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Estimation Chapter 8 Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania.

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Presentation on theme: "Estimation Chapter 8 Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania."— Presentation transcript:

1 Estimation Chapter 8 Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania

2 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 2 Estimating µ When σ is Known

3 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 3 Point Estimate An estimate of a population parameter given by a single number.

4 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 4 Margin of Error Even if we take a very large sample size, will differ from µ.

5 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 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 –z c and +z c.

6 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 6 Critical Values

7 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 7 Common Confidence Levels

8 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 8 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:

9 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 9

10 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 10 A Probability Statement In words, c is the probability that the sample mean will differ from the population mean by at most

11 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 11 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:

12 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 12 Confidence Intervals

13 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 13

14 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 14 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.

15 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 15 Critical Thinking If we repeated the confidence interval process by taking multiple random samples of equal size, some intervals would capture µ and some would not! Equation states that the proportion of all intervals containing µ will be c.

16 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 16 Multiple Confidence Intervals

17 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 17 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 new distribution, the Student’s t distribution.

18 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 18 The t Distribution

19 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 19 The t Distribution

20 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 20 The t Distribution Use Table 6 of Appendix II to find the critical values t c for a confidence level c. The figure to the right is a comparison of two t distributions and the standard normal distribution.

21 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 21 Using Table 6 to Find Critical Values Degrees of freedom, df, are the row headings. Confidence levels, c, are the column headings.

22 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 22 Maximal Margin of Error If we are using the t distribution:

23 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 23

24 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 24 What Distribution Should We Use?

25 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 25 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 > 5 and nq = n(1-p) > 5.

26 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 26 Point Estimates in the Binomial Case

27 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 27 Margin of Error The magnitude of the difference between the actual value of p and its estimate is the margin of error.

28 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 28 The Distribution of The distribution is well approximated by a normal distribution.

29 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 29 A Probability Statement With confidence level c, as before.

30 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 30

31 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 31 Public Opinion Polls

32 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 32 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.

33 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 33 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!!

34 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 34 Sample Size for Estimating If we have no preliminary estimate for p, use the following modification:

35 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 35 Independent Samples Two samples are independent if sample data drawn from one population is completely unrelated to the selection of a sample from the other population. –Occurs when we draw two random samples

36 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 36 Dependent Samples Two samples are dependent if each data value in one sample can be paired with a corresponding value in the other sample. –Occur naturally when taking the same measurement twice on one observation Example: your weight before and after the holiday season.

37 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 37 Confidence Intervals for μ 1 – μ 2 when σ 1, σ 2 known

38 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 38 Confidence Intervals for μ 1 – μ 2 when σ 1, σ 2 known

39 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 39

40 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 40 Confidence Intervals for μ 1 – μ 2 when σ 1, σ 2 unknown If σ 1, σ 2 are unknown, we use the t distribution (just like the one-sample problem).

41 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 41

42 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 42 What if σ 1 = σ 2 ? If the sample standard deviations s 1 and s 2 are sufficiently close, then it may be safe to assume that σ 1 = σ 2. –Use a pooled standard deviation. –See Section 8.4, problem 27.

43 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 43

44 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 44 Summarizing Intervals for Differences in Population Means

45 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 45 Estimating the Difference in Proportions We consider two independent binomial distributions. For distribution 1 and distribution 2, respectively, we have: n 1 p 1 q 1 r 1 n 2 p 2 q 2 r 2 We assume that all the following are greater than 5:

46 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 46 Estimating the Difference in Proportions

47 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 47

48 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.8 | 48 Critical Thinking


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