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Chapter 12 Confidence Intervals for Means Copyright © 2012 Pearson Education. All rights reserved. Copyright © 2012 Education Pearson Education. All rights reserved. © 2010 Pearson 12-1 12.1 The Sampling Distribution for the Mean We found confidence intervals for proportions to be pˆ ME where the ME was equal to a critical value, z*,times SE( p̂ ). Our confidence intervals for means will be y ME where the ME will be a critical value times SE( y ). Copyright © 2012 Pearson Education. All rights reserved. 12-2 12.1 The Sampling Distribution for the Mean The standard deviation of the sample mean is given below. SD( y ) n So we need know the true value of the population standard deviation σ. Instead of σ, we will use s, the sample standard deviation from the data. We get the following formula for standard error. s SE ( y ) n Copyright © 2012 Pearson Education. All rights reserved. 12-3 12.1 The Sampling Distribution for the Mean Gosset’s t William S. Gosset discovered that when he used the standard error s / n , the shape of the curve was no longer Normal. He called the new model the Student’s t, which is a model that is always bell-shaped, but the details change with the sample sizes. The Student’s t-models form a family of related distributions depending on a parameter known as degrees of freedom. Copyright © 2012 Pearson Education. All rights reserved. 12-4 12.1 The Sampling Distribution for the Mean Student’s t-models are unimodal, symmetric, and bell-shaped, just like the Normal model. But t-models (solid curve below) with only a few degrees of freedom have a narrower peak than the Normal model (dashed curve below) and have much fatter tails. As the degrees of freedom increase, the t-models look more and more like the Normal model. Copyright © 2012 Pearson Education. All rights reserved. 12-5 12.1 The Sampling Distribution for the Mean Example: Find the standard error Data from a survey of 25 randomly selected customers found a mean age of 31.84 years and the standard deviation was 9.84 years. What is the standard error of the mean? How would the standard error change if the sample size had been 100 instead of 25? (Assume that s = 9.84 years.) Copyright © 2012 Pearson Education. All rights reserved. 12-6 12.1 The Sampling Distribution for the Mean Example: Find the standard error Data from a survey of 25 randomly selected customers found a mean age of 31.84 years and the standard deviation was 9.84 years. What is the standard error of the mean? s 9.84 SE ( y ) 1.968 n 25 How would the standard error change if the sample size had been 100 instead of 25? (Assume that s =9.84 years.) s 9.84 SE ( y ) 0.984, which is half as large n 100 Copyright © 2012 Pearson Education. All rights reserved. 12-7 12.2 A Confidence Interval for Means Copyright © 2012 Pearson Education. All rights reserved. 12-8 12.2 A Confidence Interval for Means Copyright © 2012 Pearson Education. All rights reserved. 12-9 12.2 A Confidence Interval for Means Finding t-Values The Student’s t-model is different for each value of degrees of freedom. Typically we limit ourselves to 80%, 90%, 95%, and 99% confidence levels. We can use technology to give critical values for any number of degrees of freedom and for any confidence levels we need. More precision won’t necessarily help make good business decisions. Copyright © 2012 Pearson Education. All rights reserved. 12-10 12.2 A Confidence Interval for Means Finding t-Values A typical t-table is shown here. The table shows the critical values for varying degrees of freedom, df, and for varying confidence intervals. Since the t-models get closer to the normal as df increases, the final row has critical values from the Normal model and is labeled “∞”. Copyright © 2012 Pearson Education. All rights reserved. 12-11 12.2 A Confidence Interval for Means Finding t-Values For example, suppose we’ve performed a one-sample t-test with 19 df and a critical value of 1.639, and we want the upper tail P-value. From the table, we see that 1.639 falls between 1.328 and 1.729. All we can say is that the P-value lies between P-values of these two critical values, so 0.05 < P < 0.10. Copyright © 2012 Pearson Education. All rights reserved. 12-12 12.2 A Confidence Interval for Means Example: Construct a Confidence Interval Data from a survey of 25 randomly selected customers found a mean age of 31.84 years and the standard deviation was 9.84 years. Construct a 95% confidence interval for the mean. Interpret the interval. Copyright © 2012 Pearson Education. All rights reserved. 12-13 12.2 A Confidence Interval for Means Example: Construct a Confidence Interval Data from a survey of 25 randomly selected customers found a mean age of 31.84 years and the standard deviation was 9.84 years. Construct a 95% confidence interval for the mean. y t * SE ( y ) 31.84 (2.064)(1.968) 31.84 4.062 (27.78, 35.90) Interpret the interval. We’re 95% confident the true mean age of all customers is between 27.78 and 35.90 years. Copyright © 2012 Pearson Education. All rights reserved. 