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Chapter Eight

Chapter Eight. Estimation. Estimating  When  is Known. Assumptions: We have a simple random sample of size n . If the distribution is normal , methods work for any sample size. If the distribution is unknown , a sample size of at least 30 (sometimes even more) is required.

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Chapter Eight

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  1. Chapter Eight Estimation

  2. Estimating  When  is Known Assumptions: • We have a simple random sample of size n. • If the distribution is normal, methods work for any sample size. • If the distribution is unknown, a sample size of at least 30 (sometimes even more) is required.

  3. Point Estimate • an estimate of a population parameter given by a single number

  4. Examples of Point Estimates • x is used as a point estimate for m . • s is used as a point estimate for s.

  5. Margin of Error • the magnitude of the difference between the point estimate and the true parameter value

  6. The margin of error using xas a point estimate for m is Margin of Error

  7. Confidence Level • A confidence level, c, is a measure of the degree of assurance we have in our results. • The value of c may be any number between zero and one. • Typical values for c include 0.90, 0.95, and 0.99.

  8. Critical Value for a Confidence Level, c • the value zc such that the area under the standard normal curve falling between – zc and zc is equal to c.

  9. Confidence Level

  10. Find z0.90 such that 90% of the area under the normal curve lies between z0.90 and z0.90 P(– z0.90 < z < z0.90 ) = 0.90

  11. Find z0.90 such that 90% of the area under the normal curve lies between –z0.90 and z0.90 • P(z < z0.90 ) = (1 – 0.90)/2 = .05

  12. Find z0.90 such that 90% of the area under the normal curve lies between –z0.90 and z0.90 • According to Appendix Table 3, 0.0500 lies exactly halfway between two values in the table (.0505 and .0495). • Averaging the z values associated with areas gives z0.90 = 1.645. • z0.90 = 1.645.

  13. Common Levels of Confidence and Their Corresponding Critical Values

  14. Confidence Interval • A c confidence interval for  is an interval computed from sample data. • In a c confidence interval for , c is the probability of generating an interval containing the actual value of .

  15. To Find a Confidence Interval for  When  is Known: • Let x represent the appropriate random variable. • Obtain a simple random sample (of size n) of x values • Compute the sample mean, • If you cannot assume x has a normal distribution, use a sample size of 30 or more.

  16. Confidence Interval for When  is Known:

  17. Create a 95% confidence interval for the mean driving time between Philadelphia and Boston. Assume that the mean driving time of 64 trips was 6.4 hours and that the standard deviation is 0.9 hours.

  18. Creating a 95% confidence interval x = 6.4 hours = 0.9 hoursc = 95%, so zc = 1.96n = 64

  19. x = 6.4 hours = 0.9 hours95% confidence interval will be from

  20. x = 6.4 hours = 0.9 hoursc = 95%, so zc = 1.96n = 64

  21. 95% Confidence Interval: 6.4 – .2205 < m < 6.4 + .2205 6.1795 < m < 6.6205 We are 95% sure that the true time is between 6.18 and 6.62 hours.

  22. We may get different confidence intervals for different samples.

  23. We may get different confidence intervals for different samples. • For each sample the c confidence interval goes from • If we select many samples of the same size and find the corresponding confidence intervals, then the proportion of these intervals that actually contain  is c.

  24. When estimating the mean, how large a sample must be used in order to assure a given level of confidence? Use the formula:

  25. How do we determine the value of the population standard deviation, s? • Use the standard deviation, s, of a preliminary sample of size 30 or larger to estimate s.

  26. Determine the sample size necessary to determine (with 99% confidence) the mean time it takes to drive from Philadelphia to Boston. We wish to be within 15 minutes of the true time. Assume that a preliminary sample of 45 trips had a standard deviation of 0.8 hours.

  27. ... determine with 99% confidence... • z0.99 = 2.58

  28. ... We wish to be within 15 minutes of the true time. ... • E = 15 minutes = 0.25 hours

  29. ...a preliminary sample of 45 trips had a standard deviation of 0.8 hours. • Since the preliminary sample is large enough, we can assume that the population standard deviation is approximately equal to 0.8 hours.

  30. Minimum Sample Size =

  31. Rounding Sample Size • Any fractional value of n is always rounded to the next higher whole number.

  32. Minimum Sample Size • n» 68.16 • Round to the next higher whole number. • To be 99% confident in our results, the minimum sample size = 69.

  33. Estimating  When  is Unknown • Apply the Student’s t distribution.

  34. Student’s t Variable

  35. The shape of the t distribution depends only on the sample size, n, if the basic variable x has a normal distribution. When using the t distribution, we will assume that the x distribution is normal.

  36. Appendix Table 4(Page A8)gives values of the variable t corresponding to the number of degrees of freedom (d.f.)

  37. Degrees of Freedom • d.f. = n – 1 • where n = sample size

  38. The t Distribution has a Shape Similar to that of the the Normal Distribution

  39. Properties of a Student’s t Distribution • Symmetric about the mean 0. • Depends on the degrees of freedom. • Bell-shaped with thicker tails than the normal distribution. • As the degrees of freedom increase, the t distribution approaches the standard normal distribution

  40. Appendix Table 4 • Gives various t values for different degrees of freedom

  41. Using Table 4 to Find Critical Values tc for a c Confidence Level

  42. If the required d.f. are not in the table: • Use the closest d.f. that is smaller than the needed d.f. • This results in a larger critical value tc. • The resulting confidence interval will be longer and have a probability slightly higher than c.

  43. Using Table 4 to Find Critical Values of tc • Find the column in the table with the given c heading • Compute the number of degrees of freedom: d.f. = n 1 • Read down the column under the appropriate c value until we reach the row headed by the appropriate d.f.

  44. To find the critical value tc for a 95% confidence interval if n = 8. • Find the column in the table with c heading 0.950

  45. To find the critical value tc for a 95% confidence interval if n = 8. • Compute the number of degrees of freedom: d.f. = n 1 = 8  1 = 7

  46. To find the critical value tc for a 95% confidence interval if n = 8. • Read down the column under the appropriate c value until we reach the row headed by d.f. = 7

  47. Find the critical value tc for a 95% confidence interval if n = 8. tc = 2.365

  48. Finding Confidence Intervals for  When  is Unknown c = confidence level (0 < c < 1) tc = critical value for confidence level c, and degrees of freedom = n 1

  49. The mean weight of eight fish caught in a local lake is 15.7 ounces with a standard deviation of 2.3 ounces. Construct a 90% confidence interval for the mean weight of the population of fish in the lake.

  50. Mean = 15.7 ounces Standard deviation = 2.3 ounces. • n = 8, so d.f. = n – 1 = 7 • For c = 0.90, Appendix Table 4 gives t0.90 = 1.895.

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