1 / 39

Chapter 8

Chapter 8. Confidence Intervals. Confidence Intervals. 8.1 z-Based Confidence Intervals for a Population Mean: σ Known 8.2 t-Based Confidence Intervals for a Population Mean: σ Unknown 8.3 Sample Size Determination 8.4 Confidence Intervals for a Population Proportion

walker-higgins
Télécharger la présentation

Chapter 8

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 8 Confidence Intervals

  2. Confidence Intervals 8.1 z-Based Confidence Intervals for a Population Mean: σ Known 8.2 t-Based Confidence Intervals for a Population Mean: σ Unknown 8.3 Sample Size Determination 8.4 Confidence Intervals for a Population Proportion 8.5 Confidence Intervals for Parameters of Finite Populations (Optional) 8.6 A Comparison of Confidence Intervals and Tolerance Intervals (Optional)

  3. z-Based Confidence Intervals for a Mean: σ Known • The starting point is the sampling distribution of the sample mean • Recall that if a population is normally distributed with mean m and standard deviation σ, then the sampling distribution of x is normal with mean mx= m and standard deviation • Use a normal curve as a model of the sampling distribution of the sample mean • Exactly, because the population is normal • Approximately, by the Central Limit Theorem for large samples

  4. The Empirical Rule • 68.26% of all possible sample means are within one standard deviation of the population mean • 95.44% of all possible observed values of x are within two standard deviations of the population mean • 99.73% of all possible observed values of x are within three standard deviations of the population mean

  5. Example 8.1: The Car Mileage Case • Assume a sample size (n) of 5 • Assume the population of all individual car mileages is normally distributed • Assume the population standard deviation (σ) is 0.8 • The probability that x with be within ±0.7 of µ is 0.9544

  6. Example 8.1 The Car Mileage Case Continued • Assume the sample mean is 31.3 • That gives us an interval of [31.3 ± 0.7] = [30.6, 32.0] • The probability is 0.9544 that the interval [x ± 2σ] contains the population mean µ

  7. Example 8.1: The Car Mileage Case #3 • That the sample mean is within ±0.7155 of µ is equivalent to… • x will be such that the interval [x ± 0.7115] contains µ • Then there is a 0.9544 probability that x will be a value so that interval [x ± 0.7115] contains µ • In other wordsP(x – 0.7155 ≤ µ ≤ x + 0.7155) = 0.9544 • The interval [x ± 0.7115] is referred to as the 95.44% confidence interval for µ

  8. Example 8.1: The Car Mileage Case #4 Three 95.44% Confidence Intervals for m • Thee intervals shown • Two contain µ • One does not

  9. Example 8.1: The Car Mileage Case #5 • According to the 95.44% confidence interval, we know before we sample that of all the possible samples that could be selected … • There is 95.44% probability the sample mean is such the interval [x ± 0.7155] will contain the actual (but unknown) population mean µ • In other words, of all possible sample means, 95.44% of all the resulting intervals will contain the population mean µ • Note that there is a 4.56% probability that the interval does not contain µ • The sample mean is either too high or too low

  10. Generalizing • In the example, we found the probability that m is contained in an interval of integer multiples of sx • More usual to specify the (integer) probability and find the corresponding number of sx • The probability that the confidence interval will not contain the population mean m is denoted by • In the mileage example, a = 0.0456

  11. Generalizing Continued • The probability that the confidence interval will contain the population mean m is denoted by 1 - a • 1 – a is referred to as the confidence coefficient • (1 – a)  100% is called the confidence level • Usual to use two decimal point probabilities for 1 – a • Here, focus on 1 – a = 0.95 or 0.99

  12. General Confidence Interval • In general, the probability is 1 – a that the population mean m is contained in the interval • The normal point za/2 gives a right hand tail area under the standard normal curve equal to a/2 • The normal point - za/2 gives a left hand tail area under the standard normal curve equal to a/2 • The area under the standard normal curve between -za/2 and za/2 is 1 – a

  13. Sampling Distribution Of All Possible Sample Means

  14. z-Based Confidence Intervals for aMean with σ Known • If a population has standard deviation s (known), • and if the population is normal or if sample size is large (n 30), then … • … a (1-a)100% confidence interval for m is

  15. 95% Confidence Level • For a 95% confidence level, 1 – a = 0.95, so a = 0.05, and a/2 = 0.025 • Need the normal point z0.025 • The area under the standard normal curve between -z0.025 and z0.025 is 0.95 • Then the area under the standard normal curve between 0 and z0.025 is 0.475 • From the standard normal table, the area is 0.475 for z = 1.96 • Then z0.025 = 1.96

  16. 95% Confidence Interval • The 95% confidence interval is

  17. 99% Confidence Interval • For 99% confidence, need the normal point z0.005 • Reading between table entries in the standard normal table, the area is 0.495 for z0.005 = 2.575 • The 99% confidence interval is

