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Chapter 8

Chapter 8. Confidence Intervals Introduction Large Sample Confidence Intervals for a Population Mean Small Sample Confidence Intervals for a Population Mean Sample Size Determination Confidence Intervals for a Population Proportion. Introduction.

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Chapter 8

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  1. Chapter 8 Confidence Intervals • Introduction • Large Sample Confidence Intervals for a Population Mean • Small Sample Confidence Intervals for a Population Mean • Sample Size Determination • Confidence Intervals for a Population Proportion

  2. Introduction • As discussed in the previous chapter, point estimates are often used to estimate population parameters. • The absolute value of the difference between a point estimator and the population parameter it estimates is called the sampling error. • For the case of a sample mean estimating a population mean, the sampling error is Sampling Error = • For the case of a sample proportion estimating a population proportion, the sampling error is Sampling Error =

  3. Probability Statements About the Sampling Error • Knowledge of the sampling distribution of enables us to make probability statements about the sampling error even though the population mean  is not known. • Knowledge of the sampling distribution of enables us to make probability statements about the sampling error even though the population proportion p is not known. • A probability statement about the sampling error is a precision statement. Anderson, Sweeney, and Williams

  4. Precision StatementAbout the SamplingDistribution of There is a 1 -  probability that the value of a sample mean will provide a sampling error of or less. z/2 is the z value that has an upper tail (or lower tail) area of /2 1 -  of all values /2 /2 Anderson, Sweeney, and Williams  -z/2 +z/2

  5. Precision StatementAbout the SamplingDistribution of There is a 1 -  probability that the value of a sample proportion will provide a sampling error of or less. z/2 is the z value that has an upper tail area of /2 1 -  of all values /2 /2 Anderson, Sweeney, and Williams p -z/2 +z/2

  6. Precision statements provide confidence intervals in which we are 1 -  confident that the true population parameter will fall within these limits.

  7. General Process for Forming Confidence Intervals Original Population A simple random sample of n elements is selected from the population. The sample data provide a point estimate for the population parameter. The value of the point estimate is used to make inferences about the actual population parameter

  8. In the previous slide the last step uses the value of the point estimate to make inferences about the actual population parameter. • Specifically in this step we conclude that we are 1- confident that the true population parameter falls in the interval formed by [(point estimate) + error, (point estimate) – error] The value of the error depends upon the population parameter that you are estimating and the size of the sample. Additional information may also be required.

  9. Common Examples of ConfidenceIntervals These are confidence intervals (confidence coefficient not given)

  10. Interval Estimation of a Population Mean:Large-Sample case with  known If the sample size is large (n  30) and  is known, a(1- )100% confidence interval for  is 1- is called the confidence coefficient

  11. Interval Estimation of a Population Mean: Small-Sample case (n < 30), normal population and  known If the sample size is small (n < 30), the sampled population is normal, and is known, a (1-)100% confidence interval for  is 1- is called the confidence coefficient

  12. Steps for Computing z/2 (The z value with an upper tail area of /2. • Compute  =1 – (Confidence Coefficient). • Compute /2. • In order to determine the z value with an upper tail area of /2, we need the area beneath the normal curve to the left of the z value of interest. area= 1- /2 /2 4.Go to the area section of the standard normal table and find the area closest to the area computed in 3. The corresponding z value is z/2 .

  13. Example 8.1 National Discount has 260 retail outlets throughout the United States. National evaluates each potential location for a new retail outlet in part on the mean annual income of the individuals in the marketing area of the new location. Sampling can be used to develop an interval estimate of the mean annual income for individuals in a potential marketing area for National Discount. A sample of size n = 36 was taken. The sample mean, , is $21,100 and the population standard deviation, , is $4,500. We will use .95 as the confidence coefficient in our interval estimate. Anderson, Sweeney, and Williams

  14. Example 8.1 This is a large sample with  known. Therefore, we are 95% confident that the interval contains the population mean. Find z/2. •  =1 - .95 = .05 • /2 = .025 • area = 1 - .025 = .975 • The standard normal table reveals that the area under f(x) to the left of z = 1.96 is .975. Thus, z/2=1.96

  15. Furthermore, , therefore .

  16. Interval Estimate of  is: $21,100 + $1,470 or $19,630 to $22,570 We are 95% confident that the interval contains the population mean. Anderson, Sweeney, and Williams

  17. Example 8.2 A consumer reports article tested theaverage miles per gallon for a popular midsize vehicle. In order to conduct this test, they test drove 49 vehicles of this type and computed a mean mileage of 31.5531 and a population standard deviation of .7992.

  18. 95% Confidence Interval =[31.33, 31.78] 99% Confidence Interval Bowerman, O’Connell, and Hand

  19. The Effect of a on Confidence Interval Width Bowerman, O’Connell, and Hand

  20. Interval Estimation of a Population Mean:Small-Sample case (n < 30) • Population is Not Normally Distributed The only option is to increase the sample size to n> 30 and use the large-sample interval-estimation procedures.

  21. Interval Estimation of a Population Mean:Small-Sample case (n < 30), normal population and  unknown Population is Normally Distributed:  Estimated by s The appropriate interval estimate is based on a probability distribution known as the t distribution.

