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Chapter 12 Inference About A Population Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.1 Inference About A Population… Population Sample Inference Statistic Parameter • Identify the parameter to be estimated or tested. • Specify the parameter’s estimator and its sampling distribution. • Derive the interval estimator and test statistic. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.2 Inference About A Population… Population Sample Inference Statistic Parameter We will develop techniques to estimate and test three population parameters: Population Mean Population Variance Population Proportion p Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.3 Inference With Variance Unknown… Previously, we looked at estimating and testing the population mean when the population standard deviation ( ) was known or given: But how often do we know the actual population variance? Instead, we use the Student t-statistic, given by: Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.4 Inference With Variance Unknown… When is unknown, we use its point estimator s and the z-statistic is replaced by the the t-statistic, where the number of “degrees of freedom” , is n–1. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.5 Testing when is unknown… When the population standard deviation is unknown and the population is normal, the test statistic for testing hypotheses about is: which is Student t distributed with = n–1 degrees of freedom. The confidence interval estimator of is given by: Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.6 Example 12.1… Will new workers achieve 90% of the level of experienced workers within one week of being hired and trained? Experienced workers can process 500 packages/hour, thus if our conjecture is correct, we expect new workers to be able to process .90(500) = 450 packages per hour. Given the data, is this the case? Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.7 Example 12.1… IDENTIFY Our objective is to describe the population of the numbers of packages processed in 1 hour by new workers, that is we want to know whether the new workers’ productivity is more than 90% of that of experienced workers. Thus we have: H1: > 450 Therefore we set our usual null hypothesis to: H0: = 450 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.8 Example 12.1… COMPUTE Our test statistic is: With n=50 data points, we have n–1=49 degrees of freedom. Our hypothesis under question is: H1: > 450 Our rejection region becomes: Thus we will reject the null hypothesis in favor of the alternative if our calculated test static falls in this region. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.9 Example 12.1… From the data, we calculate COMPUTE = 460.38, s =38.83 and thus: Since we reject H0 in favor of H1, that is, there is sufficient evidence to conclude that the new workers are producing at more than 90% of the average of experienced workers. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.10 Example 12.1… COMPUTE : : Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. rejection region Alternatively, we can use t-test:Mean from Tools > Data Analysis Plus in Excel… 12.11 Example 12.1… COMPUTE p-value In addition to looking at the computed t-statistic and the critical value of t (one tail), we could look at the p-value (0.0323) and see that it is “small” (~3%), so again, we reject the null hypothesis in favor of the alternative… Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.12 Example 12.2… IDENTIFY Can we estimate the return on investment for companies that won quality awards? We have are given a random sample of n = 83 such companies. We want to construct a 95% confidence interval for the mean return, i.e. what is: ?? Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.13 Example 12.2… COMPUTE From the data, we calculate: For this term and so: Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.14 Example 12.2… INTERPRET We are 95% confident that the population mean, , i.e. the mean return of all publicly traded companies that win quality awards, lies between 13.20% and 16.84% Tools > Data Analysis Plus > t-Estimate: Mean is an alternative to the manual calculation… Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.15 Check Requisite Conditions… The Student t distribution is robust, which means that if the population is nonnormal, the results of the t-test and confidence interval estimate are still valid provided that the population is “not extremely nonnormal”. To check this requirement, draw a histogram of the data and see how “bell shaped” the resulting figure is. If a histogram is extremely skewed (say in the case of an exponential distribution), that could be considered “extremely nonnormal” and hence t-statistics would be not be valid in this case. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.16 Estimating Totals of Finite Populations… Large populations are defined as “populations that are at least 20 times the sample size” We can use the confidence interval estimator of a mean to produce a confidence interval estimator of the population total: Where N is the size of the finite population. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.17 Estimating Totals of Finite Populations… For example, a sample of 500 households (in a city of 1 million households) reveals a 95% confidence interval estimate that the household mean spent on Halloween candy lies between $20 & $30. We can estimate the total amount spent in the city by multiplying these lower and upper confidence limits by the total population: Thus we estimate that the total amount spent on Halloween in the city lies between $20 million and $30 million. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.18 Identifying Factors… Factors that identify the t-test and estimator of Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. : 12.19 Inference About Population Variance… If we are interested in drawing inferences about a population’s variability, the parameter we need to investigate is the population variance: The sample variance (s2) is an unbiased, consistent and efficient point estimator for . Moreover, the statistic, , has a chi-squared distribution, with n–1 degrees of freedom. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.20 Testing & Estimating Population Variance The test statistic used to test hypotheses about (which is chi-squared with Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. is: = n–1 degrees of freedom). 