  18. The Effect of a on Confidence Interval Width za/2 = z0.025 = 1.96 za/2 = z0.005 = 2.575

  19. Example 8.2: Car Mileage Case • Given • x = 31.56 • σ = 0.8 • n = 50 • Will calculate • 95 Percent confidence interval • 99 Percent confidence interval

  20. Example 8.2: 95 Percent Confidence Interval

  21. Example 8.2: 99 Percent Confidence Interval

  22. Notes on the Example • The 99% confidence interval is slightly wider than the 95% confidence interval • The higher the confidence level, the wider the interval • Reasoning from the intervals: • The target mean should be at least 31 mpg • Both confidence intervals exceed this target • According to the 95% confidence interval, we can be 95% confident that the mileage is between 31.33 and 31.78 mpg • We can be 95% confident that, on average, the mean exceeds the target by at least 0.33 and at most 0.78 mpg

  23. t-Based Confidence Intervals for aMean: s Unknown • If s is unknown (which is usually the case), we can construct a confidence interval for m based on the sampling distribution of • If the population is normal, then for any sample size n, this sampling distribution is called the t distribution

  24. The t Distribution • The curve of the t distribution is similar to that of the standard normal curve • Symmetrical and bell-shaped • The t distribution is more spread out than the standard normal distribution • The spread of the t is given by the number of degrees of freedom • Denoted by df • For a sample of size n, there are one fewer degrees of freedom, that is, df = n – 1

  25. Degrees of Freedom and thet-Distribution As the number of degrees of freedom increases, the spread of the t distribution decreases and the t curve approaches the standard normal curve

  26. The t Distribution and Degrees of Freedom • As the sample size n increases, the degrees of freedom also increases • As the degrees of freedom increase, the spread of the t curve decreases • As the degrees of freedom increases indefinitely, the t curve approaches the standard normal curve • If n ≥ 30, so df = n – 1 ≥ 29, the t curve is very similar to the standard normal curve

  27. t and Right Hand Tail Areas • Use a t point denoted by ta • ta is the point on the horizontal axis under the t curve that gives a right hand tail equal to a • So the value of ta in a particular situation depends on the right hand tail area a and the number of degrees of freedom • df = n – 1 • a = 1 – a , where 1 – a is the specified confidence coefficient

  28. t and Right Hand Tail Areas

  29. Using the t Distribution Table • Rows correspond to the different values of df • Columns correspond to different values of a • See Table 8.3, Tables A.4 and A.20 in Appendix A and the table on the inside cover • Table 8.3 and A.4 gives t points for df 1 to 30, then for df = 40, 60, 120 and ∞ • On the row for ∞, the t points are the z points • Table A.20 gives t points for df from 1 to 100 • For df greater than 100, t points can be approximated by the corresponding z points on the bottom row for df = ∞ • Always look at the accompanying figure for guidance on how to use the table

  30. Using the t Distribution • Example: Find t for a sample of size n=15 and right hand tail area of 0.025 • For n = 15, df = 14 •  = 0.025 • Note that a = 0.025 corresponds to a confidence level of 0.95 • In Table 8.3, along row labeled 14 and under column labeled 0.025, read a table entry of 2.145 • So t = 2.145

  31. Using the t Distribution Continued

  32. t-Based Confidence Intervals for aMean: s Unknown • If the sampled population is normally distributed with mean , then a (1­a)100% confidence interval for  is • ta/2 is the t point giving a right-hand tail area of /2 under the t curve having n­1 degrees of freedom

  33. Example 8.4 Debt-to-Equity Ratios • Estimate the mean debt-to-equity ratio of the loan portfolio of a bank • Select a random sample of 15 commercial loan accounts • x = 1.34 • s = 0.192 • n = 15 • Want a 95% confidence interval for the ratio • Assume all ratios are normally distributed but σ unknown

  34. Example 8.4 Debt-to-Equity RatiosContinued • Have to use the t distribution • At 95% confidence, 1 –  = 0.95 so  = 0.05 and /2 = 0.025 • For n = 15, df = 15 – 1 = 14 • Use the t table to find t/2 for df = 14, t/2 = t0.025 = 2.145 • The 95% confidence interval:

  35. Sample Size Determination (z) If s is known, then a sample of sizeso that x is within B units of , with 100(1-)% confidence

  36. Sample Size Determination (t) If σ is unknown and is estimated from s, then a sample of sizeso that x is within B units of , with 100(1-)% confidence. The number of degrees of freedom for the ta/2 point is the size of the preliminary sample minus 1

  37. Example 8.6: The Car Mileage Case • What sample size is needed to make the margin of error for a 95 percent confidence interval equal to 0.3? • Assume we do not know σ • Take prior example as preliminary sample • s = 0.7583 • z/2 = z0.025 = 1.96 • t/2 = t0.025 = 2.776 based on n-1 = 4 d.f.

  38. Example 8.6: The Car Mileage Case Continued

  39. Example 8.7: The Car Mileage Case • Want to see that the sample of 50 mileages produced a 95 percent confidence interval with a margin or error of ±0.3 • The margin of error is 0.227, which is smaller than the 0.3 desired

More Related