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Chapter 7 Estimates and Sample Sizes 7-1 Review and Preview 7-2 Estimating a Population Proportion 7-3 Estimating a Population Mean: σ Known (OMIT) 7-4 Estimating a Population Mean: σ Not Known 7-5 Estimating a Population Variance (OMIT) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 1 Section 7-1 Review and Preview Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 2 Review Chapters 2 & 3 we used “descriptive statistics” when we summarized data using tools such as graphs, and statistics such as the mean and standard deviation. Chapter 6 we introduced critical values: z denotes the z score with an area of to its right. If = 0.025, the critical value is z0.025 = 1.96. That is, the critical value z0.025 = 1.96 has an area of 0.025 to its right. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 3 Preview This chapter presents the beginning of inferential statistics. The two major activities of inferential statistics are (1) to use sample data to estimate values of a population parameters, and (2) to test hypotheses or claims made about population parameters. We introduce methods for estimating values of these important population parameters: proportions, means, and variances. We also present methods for determining sample sizes necessary to estimate those parameters. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 4 Section 7-2 Estimating a Population Proportion Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 5 Key Concept In this section we present methods for using a sample proportion to estimate the value of a population proportion. • The sample proportion is the best point estimate of the population proportion. • We can use a sample proportion to construct a confidence interval to estimate the true value of a population proportion, and we should know how to interpret such confidence intervals. • We should know how to find the sample size necessary to estimate a population proportion. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 6 Definition A point estimate is a single value (or point) used to approximate a population parameter. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 7 Definition ˆ The sample proportion p is the best point estimate of the population proportion p. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 8 Example: In the Chapter Problem (page 314) we noted that in a Pew Research Center poll, 70% of 1501 randomly selected adults in the United States believe in global warming, so the sample proportion is p̂ = 0.70. Find the best point estimate of the proportion of all adults in the United States who believe in global warming. Because the sample proportion is the best point estimate of the population proportion, we conclude that the best point estimate of p is 0.70. When using the sample results to estimate the percentage of all adults in the United States who believe in global warming, the best estimate is 70%. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 9 Definition A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated as CI. Here is an example of a confidence interval for the population proportion parameter: 0.677 p 0.723 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 10 NOTE We will learn how to construct confidence intervals from a sample statistic later using a formula. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 11 Interpreting a Confidence Interval We must be careful to interpret confidence intervals correctly. There is a correct interpretation and many different and creative incorrect interpretations of the confidence interval a < p < b Typically, we interpret the 95% confidence interval as follows: “We are 95% confident that the interval from a to b actually does contain the true value of the population proportion p.” Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 12 Interpreting a Confidence Interval This means that if we were to select many different samples of the same size and construct the corresponding confidence intervals, 95% of them would actually contain the value of the population proportion p. (Note that in this correct interpretation, the level of 95% refers to the success rate of the process being used to estimate the proportion.) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 13 Interpreting a Confidence Interval For example, if we calculate the 95% confidence intervals for 20 different samples of a population, we expect that 95% of the 20 samples, or 19 samples, would have confidence intervals that contain the true value of p. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 14 Interpreting a Confidence Interval Consider the chapter problem example (global warming) again. Suppose we know the true proportion of all adults who believe in global warming is p=0.75 With 95% confidence interval, if we sample 20 times we may compute a confidence interval which does not actually contain p=0.75, such as, 0.677 p 0.723 but, 19 times out of 20 we would find confidence intervals that do contain p=0.75. This is illustrated in Figure 7-1. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 15 Interpreting a Confidence Interval page 319, Figure 7-1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 16 Caution Know the correct interpretation of a confidence interval. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 17 NEXT We will learn now discuss how to construct confidence intervals from a sample statistic. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 18 Critical Values A standard z score can be used to distinguish between sample statistics that are likely to occur and those that are unlikely to occur. Such a z score is called a critical value. Critical values are based on the following observations: Under certain conditions, the sampling distribution of sample proportions can be approximated by a normal distribution. