1 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Chapter 6 Estimates and Sample Sizes

2 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Chapter 6 Estimates and Sample Sizes 6-1 Overview 6-2 Estimating a Population Mean: Large Samples 6-3 Estimating a Population Mean: Small Samples 6-4Sample Size Required to Estimate µ 6-5 Estimating a Population Proportion 6-6 Estimating a Population Variance

3 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 6-1 Overview  methods for estimating population means, proportions, and variances  methods for determining sample sizes This chapter presents:

4 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 6-2 Estimating a Population Mean: Large Samples

5 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Assumptions  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.

6 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Assumptions  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. Data collected carelessly can be absolutely worthless, even if the sample is quite large.

7 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Definitions  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

8 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman  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 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Definition Confidence Interval (or Interval Estimate) a range (or an interval) of values used to estimate the true value of the population parameter

10 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Definition 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 #

11 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Definition 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 # As an example Lower # <  < Upper #

12 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Interpreting a Confidence Interval Correct: We are 95% confident that the interval from to 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 and o < µ < o

14 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Figure 6-1 Confidence Intervals from 20 Different Samples

15 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman the number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur. The number z  /2 is a critical value that is a z score with the property that it separates an area  / 2 in the right tail of the standard normal distribution. Definition Critical Value

16 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman The Critical Value z=0 Found by using invNorm(   2 ) (corresponds to area of   2 ) z  2 -z  2  2

17 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Finding z  2 for 95% Degree of Confidence

18 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman -z  2 z  2 95%  2 = 2.5% =.025  = 5% Finding z  2 for 95% Degree of Confidence

19 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman -z  2 z  2 95%  2 = 2.5% =.025  = 5% Critical Values Finding z  2 for 95% Degree of Confidence

20 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Finding z  2 for 95% Degree of Confidence

21 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman.025  =  = 0.05 Finding z  2 for 95% Degree of Confidence.025

22 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Finding z  2 for 95% Degree of Confidence z  2 = 1.96  .025 Use invNorm(0.025) to find a z score of  =  = 0.05

23 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Margin of Error Definition

24 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Margin of Error is the maximum likely difference observed between sample mean x and true population mean µ. denoted by E Definition

25 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Margin of Error is the maximum likely difference observed between sample mean x and true population mean µ. denoted by E µ x + E x - E Definition

26 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Margin of Error is the maximum likely difference observed between sample mean x and true population mean µ. denoted by E µ x + E x - E x -E < µ < x +E Definition

27 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Margin of Error is the maximum likely difference observed between sample mean x and true population mean µ. denoted by E µ x + E x - E x -E < µ < x +E lower limit Definition upper limit

28 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Definition Margin of Error µ x + E x - E E = z  /2 Formula 6-1  n

29 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Definition Margin of Error µ x + E x - E also called the maximum error of the estimate E = z  /2 Formula 6-1  n

30 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Calculating E When  Is Unknown  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.

31 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Confidence Interval (or Interval Estimate) for Population Mean µ (Based on Large Samples: n >30) x - E < µ < x + E

32 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman x - E < µ < x + E µ = x + E Confidence Interval (or Interval Estimate) for Population Mean µ (Based on Large Samples: n >30)

33 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman x - E < µ < x + E µ = x + E (x + E, x - E) Confidence Interval (or Interval Estimate) for Population Mean µ (Based on Large Samples: n >30)

34 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Procedure for Constructing a Confidence Interval for µ ( Based on a Large Sample: n > 30 )

35 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Procedure for Constructing a Confidence Interval for µ ( Based on a Large Sample: n > 30 ) 1. Find the critical value z  2 that corresponds to the desired degree of confidence.

36 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Procedure for Constructing a Confidence Interval for µ ( Based on a Large Sample: n > 30 ) 1. Find the critical value z  2 that corresponds to the desired degree of confidence. 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.

37 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Procedure for Constructing a Confidence Interval for µ ( Based on a Large Sample: n > 30 ) 1. Find the critical value z  2 that corresponds to the desired degree of confidence. 3. Find the values of x - E and x + E. Substitute those values in the general format of the confidence interval: x - E < µ < x + E 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.

38 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Procedure for Constructing a Confidence Interval for µ ( Based on a Large Sample: n > 30 ) 1. Find the critical value z  2 that corresponds to the desired degree of confidence. 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. x - E < µ < x + E 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.

39 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Round-Off Rule for Confidence Intervals Used to Estimate µ 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.

40 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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

41 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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.

42 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman n = 106 x = 98.2 o s = 0.62 o  = 0.05  / 2 = 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.

43 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman n = 106 x = 98.2 o s = 0.62 o  = 0.05  / 2 = z  / 2 = 1.96 E = z  / 2  = = 0.12 n 106 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.

44 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman n = 106 x = 98.2 o s = 0.62 o  = 0.05  / 2 = z  / 2 = 1.96 E = z  / 2  = = 0.12 n 106 x - E <  < x + E 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.

45 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman n = 106 x = 98.2 o s = 0.62 o  = 0.05  / 2 = z  / 2 = 1.96 E = z  / 2  = = 0.12 n 106 x - E <  < x + E o <  < o 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.

46 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman n = 106 x = 98.2 o s = 0.62 o  = 0.05  / 2 = z  / 2 = 1.96 E = z  / 2  = = 0.12 n 106 x - E <  < x + E o <  < o o <  < o 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.

47 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman n = 106 x = 98.2 o s = 0.62 o  = 0.05  / 2 = z  / 2 = 1.96 E = z  / 2  = = 0.12 n 106 x - E <  < x + E o <  < o 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. Based on the sample provided, the confidence interval for the population mean is o <  < o. If we were to select many different samples of the same size, 95% of the confidence intervals would actually contain the population mean .

48 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Finding the Point Estimate and E from a Confidence Interval Point estimate of µ : x = (upper confidence interval limit) + (lower confidence interval limit) 2

49 Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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