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M ARIO F . T RIOLA

S TATISTICS. E LEMENTARY. Chapter 6 Estimates and Sample Sizes. M ARIO F . T RIOLA. E IGHTH. E DITION. 6-1 Overview 6-2 Estimating a Population Mean: Large Samples 6-3 Estimating a Population Mean: Small Samples 6-4 Sample Size Required to Estimate µ

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M ARIO F . T RIOLA

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  1. STATISTICS ELEMENTARY Chapter 6 Estimates and Sample Sizes MARIO F. TRIOLA EIGHTH EDITION

  2. 6-1 Overview 6-2 Estimating a Population Mean: Large Samples 6-3 Estimating a Population Mean: Small Samples 6-4 Sample Size Required to Estimate µ 6-5 Estimating a Population Proportion 6-6 Estimating a Population Variance Chapter 6Estimates and Sample Sizes

  3.   methods for estimating population   means, proportions, and variances   methods for determining sample sizes 6-1 Overview This chapter presents:

  4. Estimating a Population Mean: Large Samples 6-2

  5. n > 30 The sample must have more than 30 values. Simple Random Sample All samples of the same size have an equal chance of being selected. Assumptions

  6. n > 30 The sample must have more than 30 values. Simple Random Sample All samples of the same size have an equal chance of being selected. Assumptions Data collected carelessly can be absolutely worthless, even if the sample is quite large.

  7. Estimator a formula or process for using sample data to estimate a population parameter Estimate a specific value or range of values used to approximate some population parameter Point Estimate a single value (or point) used to approximate a population parameter Definitions

  8. Estimator a formula or process for using sample data to estimate a population parameter Estimate a specific value or range of values used to approximate some population parameter Point Estimate a single value (or point) used to approximate a population parameter The sample mean x is the best point estimate of the population mean µ. Definitions

  9. Confidence Interval (or Interval Estimate) a range (or an interval) of values used to estimate the true value of the population parameter Definition

  10. Confidence Interval (or Interval Estimate) a range (or an interval) of values used to estimate the true value of the population parameter Lower # < population parameter < Upper # Definition

  11. Confidence Interval (or Interval Estimate) a range (or an interval) of values used to estimate the true value of the population parameter Lower # < population parameter < Upper # Definition As an example Lower # <  < Upper #

  12. the probability 1 -  (often expressed as the equivalent percentage value) that is the relative frequency of times the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times usually 90%, 95%, or 99% ( = 10%), ( = 5%), ( = 1%) Definition Degree of Confidence • (level of confidence or confidence coefficient)

  13. Correct: We are 95% confident that the interval from 98.08 to 98.32 actually does contain the true value of . This means that if we were to select many different samples of size 106 and construct the confidence intervals, 95% of them would actually contain the value of the population mean . Wrong: There is a 95% chance that the true value of  will fall between 98.08 and 98.32. Interpreting a Confidence Interval 98.08o <µ < 98.32o

  14. Confidence Intervals from 20 Different Samples Figure 6-1

  15. the number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur. The number z/2is a critical value that is a zscore with the property that it separates an area /2in the right tail of the standard normal distribution. Definition Critical Value

  16. z2 The Critical Value 2 2 z2 -z2 z=0 Found from Table A-2 (corresponds to area of 0.5 - 2 ) Figure 6-2

  17. Finding z2for 95% Degree of Confidence 95%  = 5% 2 = 2.5% = .025 .95 .025 .025 z2 -z2

  18. Finding z2for 95% Degree of Confidence 95%  = 5% 2 = 2.5% = .025 .95 .025 .025 z2 -z2 Critical Values

  19. Finding z2for 95% Degree of Confidence  = 0.05  = 0.025 .4750 .025 Use Table A-2 to find a z score of 1.96

  20. Finding z2for 95% Degree of Confidence  = 0.05  = 0.025 .4750 .025 Use Table A-2 to find a z score of 1.96  z2 = 1.96  .025 .025 - 1.96 1.96

  21. Margin of Error Definition is the maximum likely difference observed between sample mean x and true population mean µ. denoted by E

  22. Margin of Error x - E Definition is the maximum likely difference observed between sample mean x and true population mean µ. denoted by E µ x + E

  23. Margin of Error x - E Definition is the maximum likely difference observed between sample mean x and true population mean µ. denoted by E µ x + E x -E < µ < x +E

  24. Margin of Error x - E Definition is the maximum likely difference observed between sample mean x and true population mean µ. denoted by E µ x + E x -E < µ < x +E upper limit lower limit