12-14 12.3 Assumptions and Conditions Independence Assumption There is no way to check independence of the data, but we should think about whether the assumption is reasonable. Randomization Condition: The data arise from a random sample or suitably randomized experiment. 10% Condition: The sample size should be no more than 10% of the population. For means our samples generally are, so this condition will only be a problem if our population is small. Copyright © 2012 Pearson Education. All rights reserved. 12-15 12.3 Assumptions and Conditions Normal Population Assumption Student’s t-models won’t work for data that are badly skewed. We assume the data come from a population that follows a Normal model. Data being Normal is idealized, so we have a “nearly normal” condition we can check. Nearly Normal Condition: The data come from a distribution that is unimodal and symmetric. This can be checked by making a histogram. Copyright © 2012 Pearson Education. All rights reserved. 12-16 12.3 Assumptions and Conditions Normal Population Assumption Nearly Normal Condition: •For very small samples (n < 15), the data should follow a Normal model very closely. If there are outliers or strong skewness, t methods shouldn’t be used. •For moderate sample sizes (n between 15 and 40), t methods will work well as long as the data are unimodal and reasonably symmetric. •For sample sizes larger than 40 or 50, t methods are safe to use unless the data are extremely skewed. If outliers are present, analyses can be performed twice, with the outliers and without. Copyright © 2012 Pearson Education. All rights reserved. 12-17 12.3 Assumptions and Conditions Normal Population Assumption In business, the mean is often the value of consequence. Even when we must sample from a very skewed distribution, the Central Limit Theorem tells us that the sampling distribution of our sample mean will be close to Normal. We can use Student’s t methods without much worry as long as the sample size is large enough. Copyright © 2012 Pearson Education. All rights reserved. 12-18 12.3 Assumptions and Conditions Normal Population Assumption The histogram below displays the compensation of 500 CEO’s. We see an extremely skewed distribution. Copyright © 2012 Pearson Education. All rights reserved. 12-19 12.3 Assumptions and Conditions Normal Population Assumption Taking many samples of 100 CEO’s, we obtain the nearly Normal plot below for the sample means. Copyright © 2012 Pearson Education. All rights reserved. 12-20 12.2 A Confidence Interval for Means Example: Check Assumptions and Conditions Data from a survey of 25 randomly selected customers found a mean age of 31.84 years and the standard deviation was 9.84 years. A 95% confidence interval for the mean is (27.78, 25.90). Check conditions for this interval. Copyright © 2012 Pearson Education. All rights reserved. 12-21 12.2 A Confidence Interval for Means Example: Check Assumptions and Conditions Data from a survey of 25 randomly selected customers found a mean age of 31.84 years and the standard deviation was 9.84 years. A 95% confidence interval for the mean is (27.78, 25.90). Check conditions for this interval. Independence: Data were gathered from a random sample and should be independent. 10% Condition: These customers are fewer than 10% of the customer population. Nearly Normal: The histogram is unimodal and approximately symmetric. Copyright © 2012 Pearson Education. All rights reserved. 12-22 12.4 Cautions About Interpreting Confidence Intervals Confidence intervals for means offer new, tempting, wrong interpretations. Here are some ways to keep from going astray: • Don’t say, “95% of all the policies sold by this sales rep have profits between $942.48 and $1935.32.” The confidence interval is about the mean, not about the measurements of individual policies. • Don’t say, “We are 95% confident that a randomly selected policy will have a net profit between $942.48 and $1935.32.” This false interpretation is also about individual policies rather than about the mean of the policies. Copyright © 2012 Pearson Education. All rights reserved. 12-23 12.4 Cautions About Interpreting Confidence Intervals • Don’t say, “The mean profit is $1438.90 95% of the time.” That’s about means, but still wrong. It implies that the true mean varies, when in fact it is the confidence interval that would have been different had we gotten a different sample. • Don’t say, “95% of all samples will have mean profits between $942.48 and $1935.32.” That statement suggests that this interval somehow sets a standard for every other interval. In fact, this interval is no more (or less) likely to be correct than any other. Copyright © 2012 Pearson Education. All rights reserved. 12-24 12.4 Cautions About Interpreting Confidence Intervals • If the confidence interval is for the mean, then do not interpret the results in terms of individuals. • Don’t forget that the true mean does not vary, but the confidence interval will vary based on the sample. • Don’t suggest that a particular confidence interval somehow sets the standard for every other interval. Copyright © 2012 Pearson Education. All rights reserved. 