Chapter 8 Estimation • • • • Estimating µ when σ is know Estimating µ when σ is unknow Estimating p in the binomial distribution Estimating µ1- µ2 and p1-p2 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8|1 8.1 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, x will differ from µ. 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 Example Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8|7 Common Confidence Levels 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: Mean of x x x Standard Deviation of x x x Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. n 8|9 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 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 11 Maximal Margin of Error • Since µ is unknown, the margin of error | x - µ| is unknown. • Using confidence level c, we can say that x differs from µ by at most: Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 12 Confidence Intervals Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 13 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 14 Example Julia enjoys jogging. She has been jogging over a period of several years, during which time her physical condition has remained constantly good. Usually, she jogs 2 miles per day. The standard deviation of her times is 1.80 minutes. During the past year, Julia has recorded her times to run 2 miles. She has a random sample of 90 of these times. For these 90 times, the mean was 15.60 minutes. Let µ be the mean jogging time for the entire distribution of Julia’s 2-mile running times (taken over the past year). Find a 0.95 confidence interval for µ . Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 15 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 | 16 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 | 17 Multiple Confidence Intervals Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 18 Sample Size for Estimating the Mean µ x Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 19 Sample Size for Estimating the Mean µ If n is not a whole number, increase n to the next higher whole number. Note that n is the minimal sample size for a specified confidence level and maximal error of estimate E. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 20 Example A wildlife study is designed to find the mean weight of salmon caught by an Alaskan fishing company. A preliminary study of a random sample of 50 salmon showed s=2.15 pounds. How large a sample should be taken to be 99% confi dent that the sample mean is within 0.20 pound of the true mean weight µ? Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 21 8.2 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 | 22 The t Distribution Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 23 The t Distribution Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 24 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 | 25 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 | 26 Example Find the critical value t c for a 0.99 confidence level for a t distribution with sample size n=5. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 27 Maximal Margin of Error • If we are using the t distribution: Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 28 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 29 Example Suppose an archaeologist discovers only seven fossil skeletons from a previously unknown species of miniature horse. Reconstructions of the skeletons of these seven miniature horses show the shoulder heights (in centimeters) to be 45.3 47.1 44.2 46.8 46.5 45.5 47.6 For these sample data, the mean is 46.14 and the sample standard deviation is s=1.19. Let µ be the mean shoulder height (in centimeters) for this entire species of miniature horse, and assume that the population of shoulder heights is approximately normal. Find a 99% confidence interval for µ, the mean shoulder height of the entire population of such horses. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 30 What Distribution Should We Use? Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 31 8.3 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 | 32 Point Estimates in the Binomial Case Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 33 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 | 34 The Distribution of pˆ • The distribution is well approximated by a normal distribution. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 35 A Probability Statement With confidence level c, as before. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 36 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 37 Public Opinion Polls Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 38 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 | 39 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 | 40 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 | 41 8.4 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 | 42 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 | 43 Confidence Intervals for μ1 – μ2 when σ1, σ2 known Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 44 Confidence Intervals for μ1 – μ2 when σ1, σ2 known Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 45 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 46 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 | 47 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 48 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 | 49 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 50 Summarizing Intervals for Differences in Population Means Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 51 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 | 52 Estimating the Difference in Proportions Then r1 n1 r2 has the following properties : n2 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 53 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 54 Critical Thinking Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 8 | 55