Chapter 8 Estimation Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania Estimating µ When σ is Known Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8|2 Point Estimate • An estimate of a population parameter given by a single number. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8|3 Margin of Error • Even if we take a very large sample size, will differ from µ. x Margin of Error x Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8|4 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. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8|5 Critical Values Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8|6 Common Confidence Levels Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8|7 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: Mean of x x x Standard Deviationof x x x Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. n 8|8 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8|9 A Probability Statement • In words, c is the probability that the sample mean will differ from the population mean by at most Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 10 Maximal Margin of Error • Since µ is unknown, the margin of error | x - µ| is unknown. • Using confidence level c, we can say that differs from µ by at most: Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. x 8 | 11 Confidence Intervals Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 12 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 13 Critical Thinking • Since x is a random variable, so are the endpoints x E • After the confidence interval is numerically fixed for a specific sample, it either does or does not contain µ. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 14 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. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 15 Multiple Confidence Intervals Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 16 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 x will follow a new distribution, the Student’s t distribution. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 17 The t Distribution Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 18 The t Distribution Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 19 The t Distribution • Use Table 6 of Appendix II to find the critical values tc for a confidence level c. • The figure to the right is a comparison of two t distributions and the standard normal distribution. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 20 Using Table 6 to Find Critical Values • Degrees of freedom, df, are the row headings. • Confidence levels, c, are the column headings. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 21 Maximal Margin of Error • If we are using the t distribution: Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 22 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 23 What Distribution Should We Use? Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 24 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. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 25 Point Estimates in the Binomial Case Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 26 Margin of Error • The magnitude of the difference between the actual value of p and its estimate pˆ is the margin of error. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 27 The Distribution of pˆ • The distribution is well approximated by a normal distribution. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 28 A Probability Statement With confidence level c, as before. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 29 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 30 Public Opinion Polls Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 31 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. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 32 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!! Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 33 Sample Size for Estimating pˆ If we have no preliminary estimate for p, use the following modification: Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 34 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 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 35 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. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 36 Confidence Intervals for μ1 – μ2 when σ1, σ2 known Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 37 Confidence Intervals for μ1 – μ2 when σ1, σ2 known Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 38 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 39 Confidence Intervals for μ1 – μ2 when σ1, σ2 unknown • If σ1, σ2 are unknown, we use the t distribution (just like the one-sample problem). Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 40 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 41 What if σ1 = σ2 ? • If the sample standard deviations s1 and s2 are sufficiently close, then it may be safe to assume that σ1 = σ2. – Use a pooled standard deviation. – See Section 8.4, problem 27. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 42 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 43 Summarizing Intervals for Differences in Population Means Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 44 Estimating the Difference in Proportions • We consider two independent binomial distributions. • For distribution 1 and distribution 2, respectively, we have: n1 p1 q1 r1 n2 p2 q2 r2 • We assume that all the following are greater than 5: Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 45 Estimating the Difference in Proportions r1 r 2 Then has the following properties : n1 n 2 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 46 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 47 Critical Thinking Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 48