  25. Margin of Error E = z/2 •Formula 6-1  n x - E Definition µ x + E

  26. Margin of Error E = z/2 •Formula 6-1  n x - E Definition µ x + E also called the maximum error of the estimate

  27.   If n> 30, we can replace  in Formula 6-1   by the sample standard deviation s.   If n 30, the population must have a   normal distribution and we must know    to use Formula 6-1. Calculating E When  Is Unknown

  28. Confidence Interval (or Interval Estimate) for Population Mean µ(Based on Large Samples: n >30) x - E < µ < x + E

  29. Confidence Interval (or Interval Estimate) for Population Mean µ(Based on Large Samples: n >30) x - E < µ < x + E µ = x + E

  30. Confidence Interval (or Interval Estimate) for Population Mean µ(Based on Large Samples: n >30) x - E < µ < x + E µ = x + E (x + E, x - E)

  31. 1. Find the critical value z2 that corresponds to the desired degree of confidence. Procedure for Constructing a Confidence Interval for µ( Based on a Large Sample: n > 30 )

  32. 1. Find the critical value z2 that corresponds to the desired degree of confidence. Procedure for Constructing a Confidence Interval for µ( Based on a Large Sample: n > 30 ) 2. Evaluate the margin of error E = z2 •  / n . If the population standard deviation  is unknown,  use the value of the sample standard deviation s  provided that n > 30.

  33. 1. Find the critical value z2 that corresponds to the desired degree of confidence. x - E < µ < x + E Procedure for Constructing a Confidence Interval for µ( Based on a Large Sample: n > 30 ) 2. Evaluate the margin of error E = z2 •  / n . If the population standard deviation  is unknown,  use the value of the sample standard deviation s  provided that n > 30. 3. Find the values of x - E and x + E. Substitute those values in the general format of the confidence interval:

  34. 1. Find the critical value z2 that corresponds to the desired degree of confidence. x - E < µ < x + E Procedure for Constructing a Confidence Interval for µ( Based on a Large Sample: n > 30 ) 2. Evaluate the margin of error E = z2 •  / n . If the population standard deviation  is unknown,  use the value of the sample standard deviation s  provided that n > 30. 3. Find the values of x - E and x + E. Substitute those values in the general format of the confidence interval: 4. Round using the confidence intervals roundoff rules.

  35. 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. Round-Off Rule for Confidence Intervals Used to Estimate µ

  36. 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 Rule for Confidence Intervals Used to Estimate µ

  37. Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval.

  38. n = 106 x = 98.2o s = 0.62o  = 0.05 /2 = 0.025 z/ 2= 1.96 Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval.

  39. n = 106 x = 98.20o s = 0.62o  = 0.05 /2 = 0.025 z/ 2= 1.96 Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval. E = z/ 2•  = 1.96 • 0.62 = 0.12 n 106

  40. n = 106 x = 98.20o s = 0.62o  = 0.05 /2 = 0.025 z/ 2= 1.96 Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval. E = z/ 2•  = 1.96 • 0.62 = 0.12 n 106 x - E <  < x + E

  41. n = 106 x = 98.20o s = 0.62o  = 0.05 /2 = 0.025 z/ 2= 1.96 Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval. E = z/ 2•  = 1.96 • 0.62 = 0.12 n 106 x - E <  < x + E 98.20o - 0.12 <<98.20o + 0.12

  42. n = 106 x = 98.20o s = 0.62o  = 0.05 /2 = 0.025 z/ 2= 1.96 Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval. E = z/ 2•  = 1.96 • 0.62 = 0.12 n 106 x - E <  < x + E 98.20o - 0.12 <<98.20o + 0.12 98.08o << 98.32o

  43. n = 106 x = 98.20o s = 0.62o  = 0.05 /2 = 0.025 z/ 2= 1.96 Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval. E = z/ 2•  = 1.96 • 0.62 = 0.12 n 106 x - E <  < x + E 98.08o << 98.32o Based on the sample provided, the confidence interval for the population mean is 98.08o<<98.32o. If we were to select many different samples of the same size, 95% of the confidence intervals would actually contain the population mean .

  44. Finding the Point Estimate and E from a Confidence Interval Point estimate of µ: x =(upper confidence interval limit) + (lower confidence interval limit) 2

  45. Finding the Point Estimate and E from a Confidence Interval Point estimate of µ: x =(upper confidence interval limit) + (lower confidence interval limit) 2 Margin of Error: E = (upper confidence interval limit) - (lower confidence interval limit) 2

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