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Chapter 8

Chapter 8. Confidence Intervals. Confidence Interval. A confidence interval for the population mean is an interval constructed around the sample mean so that the interval will contain the population mean with certain probability.

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Chapter 8

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  1. Chapter 8 Confidence Intervals

  2. Confidence Interval • A confidence interval for the population mean is an interval constructed around the sample mean so that the interval will contain the population mean with certain probability. • The probability that the interval contains the population parameter is called a confidence level. • The probability for the interval not to contain the population mean is 1- confidence level, denoted by a. • a=1- confidence level • confidence level=1-a

  3. Example 8.1: The Car Mileage Case Suppose an automaker conducts mileage tests on a sample of 50 of its new mid-size cars and obtains the sample mean with x =31.56. Assuming population standard deviation σ=0.8. Please compute the 95 percent confidence interval of the population mean.

  4. Confidence Interval vs a • The probability that the confidence interval will contain the population mean, the confidence level , is denoted by 1 - a • If a =0.05, what is the confidence level? • 1-0.05=0.95 • If the confidence level is 99%, a =0.01?

  5. z-Based Confidence Intervals for a Population Mean: Population σ is Known • 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 • x is approximately normally distributed if population mean m standard deviation σ both exist and n≥30.

  6. z-Based Confidence Intervals for a Population Mean: Population σ is 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) confidence interval for m is

  7. t and Right Hand Tail Areas The definition of the critical value Zα The area to the right if 1-α Zα Zαis the percentile such that P(Z< Zα) =1-α

  8. Find out the critical values Zα/2

  9. Find out the critical values Zα/2

  10. 1- a Confidence Interval • If the population is normal or if n≥30, and population standard deviation is know, the 1- a confidence interval for population mean is • The normal point za/2 gives a right hand tail area under the standard normal curve equal to a/2

  11. Example 8.1: The Car Mileage Case Suppose an automaker conducts mileage tests on a sample of 50 of its new mid-size cars and obtains the sample mean with x =31.56. Assuming population standard deviation σ=0.8. Please compute the 95 percent confidence interval of the population mean. x = 31.56;σ = 0.8;n = 50

  12. 99% Confidence Interval • A bank manager developed a new system to reduce the service time. Suppose the new service time has a normal distribution with known standard deviation 2.47 minutes. The mean of a sample of 10 randomly selected customers is 5.46. Please construct an 99% confidence interval of the population mean.

  13. 99% Confidence Interval • For a 99% confidence level, 1 – a = 0.99, so a = 0.01, and a/2 = 0.005 • Reading between table entries z0.005 = 2.575 • The 99% confidence interval is

  14. 99% Confidence Interval x = 5.46;σ = 2.47;n = 10

  15. Notes on the Example • The confidence interval can be expression as margin of error • The length of confidence interval is equal to 2E. • The 99% confidence interval is slightly wider than the 95% confidence interval • The higher the confidence level, the bigger the critical value Zα/2, the wider the interval.

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

  17. 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

  18. 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 • The result applies if sample size is ≥30 • ta/2 is the t point giving a right-hand tail area of /2 under the t curve having n­1 degrees of freedom

  19. 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

  20. 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

  21. 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

  22. 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

  23. Definition of the critical value tα

  24. 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, t0.025=2.145 •  = 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

  25. Using the t Distribution Continued

  26. Find out the critical values tα/2

  27. Find out the critical values Zα/2

  28. Suppose that, in order to reduce risk, a large bank has decided to initiate a policy limiting the mean debt-to-equity ratio for its portfolio of commercial loans to 1.5. In order to estimate the mean debt-to-equity ratio of its load portfolio, the bank randomly selects a sample of 15 of its commercial loan accounts. The sample mean and standard deviation of the sample is 1.3433 and 0.1921. The population has normal distribution. Please construct a 95% confidence interval for the mean of the debt-to-equity ratio.

  29. 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 = 1.3433 • s = 0.1921 • n = 15 • Want a 95% confidence interval for the ratio • Assume all ratios are normally distributed but σ unknown

  30. 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:

  31. Length of Confidence Interval • The confidence interval can be expression as margin of error equal to or • The length of confidence interval is equal to 2E.

  32. Sample Size Determination (z) If σ is known, then a sample of size of at least will result in a confidence interval such so that x is within E units of  with probability 100(1-)% .

  33. Sample Size Determination (t) If σ is unknown and is estimated from s, then a sample of size of at least will give an interval so that x is within E 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.

  34. A bank manager developed a new system to reduce the service time. Suppose the new service time has a normal distribution with known standard deviation 2.47 minutes. How large the sample should be if the manager wants to be 99% confident that sample mean is within 0.5 minute of mu (the population mean) . • round up to 162

  35. A new alert system is installed for air traffic controllers. It is hoped that the mean “alert time” for the new equipment is less than 8 seconds. In order to test the equipment 15 randomly selected air traffic controllers are trained to use the machine and their alert times for a simulated collision course are recorded. The sample alert times has a mean of 7.4 seconds and s=1.026. Supposed the alert times are normally distributed, please construct 95% confidence interval for the mean alert time of the machine. Can we be 95% confident that mu is less than 8 seconds? Determine the sample size needed to make us 95% confident that the sample mean is within a margin of error of 0.3 second of mu. • t0.025 =2.145, for degree of freedom 14; rockville

  36. Selecting an Appropriate Confidence Interval for a Population Mean

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