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 19 Critical Values DEFINE: A z score of z / 2 associated with a sample proportion has a probability of /2 of falling in the right tail. Therefore: To find z / 2 , find the z-score in Table A-2 that corresponds to an area of 1 / 2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 20 Example Page 328, problem 7 Find z/2 for =0.10 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 21 Example Page 328, problem 7 ANSWER: Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 22 Definition A critical value is the number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur. The number z/2 is a critical value that is a z score with the property that it separates an area of /2 in the right tail of the standard normal distribution. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 23 Critical Value Because the standard normal distribution is symmetric about the value of z=0, the value of –z/2 is at the vertical boundary for the area of /2 in the left tail Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 24 The Critical Value z 2 -z/2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 25 Definition A confidence level is the probability 1 – (often expressed as the equivalent percentage value) that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times. (The confidence level is also called degree of confidence, or the confidence coefficient.) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 26 Definition Common choices for confidence levels are: 90% confidence level where = 10%, 95% confidence level where = 5%, 99% confidence level where = 1%, Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 27 Confidence Level For the standard normal distribution a confidence level of P % corresponds to P percent of the area between the values –z/2 and z/2 For example, if = 5% , the confidence level is 1-0.05=0.95=95% which gives the z-score z / 2 z0.025 1.96 and 95% of the area lies between -1.96 and 1.96 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 28 z2 for a 95% Confidence Level = 0.05 z2 = 1.96 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 29 z2 for a 95% Confidence Level = 5% 2 = 2.5% = .025 z2 -z2 Critical Values Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 30 Definition When data from a simple random sample are used to estimate a population proportion p, the margin of error, denoted by E, is the maximum likely difference (with probability 1 – , such as 0.95) between the observed proportion p̂ and the true value of the population proportion p. The margin of error E is also called the maximum error of the estimate and can be found by multiplying the critical value and the standard deviation of the sample proportions: Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 31 Margin of Error for Proportions E z 2 NOTE: ˆˆ pq n qˆ 1 pˆ Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 32 Confidence Interval for Estimating a Population Proportion p p = population proportion p̂ = sample proportion n = number of sample values E = margin of error z/2 = z score separating an area of /2 in the right tail of the standard normal distribution 7.1 - 33 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Requirements for Using a Confidence Interval for Estimating a Population Proportion p 1. The sample is a simple random sample. 2. The conditions for the binomial distribution are satisfied: there is a fixed number of trials, the trials are independent, there are two categories of outcomes, and the probabilities remain constant for each trial. 3. There are at least 5 successes and 5 failures. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 34 Confidence Interval for Estimating a Population Proportion p pˆ – E < p < p̂ + E where E z 2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. ˆˆ pq n 7.1 - 35 Confidence Interval for Estimating a Population Proportion p ˆ – E < p < pˆ + E p ˆ + E p ˆ + E) (pˆ – E, p Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 36 Round-Off Rule for Confidence Interval Estimates of p Round the confidence interval limits for p to three significant digits. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 37 Example Page 328 Problem 18: Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 38 Example Page 328, problem 18 ANSWER: compute the critical value z/2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 39 Example Page 328, problem 18 ANSWER: compute the sample proportion p̂ and q̂ qˆ 1 pˆ 0.56 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 40 Example Page 328, problem 18 ANSWER: compute the margin of error E Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 41 Calculator Use Here is what the solution manual suggests Calculate z/2 first To use the formula on a TI calculator: Calculate p̂ , press the multiply key, the parentheses key, then 1- ANS (ANS is the 2nd (-) key on bottom row) , then ENTER. This will give pˆ qˆ Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 42 Example Press the divide key and input the value of n then ENTER. This will give pˆ qˆ / n Press the square root key, then ANS, then ENTER. This will give pˆ qˆ / n Finally press the multiply key then input the value of z/2 then ENTER. This will give E Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 43 Procedure for Constructing a Confidence Interval for p 1. Verify that the required assumptions are satisfied. (The sample is a simple random sample, the conditions for the binomial distribution are satisfied, and the normal distribution can be used to approximate the distribution of sample proportions because np 5, and nq 5 are both satisfied.) 2. Refer to Table A-2 and find the critical value z corresponds to the desired confidence level. 