  22. t Distribution • The t distribution is a family of similar probability distributions. • A specific t distribution depends on a parameter known as the degrees of freedom. • As the number of degrees of freedom increases, the difference between the t distribution and the standard normal probability distribution becomes smaller and smaller. • A t distribution with more degrees of freedom has less dispersion. • The mean of the t distribution is zero. Anderson, Sweeney, and Williams

  23. Standard normal distribution t distribution (20 degrees of freedom) t distribution (10 degrees of freedom) z, t 0 t Distribution Anderson, Sweeney, and Williams

  24. Interval Estimation of a Population Mean:Small-Sample Case (n < 30), normal population and  unknown If the population is normal, the sample size is small (n <30) and  is unknown, estimate  using the sample standard deviation s. A (1- )100% confidence interval for  is where 1 - = the confidence coefficient t/2 = the t value providing an area of /2 in the upper tail of a t distribution with n - 1 degrees of freedom s = the sample standard deviation

  25. Interval Estimation of a Population Mean:Large-Sample Case (n>30) and  unknown If the sample size is large (n >30) and  is unknown, estimate  using the sample standard deviation s. A (1- )100% confidence interval for  is where 1 - = the confidence coefficient t/2 = the t value providing an area of /2 in the upper tail of a t distribution with n - 1 degrees of freedom s = the sample standard deviation

  26. Steps for Computing t/2 (The t value with an upper tail area of /2. • Compute  =1 – (Confidence Coefficient). • Compute upper tail area= /2. • Degrees of freedom= n-1. • Go to t chart and find the t value with the upper tail area (computed in step 2), and the degrees of freedom (computed in step 3). This is t/2 .

  27. Example 8.3 A reporter for a student newspaper is writing an article on the cost of off-campus housing. A sample of 10 one-bedroom units within a half-mile of the campus resulted in a sample mean of $550 per month and a sample standard deviation of $60. Let us provide a 95% confidence interval estimate of the mean rent per month for the population of one-bedroom units within a half-mile of campus. We’ll assume this population to be normally distributed. Anderson, Sweeney, and Williams

  28. Example 8.3 At 95% confidence,  =1 - .95 =.05, and /2 = .025. t.025 is based on n - 1 = 10 - 1 = 9 degrees of freedom. In the t distribution table, we see that t.025 = 2.262. Anderson, Sweeney, and Williams

  29. Example 8.3 550 + 42.92 or $507.08 to $592.92 We are 95% confident that the mean rent per month for the population of one-bedroom units within a half-mile of campus is between $507.08 and $592.92. Anderson, Sweeney, and Williams

  30. Summary of Interval Estimation Procedures for a Population Mean Yes No n > 30 ? No Popul. approx. normal ?  known ? Yes Yes Use s to estimate No  known ? No Use s to estimate  Yes Increase n to > 30 Anderson, Sweeney, and Williams

  31. Determining the Sample Size for an Interval Estimate of a Population Mean • Let E = the maximum sampling error mentioned in the precision statement. • E is the amount added to and subtracted from the point estimate to obtain an interval estimate. • E is often referred to as the margin of error.

  32. Sample Size for an Interval Estimate of a Population Mean • Margin of Error • Necessary Sample Size

  33. Example 8.1 Revisited (Determining the Sample Size) Suppose that National’s 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?

  34. Example 8.1 Revisited (Determining the Sample Size) At 95% confidence, z.025 = 1.96. Recall that = 4,500. We need to sample 312 to reach a desired precision of + $500 at 95% confidence. Anderson, Sweeney, and Williams

  35. Interval Estimation of a Population Proportion If n >5 and n(1 - ) >5, then a (1- )100% confidence interval for p is where: * 1 -is the confidence coefficient * z/2 is the z value providing an area of  /2 in the upper tail of the standard normal probability distribution * is the sample proportion

  36. Example 8.4 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, interviewers ask registered voters who they would vote for if the election were held that day. In a recent election campaign, PSI found that 220 registered voters, out of 500 contacted, favored a particular candidate. PSI wants to develop a 95% confidence interval estimate for the proportion of the population of registered voters that favors the candidate. Anderson, Sweeney, and Williams

  37. Example 8.4(Interval Estimate of a Population Proportion) Recall that Provided np>5 & n(1-p)>5 For this problem: n = 500, = 220/500 = .44, {n = 220 & n(1- )= 280} z.025= 1.96 = .44 + .0435 PSI is 95% confident that the proportion of all voters that favors the candidate is between .3965 and .4835. Anderson, Sweeney, and Williams

  38. Example 8.5 Ceta County’s Board of Education conducted a poll to determine their residents feelings toward an upcoming education bond referendum.They surveyed 200 residents, and 35 of them said that they would vote against the bond. Construct a 95% confidence interval for the population proportion of residents that would vote against the bond.

  39. Example 8.5 *** FIRST VERIFY np>5 & n(1-p)>5 ***

  40. Determining Sample Size for Confidence Interval for p A sample size will yield an estimate , precise to within B (or E) units of p, with 100(1-) confidence. Note that the formula requires a preliminary estimate of p. The conservative value of p = 0.5 is generally used when there is no prior information on p.

  41. Example 8.4 Revisited (Determining the Sample Size for a Population Proportion) 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? At 99% confidence, z.005 = 2.575.

  42. The End

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