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Chapter 6

Chapter 6. 6-1. Estimates and Sample Size with One Sample. Outline. 6-2. 6-1 Introduction 6-3 Estimating a Population Mean with:  known  6-4 Estimating a Population Mean with:  unknown . Objectives. 6-4.

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Chapter 6

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  1. Chapter 6 6-1 Estimates and Sample Size with One Sample

  2. Outline 6-2 • 6-1 Introduction • 6-3 Estimating a Population Mean with:  known • 6-4 Estimating a Population Mean with:  unknown

  3. Objectives 6-4 • Find the confidence interval for the mean when  is known or n 30. • Determine the minimum sample size for finding a confidence interval for the mean. • Find the confidence interval for the mean when  is unknown and n 30.

  4. 6-3 Confidence Intervals for the Mean ( known or n 30) and Sample Size 6-6 A point estimate is a specific numerical value estimate of a parameter. The best estimate of the population mean is thesample mean .  X

  5. 6-3 Three Properties of a Good Estimator 6-7 • The estimator must be anunbiased estimator. That is, the expected value or the mean of the estimates obtained from samples of a given size is equal to the parameter being estimated.

  6. 6-3 Three Properties of a Good Estimator 6-8 • The estimator must be consistent. For aconsistent estimator, as sample size increases, the value of the estimator approaches the value of the parameter estimated.

  7. 6-3 Three Properties of a Good Estimator 6-9 • The estimator must be arelatively efficient estimator. That is, of all the statistics that can be used to estimate a parameter, the relatively efficient estimator has the smallest variance.

  8. 6-3 Confidence Intervals 6-10 • An interval estimateof a parameter is an interval or a range of values used to estimate the parameter. This estimate may or may not contain the value of the parameter being estimated.

  9. 6-3 Confidence Intervals 6-11 • Aconfidence intervalis a specific interval estimate of a parameter determined by using data obtained from a sample and the specific confidence level of the estimate.

  10. 6-3 Confidence Intervals 6-12 • Theconfidence levelof an interval estimate of a parameter is the probability that the interval estimate will contain the parameter.

  11. 6-3 Formula for the Confidence Interval of the Mean for a Specific 6-13 • Theconfidence levelis the percentage equivalent to the decimal value of 1 – .

  12. 6-3 Maximum Error of Estimate or Margin of Error (E) 6-14 • Themaximum error of estimateor margin of error (E) is the maximum difference between the point estimate of a parameter and the actual value of the parameter.

  13. 6-3 Confidence Intervals -Example 6-15 • The president of a large university wishes to estimate the average age of the students presently enrolled. From past studies, the standard deviation is known to be 2 years. A sample of 50 students is selected, and the mean is found to be 23.2 years. Find the 95% confidence interval of the population mean.

  14. Since the 95% confidence interval = is desired , z 1 . 96 . Hence , a 2 substituting in the formula s s æ ö æ ö < m < ç ÷ ç ÷ X – z X + z è ø è ø a a n n 2 2 one gets 6-3 Confidence Intervals -Example 6-16

  15. 6-3 Confidence Intervals -Example 6-17 2 2      23 . 2 (1.96) ( ) 23.2 (1.96) ( ) 50 50      23 . 2 0 . 6 23 . 2 0 . 6    22 . 6 23 . 8 or 23.20.6years. Hence , the president can say , with 95% confidence , that the average age of the students is between 22 . 6 and 23 . 8 years , based on 50 students .

  16. 6-3 Confidence Intervals -Example 6-18 • A certain medication is known to increase the pulse rate of its users. The standard deviation of the pulse rate is known to be 5 beats per minute. A sample of 30 users had an average pulse rate of 104 beats per minute. Find the 99% confidence interval of the true mean.

  17. Since the 99% confidence interval  is desired , z 2 . 58 . Hence ,  2 substituting in the formula              X – z X + z      n n 2 /2 one gets 6-3 Confidence Intervals -Example 6-19

  18. 6-3 Confidence Intervals -Example 6-20 5 5      104 (2.58) . ( ) 104 ( ) (2.58) 30 30      104 2 . 4 104 2 . 4    101 . 6 106 . 4 . Hence , one can say , with 99% confidence , that the average pulse rate is between 101 . 6 and 106.4 beats per minute, based on 30 users.

  19. 6-3 Formula for the Minimum Sample Size Needed for an Interval Estimate of the Population Mean 6-21

  20. 6-3 Minimum Sample Size Needed for an Interval Estimate of the Population Mean -Example 6-22 • The college president asks the statistics teacher to estimate the average age of the students at their college. How large a sample is necessary? The statistics teacher decides the estimate should be accurate within 1 year and be 99% confident. From a previous study, the standard deviation of the ages is known to be 3 years.

  21. a Since = 0 . 01 ( or 1 – 0 . 99 ), z = 2 . 58 , and E = 1 , substituting a 2 × s 2 z æ ö = in n gives ç ÷ a 2 è ø E 2 æ ( 2 . 58 )( 3 ) ö = » ç ÷ n = 59 . 9 60 . è ø 1 6-3 Minimum Sample Size Needed for an Interval Estimate of the Population Mean -Example 6-23

  22. 6-4 Characteristics of the t-Distribution 6-24 • The t-distribution shares some characteristics of the normal distribution and differs from it in others. The t-distribution is similar to the standard normal distribution in the following ways: • It is bell-shaped. • It is symmetrical about the mean.

  23. 6-4 Characteristics of thet-Distribution 6-25 • The mean, median, and mode are equal to 0 and are located at the center of the distribution. • The curve never touches the x axis. • The t distribution differs from the standard normal distribution in the following ways:

  24. 6-4 Characteristics of thet-Distribution 6-26 • The variance is greater than 1. • The t distribution is actually a family of curves based on the concept of degrees of freedom, which is related to the sample size. • As the sample size increases, the t distribution approaches the standard normal distribution.

  25. 6-4 Standard Normal Curve and the t Distribution 6-27

  26. 6-4 Formula for the Confidence Interval of the Mean for a Specific 6-13 • When n < 30 and s is unknown use t-distribution with degrees of freedom = n – 1.

  27. 6-4 Confidence Interval for the Mean ( unknown and n < 30) - Example 6-28 • Ten randomly selected automobiles were stopped, and the tread depth of the right front tires were measured. The mean was 0.32 inches, and the standard deviation was 0.08 inches. Find the 95% confidence interval of the mean depth. Assume that the variable is approximately normally distributed.

  28. 6-4 Confidence Interval for the Mean ( unknown and n < 30) - Example 6-29 • Since  is unknown and s must replace it, the t distribution must be used with  = 0.05. Hence, with 9 degrees of freedom, t/2 = 2.262 (see Table F in text). • From the next slide, we can be 95% confident that the population mean is between 0.26 and 0.38.

  29. 6-4 Confidence Interval for the Mean ( unknown and n < 30) - Example 6-30 Thus the 95% confidence interval of the population mean is found by substituting in  s   s           X t X t     n n   2 2 0.08 0.08             0.32 – (2.262) 0 . 32 (2.262)     10 10    0 . 26 0 . 38

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