approximated by a normal distribution whenever np > 5 and by a normal distribution whenever np > 5 and n (1 – p ) > 5. n (1 – p ) > 5.

33. 33 Slide © 2008 Thomson South-Western. All Rights Reserved  /2 Interval Estimation of a Population Proportion n Normal Approximation of Sampling Distribution of Sampling distribution of Sampling distribution of p p 1 -  of all values 1 -  of all values

34. 34 Slide © 2008 Thomson South-Western. All Rights Reserved n Interval Estimate Interval Estimation of a Population Proportion where: 1 -  is the confidence coefficient z  /2 is the z value providing an area of z  /2 is the z value providing an area of  /2 in the upper tail of the standard  /2 in the upper tail of the standard normal probability distribution normal probability distribution is the sample proportion is the sample proportion

35. 35 Slide © 2008 Thomson South-Western. All Rights Reserved Political Science, Inc. (PSI) specializes in voter polls and surveys designed to keep political office seekers informed of their position in a race. Using telephone surveys, PSI interviewers ask registered voters who they would vote for if the election were held that day. Interval Estimation of a Population Proportion n Example: Political Science, Inc.

36. 36 Slide © 2008 Thomson South-Western. All Rights Reserved In a current election campaign, PSI has just found that 220 registered voters, out of 500 contacted, favor a particular candidate. candidate. PSI wants to develop a 95% confidence interval estimate for the proportion of the population of registered voters that favor the candidate. Interval Estimation of a Population Proportion n Example: Political Science, Inc.

37. 37 Slide © 2008 Thomson South-Western. All Rights Reserved where: n = 500, = 220/500 =.44, z  /2 = 1.96 Interval Estimation of a Population Proportion PSI is 95% confident that the proportion of all voters that favor the candidate is between.3965 and.4835. PSI is 95% confident that the proportion of all voters that favor the candidate is between.3965 and.4835. =.44 +.0435

38. 38 Slide © 2008 Thomson South-Western. All Rights Reserved Solving for the necessary sample size, we get n Margin of Error Sample Size for an Interval Estimate of a Population Proportion However, will not be known until after we have selected the sample. We will use the planning value p * for..

39. 39 Slide © 2008 Thomson South-Western. All Rights Reserved Sample Size for an Interval Estimate of a Population Proportion The planning value p * can be chosen by: 1. Using the sample proportion from a previous sample of the same or similar units. 2. Selecting a preliminary sample and using the sample proportion from this sample. 3. Use judgment or best guess for p*. 4. Use p* =.50 n Necessary Sample Size

40. 40 Slide © 2008 Thomson South-Western. All Rights Reserved Sample Size for an Interval Estimate of a Population Proportion p* is frequently used when no other information is available. A larger value for the quantity p*(1-p*) will result in a larger sample size. The largest value occurs when p* =.50. Therefore, p* =.50 will provide the largest sample size recommendation. If the sample proportion turns out to be different from.50, the margin of error will be smaller than expected. Thus, in using p* =.50, we guarantee that the sample size will be sufficient to obtain the desired margin of error.

41. 41 Slide © 2008 Thomson South-Western. All Rights Reserved Suppose that PSI would like a.99 probability that the sample proportion is within +.03 of the population proportion. How large a sample size is needed to meet the required precision? (A previous sample of similar units yielded.44 for the sample proportion.) Sample Size for an Interval Estimate of a Population Proportion

42. 42 Slide © 2008 Thomson South-Western. All Rights Reserved At 99% confidence, z.005 = 2.576. Recall that  =.44. A sample of size 1817 is needed to reach a desired A sample of size 1817 is needed to reach a desired precision of +.03 at 99% confidence. precision of +.03 at 99% confidence. Sample Size for an Interval Estimate of a Population Proportion

43. 43 Slide © 2008 Thomson South-Western. All Rights Reserved Note: We used.44 as the best estimate of p in the preceding expression. If no information is available about p, then.5 is often assumed because it provides the highest possible sample size. If we had used p =.5, the recommended n would have been 1843. Sample Size for an Interval Estimate of a Population Proportion

44. 44 Slide © 2008 Thomson South-Western. All Rights Reserved End of Chapter 8