3. Evaluate the margin of error Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. /2 that ˆˆ n E z 2 pq 7.1 - 44 Procedure for Constructing a Confidence Interval for p - cont 4. Using the value of the calculated margin of error, E and the value of the sample proportion, p, find the values of p – E and p + E. Substitute those values in the general format for the confidence interval: ˆ ˆ ˆ p ˆ – E < p < pˆ + E 5. Round the resulting confidence interval limits to three significant digits. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 45 Example Page 328 Problem 22: Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 46 Example Page 328, problem 22 95% confidence interval gives = 5%, compute the critical value z/2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 47 Example Page 328, problem 22 compute the sample proportion p̂ and q̂ qˆ 1 pˆ 0.8000 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 48 Example Page 328, problem 22 compute the margin of error E E z / 2 pˆ qˆ / n 1.96 (0.2000)(0.8000) / 2000 0.0175 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 49 Example Page 328, problem 22 compute the upper and lower limits of the confidence interval upper limit pˆ z / 2 pˆ qˆ / n 0.2000 0.0175 lower limit pˆ z / 2 pˆ qˆ / n 0.2000 0.0175 0.2175 0.1825 0.218 0.183 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 50 Example Page 328, problem 22 ANSWER: write down the confidence interval using the upper and lower limits 0.183 p 0.218 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 51 Example Page 328, problem 22 The confidence interval can also be expressed as pˆ E 0.2000 0.0175 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 52 Example Page 328 Problem 10: Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 53 Example Page 328, problem 10 ANSWER: The sample proportion is the midpoint of the upper and lower limits of the confidence interval pˆ 0.720 0.780 0.750 2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 54 Example Page 328, problem 10 ANSWER: The margin of error is the difference between the upper limit of the confidence interval and the sample proportion E 0.780 0.750 0.030 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 55 Example Page 328, problem 10 The confidence interval can be expressed as pˆ E 0.750 0.030 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 56 Calculate Confidence Intervals Directly From Calculator The TI calculator will compute confidence intervals as follows: Press STAT and select TESTS Select A:1-PropZInt Enter x,n,C-Level and then calculate You should be able to calculate a confidence interval both ways: use the formula as in previous examples and directly with A:1PropZInt as above Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 57 Example Page 329, problem 30 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 58 Example Page 329, problem 30 a) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 59 Example Page 329, problem 30 b) 99% confidence interval gives = 1%, Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 60 Example Page 329, problem 30 b) compute the margin of error E E z / 2 pˆ qˆ / n 2.575 (0.8355)(0.1645) / 152 0.0774 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 61 Example Page 329, problem 30 b) compute the upper and lower limits of the confidence interval upper limit pˆ z / 2 pˆ qˆ / n 0.8355 0.0774 lower limit pˆ z / 2 pˆ qˆ / n 0.8355 0.0774 0.9129 0.7581 0.913 0.758 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 62 Example Page 329, problem 30 b) ANSWER: write down the confidence interval using the upper and lower limits 0.758 p 0.913 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 63 Example Page 329, problem 30 c) We interpret the answer to part (b) as follows: “We are 99% confident that the true value of the proportion of boys in the population will be between 0.758 and 0.913.” Therefore, Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 64 Example Page 329, problem 30 c) if the YSORT method has no effect we expect the population proportion to be p=0.5 which is not within the 99% confidence interval from part (b) and we can be 99% confident that the YSORT method is effective. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 65 Example Page 329, problem 34 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 66 Example Page 329, problem 34 a) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 67 Example Page 329, problem 34 b) We interpret the answer to part (a) as follows: “We are 99% confident that the true value of the proportion of people who say they vote in the population will be between 0.662 and 0.737.” Therefore, Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 68 Example Page 329, problem 34 c) since the true population proportion is given as p=0.61 which is not within the 99% confidence interval from part (b), we can be 99% confident that people do not tell the truth about their voting record. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 69 Analyzing Polls When analyzing polls consider: 1. The sample should be a simple random sample, not an inappropriate sample (such as a voluntary response sample). 2. The confidence level should be provided. (It is often 95%, but media reports often neglect to identify it.) 3. The sample size should be provided. (It is usually provided by the media, but not always.) 4. Except for relatively rare cases, the quality of the poll results depends on the sampling method and the size of the sample, but the size of the population is usually not a factor. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 70 Caution Never follow the common misconception that poll results are unreliable if the sample size is a small percentage of the population size. The population size is usually not a factor in determining the reliability of a poll. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 71 Sample Size Suppose we want to collect sample data in order to estimate some population proportion. The question is how many sample items must be obtained? Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 72 Determining Sample Size z 2 E= p ˆ qˆ n (solve for n by algebra) n= ( Z 2)2 p ˆ ˆq E2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 73 Sample Size for Estimating Proportion p ˆ When an estimate of p is known: n= ( z 2 )2 pˆ qˆ Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. E2 7.1 - 74 Sample Size for Estimating Proportion p ˆ When no estimate of p is known: n= ( z 2)2 0.25 E2 NOTE: here we are assuming that pˆ qˆ 0.50 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 75 Round-Off Rule for Determining Sample Size If the computed sample size n is not a whole number, round the value of n up to the next larger whole number. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 76 Example Page 331, problem 42 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 77 Example Page 331, problem 42 a) We are told that the sample percentage is within 4 percentage points of the true population percentage. This means that the margin of error is E=0.04 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 78 Example Page 331, problem 42 a) 90% confidence interval gives = 10%, which then gives the values Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 79 Example Page 331, problem 42 a) When sample proportion is unknown use: pˆ qˆ 0.50 to get Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 80 Example Page 331, problem 42 b) sample proportion is given pˆ 0.08 and qˆ 1 pˆ 0.92 and the same values for the critical value and margin of error as in part (a) to get Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 81 Recap In this section we have discussed: Point estimates. Confidence intervals. Confidence levels. Critical values. Margin of error. Determining sample sizes. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 82 Section 7-4 Estimating a Population Mean: Not Known Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 83 Key Concept This section presents methods for estimating a population mean when the population standard deviation is not known. With σ unknown, we use the Student t distribution assuming that the relevant requirements are satisfied. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 84 Sample Mean The sample mean is the best point estimate of the population mean. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 85 Notation = population mean x = sample mean s = sample standard deviation n = number of sample values Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 86 Student t Distribution If the distribution of a population is essentially normal, then the distribution of t = x-µ s n is a Student t Distribution for all samples of size n. It is often referred to as a t distribution and is used to find critical values denoted by t/2. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 87 Important Properties of the Student t Distribution 1. The Student t distribution is different for different sample sizes (see the following slide, for the cases n = 3 and n = 12). 2. The Student t distribution has the same general symmetric bell shape as the standard normal distribution but it reflects the greater variability (with wider distributions) that is expected with small samples. 3. The Student t distribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0). 4. The standard deviation of the Student t distribution varies with the sample size and is greater than 1 (unlike the standard normal distribution, which has a = 1). 5. As the sample size n gets larger, the Student t distribution gets closer to the normal distribution. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 88 Student t Distributions for n = 3 and n = 12 Figure 7-5 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 89 Definition The number of degrees of freedom for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data values. The degree of freedom is often abbreviated df. degrees of freedom = n – 1 in this section. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 90 Notation E = margin of error t/2 = critical t value separating an area of /2 in the right tail of the t distribution Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 91 Margin of Error E for Estimate of (With σ Not Known) Formula 7-6 E = t s 2 n where t2 has n – 1 degrees of freedom. Table A-3 lists values for tα/2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 92 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 93 Confidence Interval for the Estimate of μ (With σ Not Known) x–E <µ<x +E where E = t/2 s n df = n – 1 t/2 found in Table A-3 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 94 Example Use the sample statistics of n = 49, x = 0.4 and s = 21.0 to construct a 95% confidence interval estimate of the population mean. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 95 Example 95% confidence level so = 5%=0.05 With n = 49, the df = 49 – 1 = 48 Closest df in Table A-3 is 50, using two tails = 5%=0.05 using one tail /2= 2.5%=0.025 t/2 = 2.009 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 96 Example Using t/2 = 2.009, s = 21.0 and n = 49 the margin of error is: s 21.0 E t / 2 2.009 6.027 n 49 and the confidence interval is x E x E 0.4 6.027 0.4 6.027 5.6 6.4 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 97 Requirements for Using a Confidence Interval for Estimating a Population Mean µ 1. The sample is a simple random sample. 2. Either the sample is from a normally distributed population or n>30 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 98 Procedure for Constructing a Confidence Interval for µ (With σ Unknown) 1. Verify that the requirements are satisfied. 2. Using n – 1 degrees of freedom, refer to Table A-3 or use technology to find the critical value t2 that corresponds to the desired confidence level. 3. Evaluate the margin of error E = t2 • s / n . 