12.21 Testing & Estimating Population Variance Combining this statistic: With the probability statement: Yields the confidence interval estimator for lower confidence limit Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. : upper confidence limit 12.22 Example 12.3… IDENTIFY Consider a container filling machine. Management wants a machine to fill 1 liter (1,000 cc’s) so that that variance of the fills is less than 1 cc2. A random sample of n=25 1 liter fills were taken. Does the machine perform as it should at the 5% significance level? Variance is less than 1 cc2 We want to show that: H1: <1 (so our null hypothesis becomes: H0: = 1). We will use this test statistic: Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.23 Example 12.3… COMPUTE Since our alternative hypothesis is phrased as: H1: <1 We will reject H0 in favor of H1 if our test statistic falls into this rejection region: We computer the sample variance to be: s2=.8088 And thus our test statistic takes on this value… Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.24 Example 12.3… INTERPRET Since: There is not enough evidence to infer that the claim is true. Excel output can also be used for this test… compare Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.25 Example 12.4… As we saw, we cannot reject the null hypothesis in favor of the alternative. That is, there is not enough evidence to infer that the claim is true. Note: the result does not say that the variance is greater than 1, rather it merely states that we are unable to show that the variance is less than 1. We could estimate (at 99% confidence say) the variance of the fills… Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.26 Example 12.4… COMPUTE In order to create a confidence interval estimate of the variance, we need these formulae: lower confidence limit upper confidence limit we know (n–1)s2 = 19.41 from our previous calculation, and we have from Table 5 in Appendix B: Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.27 Example 12.4… COMPUTE Thus the 99% confidence interval estimate is: That is, the variance of fills lies between .426 and 1.963 cc2. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.28 Identifying Factors… Factors that identify the chi-squared test and estimator of Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. : 12.29 Inference: Population Proportion… When data are nominal, we count the number of occurrences of each value and calculate proportions. Thus, the parameter of interest in describing a population of nominal data is the population proportion p. This parameter was based on the binomial experiment. Recall the use of this statistic: where p-hat ( ) is the sample proportion: x successes in a sample size of n items. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.30 Inference: Population Proportion… When np and n(1–p) are both greater than 5, the sampling distribution of is approximately normal with mean: standard deviation: Hence: Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.31 Inference: Population Proportion… Test statistic for p: The confidence interval estimator for p is given by: (both of which require that np>5 and n(1–p)>5) Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.32 Example 12.5… IDENTIFY At an exit poll, voters are asked by a certain network if they voted Democrat (code=1) or Republican (code=2). Based on their small sample, can the network conclude that the Republican candidate will win the vote? That is: H1: p > .50 And hence our null hypothesis becomes: H0: p = .50 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.33 Example 12.5… COMPUTE Since our research hypothesis is: H1: p > .50 our rejection region becomes: Looking at the data, we count 407 (of 765) votes for code=2. Hence, we calculate our test statistic as follows… Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.34 Example 12.5… INTERPRET Since: …we reject H0 in favor of H1, that is, there is enough evidence to believe that the Republicans win the vote. Likewise from Excel: compare these… …or look at p-value Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.35 Selecting the Sample Size… The confidence interval estimator for a population proportion is: Thus the (half) width of the interval is: Solving for n, we have: Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.36 Selecting the Sample Size… For example, we want to know how many customers to survey in order to estimate the proportion of customers who prefer our brand to within .03 (with 95% confidence). I.e. our confidence interval after surveying will be ± .03, that means W=.03 Uh Oh. Since we haven’t taken a sample yet, we don’t have this sample proportion… Substituting into the equation… Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.37 Selecting the Sample Size… Two methods – in each case we choose a value for solve the equation for n. then Method 1 : no knowledge of even a rough value of is a ‘worst case scenario’ so we substitute = .50 . This Method 2 : we have some idea about the value of . This is a better scenario and we substitute in our estimated value. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.38 Selecting the Sample Size… Method 1 : no knowledge of value of Method 2 : some idea about a possible , use 50%: value, say 20%: Thus, we can sample fewer people if we already have a reasonable estimate of the population proportion before starting. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.39 Estimating Totals for Large Populations… In much the same way as we saw earlier, when a population is large and finite we can estimate the total number of successes in the population by taking the product of the size of the population (N) and the confidence interval estimator: The Nielsen Ratings (used to measure TV audiences) uses this technique. Results from a small sample audience (say 2,000 viewers) is extrapolated to the total number of TV sets (say 100 million)… Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.40 Nielsen Ratings Example… COMPUTE Problem: describe the population of television shows watched by viewers across the country (population), by examining the results from 2,000 viewers (sample). We take these values and multiply them by N=100 million to estimate that between 9.9 million and 12.7 million viewers are watching the “Tonight Show”. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.41 Identifying Factors… Factors that identify the z-test and interval estimator of p: Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.42 Flowchart of Techniques… Describe a Population Data Type? Interval Nominal Type of descriptive measurement? z test & estimator of p Central Location Variability t test & estimator of u. X2 test & estimator ofs2 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 12.43