12-25 12.5 Sample Size We know that a larger sample will almost always give better results, but more data costs money, effort, and time. We know how to find the margin of error for the mean. ME t n*1 SE ( y ) We also know how to find the standard error for the mean. s SE ( y ) n We can determine the sample size by solving this equation for n. Copyright © 2012 Pearson Education. All rights reserved. 12-26 12.5 Sample Size The equation has several values that we don’t know. We need to know s, but we won’t know s until we collect some data, and we want to calculate the sample size before we collect the data. Often a “good guess” for s is sufficient. If we have no idea what the value for s is, we could run a small pilot study to get some feeling for the size of the standard deviation. Copyright © 2012 Pearson Education. All rights reserved. 12-27 12.5 Sample Size Without knowing n, we don’t know the degrees of freedom, and we can’t find the critical value, t n*1. One common approach is to use the corresponding z* value from the Normal model. For example, if you’ve chosen a 95% confidence interval, then use 1.96 (or 2). If your estimated sample size is 60 or more, your z* was probably a good guess. If it’s smaller, use z* at first, finding n, and then replacing z* with the corresponding t n*1 and calculating the sample size once more. Copyright © 2012 Pearson Education. All rights reserved. 12-28 12.5 Sample Size Sample size calculations are never exact. The margin of error you find after collecting the data won’t match exactly the one you used to find n. Before you collect data, it’s always a good idea to know whether the sample size is large enough to give you a good chance of being able to tell you what you want to know. Copyright © 2012 Pearson Education. All rights reserved. 12-29 12.5 Sample Size Example: Check Assumptions and Conditions Data from a survey of 25 randomly selected customers found a mean age of 31.84 years and the standard deviation was 9.84 years. A 95% confidence interval for the mean is (27.78, 25.90). How large a sample is needed to cut the margin of error in half? How large a sample is needed to cut the margin of error by a factor of 10? Copyright © 2012 Pearson Education. All rights reserved. 12-30 12.5 Sample Size Example: Check Assumptions and Conditions Data from a survey of 25 randomly selected customers found a mean age of 31.84 years and the standard deviation was 9.84 years. A 95% confidence interval for the mean is (27.78, 25.90). How large a sample is needed to cut the margin of error in half? Four times as large, or n = 100. How large a sample is needed to cut the margin of error by a factor of 10? one hundred times as large, or n = 2500 Copyright © 2012 Pearson Education. All rights reserved. 12-31 12.6 Degrees of Freedom – Why n – 1? If we know the true population mean, μ, we can find the standard deviation using n instead of n – 1. s 2 ( y ) n We use yinstead of μ. For any sample, will y be as close to the data values as possible, and the population mean μ will be farther away. ( y y ) instead of ( y ) in the equation to If we use calculate s, our standard deviation will be too small. 2 2 We compensate for this by dividing by n – 1 instead of by n. Copyright © 2012 Pearson Education. All rights reserved. 12-32 First, you must decide when to use Student’s t methods. •Don’t confuse proportions and means. Use Normal models with proportions. Use Student’s t methods with means. •Be careful of interpretation when confidence intervals overlap. Don’t assume that the means of overlapping confidence intervals are equal. Copyright © 2012 Pearson Education. All rights reserved. 12-33 Student’s t methods work only when the Normal Population Assumption is true. •Beware of multimodality. If you see this, try to separate the data into groups. •Beware of skewed data. If it is skewed, try re-expressing the data •Investigate outliers. If they are clearly in error, remove them. If they can’t be removed, you might run the analysis with and without the outlier. Copyright © 2012 Pearson Education. All rights reserved. 12-34 The are other risks when doing inferences about means. • Watch out for bias. Measurements can be biased. • Make sure data are independent. Consider whether there are likely violations of independence in the data collection methods. Copyright © 2012 Pearson Education. All rights reserved. 12-35 What Have We Learned? Know the sampling distribution of the mean. • To apply the Central Limit Theorem for the mean in practical applications, we must estimate the standard deviation. This standard error is s SE ( y ) n • When we use the SE, the sampling distribution that allows for the additional uncertainty is Student’s t. Copyright © 2012 Pearson Education. All rights reserved. 12-36 What Have We Learned? Construct confidence intervals for the true mean, µ. • A confidence interval for the mean has the form y ME • The Margin of Error is ME = t*df SE( y ). Find values by technology or from tables. • When constructing confidence intervals for means, the correct degrees of freedom is n – 1. Check the Assumptions and Conditions before using any sampling distribution for inference. Write clear summaries to interpret a confidence interval. Copyright © 2012 Pearson Education. All rights reserved. 12-37