4. Find the values of x E and x E. Substitute those values in the general format for the confidence interval: x E x E 5. Round the resulting confidence interval limits. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 99 Round-Off Rule for Confidence Interval Estimates of µ If using the original set of data, round to one more decimal place than is used for the original data set. If using summary statistics x ,Sx, n round to the same number of decimal places used for the sample mean x Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 100 CALCULATOR You will not be given Table A-3 on the exam and should be able to use your calculator to compute the confidence interval for population mean with σ unknown. This is the method that will be used for the remaining slides. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 101 Example Page 354 Problem 14: NOTE: do part (b) first using the TI calculator Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 102 Example The TI calculator will compute confidence intervals for population mean with σ unknown as follows: Press STAT and select TESTS Select 8:Tinterval Arrow right to select Inpt: Stats Enter x ,Sx, n, C-Level and then calculate Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 103 Example Page 354, problem 14 ANSWER to part (b) Press STAT and select TESTS Select 8:Tinterval Arrow right to select Inpt: Stats Enter x 0.12, Sx 0.04, n 7, C - Level 0.99 Calculate then gives: (0.06395,0.17605) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 104 Example Page 354, problem 14 ANSWER to part (b) We must round two decimal places (same number of places as the sample mean) 0.06 0.18 Units are grams/mile Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 105 Finding the Point Estimate and E from a Confidence Interval Point estimate of µ: x = (upper confidence limit) + (lower confidence limit) 2 Margin of Error: E = upper confidence limit - x Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 106 Example Page 354, problem 14 ANSWER to part (a) uses part (b) 0.06 0.18 0.06 0.18 x 0.12 grams/mile 2 E 0.18 0.12 0.06 grams/mile Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 107 Example Page 354, problem 18 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 108 Example Page 354, problem 18 a) ANSWER: x 3103 grams Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 109 Example Page 354, problem 18b) Press STAT and select TESTS Select 8:Tinterval Arrow right to select Inpt: Stats Enter x 3103, Sx 696, n 186, C - Level : 0.95 then calculate to get the confidence interval (3002.3,3203.7) which rounds to: 3002 g 3204 g Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 110 Example Page 354, problem 18 c) ANSWER: Yes, since the confidence interval 2608 g 2792 g for the mean birth weight for mothers who used cocaine is entirely below the confidence interval in part (b) for mothers who did not use cocaine, it appears that cocaine use is associated with lower birth weights. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 111 Confidence Intervals for Comparing Data As in Sections 7-2 and 7-3, confidence intervals can be used informally to compare different data sets, but the overlapping of confidence intervals should not be used for making formal and final conclusions about equality of means. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 112 Example Page 355, problem 22 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 113 Example Page 355, problem 22(a) Press STAT and select TESTS Select 8:Tinterval Arrow right to select Inpt: Stats Enter x 1.8, Sx 1.4, n 142, C - Level : 0.95 then calculate to get the confidence interval 1.6 2.0 (headaches ) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 114 Example Page 355, problem 22(b) Press STAT and select TESTS Select 8:Tinterval Arrow right to select Inpt: Stats Enter x 1.6, Sx 1.2, n 80, C - Level : 0.95 then calculate to get the confidence interval 1.3 1.9 (headaches ) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 115 Example Page 355, problem 22(c) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 116 Example Page 356, problem 26 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 117 Example Page 356, problem 26 Here: x ?, Sx ?, n 7, C - Level : 0.98 and we are not given the mean and standard deviation. We must determine these from the given data. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 118 Example Compute the mean: x x = n 0.06 0.11 0.16 0.15 0.14 0.08 0.15 x 7 0.85 7 0.121 g/mile Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 119 Example Compute the standard deviation: s (x x) 2 n 1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 120 Example x xx (x x) 0.06 -0.061 0.003721 0.11 -0.011 0.000121 0.16 0.039 0.001521 0.15 0.029 0.000841 0.14 0.019 0.000361 0.08 -0.041 0.001681 0.15 0.029 0.000841 Sum of last column: (x x) 2 2 0.009087 0.009087 s 0.0015145 0.0389 g/mile 7 1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 121 •Calculator: 1) Enter the list of data values into a list using STAT 1:Edit 2) Select 2nd STAT (LIST) and arrow right to choose MATH option 3:mean( 3) Select 2nd STAT (LIST) and arrow right to choose MATH option 7:stdDev( 4) Choose the list the data is in 3.1 - Example Press STAT and select TESTS Select 8:Tinterval Arrow right to select Inpt: Stats Enter x 0.121, Sx 0.0389, n 7, C - Level : 0.98 then calculate to get the confidence interval 0.075 0.168 (grams/mil e) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 123 Example Page 356, problem 26 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 124 Recap In this section we have discussed: Student t distribution. Degrees of freedom. Margin of error. Confidence intervals for μ with σ unknown. Choosing the appropriate distribution. Point estimates. Using confidence intervals to compare data. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7.1 - 125
