Chapter 8

1. Chapter 8 Estimation Understanding Basic Statistics Fifth Edition By Brase and Brase Prepared by Jon Booze

2. Estimating µ When σ is Known

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

4. Margin of Error Even if we take a very large sample size, may differ from µ.

5. Confidence Levels A confidence level, c, is any value between 0 and 1 that corresponds to the area under the standard normal curve between –zc and +zc.

6. Critical Values

7. Critical Values Which of the following correctly expresses the confidence interval shown at right? z 2.58 –2.58 0 a). b). c). d).

8. Critical Values Which of the following correctly expresses the confidence interval shown at right? z 2.58 –2.58 0 a). b). c). d).

9. Common Confidence Levels

10. Recall From Sampling Distributions If we take samples of size n from our population, then the distribution of the sample mean has the following characteristics:

12. Maximal Margin of Error Since µ is unknown, the margin of error | – µ| is unknown. Using confidence level c, we can say that differs from µ by at most:

13. The Probability Statement In words, c is the probability that the sample mean will differ from the population mean by at most

14. Confidence Intervals

16. For a population of domesticated geese, the standard deviation of the mass is 1.3 kg. A sample of 45 geese has a mean mass of 5.7 kg. Find the confidence interval for the population mean at the 95% confidence level. a). 5.32 <  < 6.08 b). 0 <  < 2.97 c). 5.20 <  < 6.20 d). 5.38 <  < 6.02

17. For a population of domesticated geese, the standard deviation of the mass is 1.3 kg. A sample of 45 geese has a mean mass of 5.7 kg. Find the confidence interval for the population mean at the 95% confidence level. a). 5.32 <  < 6.08 b). 0 <  < 2.97 c). 5.20 <  < 6.20 d). 5.38 <  < 6.02

18. Critical Thinking Since is a random variable, so are the endpoints After the confidence interval is numerically fixed for a specific sample, it either does or does not contain µ.

19. If we repeated the confidence interval process by taking multiple random samples of equal size, some intervals would capture µ and some would not! The equation states that the proportion of all intervals containing µ will be c. Critical Thinking

20. Interpretation of the Confidence Interval At the 0.90 confidence level, 1 in 10 samples, on average, will fail to enclose the true mean  within the confidence interval.

21. Estimating µ When σ is Unknown In most cases, researchers will have to estimate σ with s (the standard deviation of the sample). The sampling distribution for will follow a non-normal distribution called the Student’s t distribution.

22. The t Distribution

23. The t Distribution Find the t-value for the following data: a). –27.62 b). –0.11 c). –8.95 d). –4.37

24. The t Distribution Find the t-value for the following data: a). –27.62 b). –0.11 c). –8.95 d). –4.37

25. The t Distribution

26. The t Distribution Use Table 6 of Appendix II to find the critical values tcfor a confidence level c. The figure to the right is a comparison of two t distributions and the standard normal distribution.

27. Using Table 6 to Find Critical Values Degrees of freedom, df, are the row headings. Confidence levels, c, are the column headings.

28. Using Table 4 to Find Critical Values Use Table 4 in the Appendix to find the critical value tc for a 0.95 confidence level for a t-distribution with sample size n = 32. a). 2.457 b). 2.438 c). 2.042 d). 2.030

29. Using Table 4 to Find Critical Values Use Table 4 in the Appendix to find the critical value tc for a 0.95 confidence level for a t-distribution with sample size n = 32. a). 2.457 b). 2.438 c). 2.042 d). 2.030

30. Maximal Margin of Error If we are using the t distribution:

32. What Distribution Should We Use?

33. Estimating p in the Binomial Distribution We will use large-sample methods in which the sample size, n, is fixed. We assume the normal curve is a good approximation to the binomial distribution if both np >5and nq = n(1 – p)> 5.

34. Point Estimates in the Binomial Case

35. Margin of Error The magnitude of the difference between the actual value of p and its estimate is the margin of error.

36. The Distribution of For large samples, the distribution is well approximated by a normal distribution.

37. A Probability Statement With confidence level c, as before.

39. Public Opinion Polls

40. Choosing Sample Sizes When designing statistical studies, it is good practice to decide in advance: The confidence level The maximal margin of error Then, we can calculate the required minimum sample size to meet these goals.

41. Sample Size for Estimating μ If σ is unknown, use σ from a previous study or conduct a pilot study to obtain s. Always round n up to the next integer!!

42. Sample Size for Estimating If we have no preliminary estimate for p, use the following modification:

43. Sample Size for Estimating How many students should be surveyed to determine the proportion of students who prefer vanilla ice cream to chocolate, accurate to 0.1 at a 90% confidence level? a). 100 b). 69 c). 52 d). 5

44. Sample Size for Estimating How many students should be surveyed to determine the proportion of students who prefer vanilla ice cream to chocolate, accurate to 0.1 at a 90% confidence level? a). 100 b). 69 c). 52 d). 5