Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 8 Statistical Inference: Confidence Intervals Section 8.1 Point and Interval Estimates of Population Parameters

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 3 Point Estimate and Interval Estimate A point estimate is a single number that is our “best guess” for the parameter. An interval estimate is an interval of numbers within which the parameter value is believed to fall. Figure 8.1 A point estimate predicts a parameter by a single number. An interval estimate is an interval of numbers that are believable values for the parameter. Question: Why is a point estimate alone not sufficiently informative?

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 4 A point estimate doesn’t tell us how close the estimate is likely to be to the parameter. An interval estimate is more useful, it incorporates a margin of error which helps us to gauge the accuracy of the point estimate. Point Estimate and Interval Estimate

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 5 Properties of Point Estimators Property 1: A good estimator has a sampling distribution that is centered at the parameter. An estimator with this property is unbiased.  The sample mean is an unbiased estimator of the population mean.  The sample proportion is an unbiased estimator of the population proportion.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 6 Property 2: A good estimator has a small standard deviation compared to other estimators.  This means it tends to fall closer than other estimates to the parameter.  The sample mean has a smaller standard error than the sample median when estimating the population mean of a normal distribution. Properties of Point Estimators

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 7 Confidence Interval A confidence interval is an interval containing the most believable values for a parameter. The probability that this method produces an interval that contains the parameter is called the confidence level. This is a number chosen to be close to 1, most commonly 0.95.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 8 The Logic behind Constructing a Confidence Interval To construct a confidence interval for a population proportion, start with the sampling distribution of a sample proportion.  Gives the possible values for the sample proportion and their probabilities. The sampling distribution:  Is approximately a normal distribution for large random samples by the CLT.  Has mean equal to the population proportion.  Has standard deviation called the standard error.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 9 Fact: Approximately 95% of a normal distribution falls within 2 standard deviations of the mean. With probability 0.95, the sample proportion falls within about 1.96 standard errors of the population proportion. The distance of 1.96 standard errors is the margin of error in calculating a 95% confidence interval for the population proportion. The Logic behind Constructing a Confidence Interval

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 10 Margin of Error The margin of error measures how accurate the point estimate is likely to be in estimating a parameter. It is a multiple of the standard error of the sampling distribution when the sampling distribution is a normal distribution. The distance of 1.96 standard errors in the margin of error for a 95% confidence interval for a parameter from a normal distribution.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 11 Example: The GSS asked 1805 respondents whether they agreed with the statement “It is more important for a wife to help her husband’s career than to have one herself”. 19% agreed. Assuming the standard error is 0.01, calculate a 95% confidence interval for the population proportion who agreed with the statement.  Margin of error =   We predict that the population proportion who would agree is somewhere between 0.17 and Example: Confidence Interval for a Proportion: A Wife's Career

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 8 Statistical Inference: Confidence Intervals Section 8.2 Constructing a Confidence Interval to Estimate a Population Proportion

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 13 Finding the 95% Confidence Interval for a Population Proportion  We symbolize a population proportion by p.  The point estimate of the population proportion is the sample proportion.  We symbolize the sample proportion by, called “p-hat”.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 14 A 95% confidence interval uses a margin of error = 1.96(standard deviation) CI = [point estimate margin of error] = Finding the 95% Confidence Interval for a Population Proportion

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 15 The exact standard deviation of a sample proportion equals: This formula depends on the unknown population proportion, p. In practice, we don’t know p, and we need to estimate the standard error as Finding the 95% Confidence Interval for a Population Proportion

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 16 A 95% confidence interval for a population proportion p is: where denotes the sample proportion based on n observations. Finding the 95% Confidence Interval for a Population Proportion

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 17 Example: Paying Higher Prices to Protect the Environment In 2010, the GSS asked subjects if they would be willing to pay much higher prices to protect the environment. Of n = 1,361 respondents, 637 were willing to do so. Find and interpret a 95% confidence interval for the population proportion of adult Americans willing to do so at the time of the survey.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 18 The sample proportion is The standard error of the sample proportion = Using this se, a 95% confidence interval for the population proportion is Example: Paying Higher Prices to Protect the Environment

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 19 Sample Size Needed for Validity of Confidence Interval for a Proportion For the 95% confidence interval for a proportion p to be valid, you should have at least 15 successes and 15 failures: At the end of Section 8.4, we’ll learn about a simple adjustment to this confidence interval formula that you should use when this guideline is violated.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 20 “95% confidence“ means that there’s a 95% chance that a sample proportion value occurs such that the confidence interval contains the unknown value of the population proportion, p. With probability 0.05, however, the method produces a confidence interval that misses p, and the inference is incorrect. How Can We Use Confidence Levels Other than 95%?

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 21 In practice, the confidence level 0.95 is the most common choice. But, some applications require greater (or less) confidence. To increase the chance of a correct inference, we can use a larger confidence level, such as Figure 8.4 A 99% Confidence Interval Is Wider Than a 95% Confidence Interval. Question: If you want greater confidence, why would you expect a wider interval? How Can We Use Confidence Levels Other than 95%?

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 22 In using confidence intervals, we must compromise between the desired margin of error and the desired confidence of a correct inference. As the desired confidence level increases, the margin of error gets larger. How Can We Use Confidence Levels Other than 95%?

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 23 Example: A Husband Choosing Not to Have Children A recent GSS asked “If the wife in a family wants children, but the husband decides that he does not want any children, is it all right for the husband to refuse to have children? Of 598 respondents, 366 said yes, and 232 said no. Calculate the 99% confidence interval.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 24 What is the Error Probability for the Confidence Interval Method? The general formula for the confidence interval for a population proportion is: Sample Proportion (z-score)(std. error) which in symbols is Table 8.2 Table 8.2 z -Scores for the Most Common Confidence Levels. The large- sample confidence interval for the population proportion is

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 25 Summary: Confidence Interval for a Population Proportion, p A confidence interval for a population proportion p is: Assumptions  Data obtained by randomization  A large enough sample size n so that the number of success,, and the number of failures,, are both at least 15.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 26 Effects of Confidence Level and Sample Size on Margin of Error The margin of error for a confidence interval:  Increases as the confidence level increases  Decreases as the sample size increases For instance, a 99% confidence interval is wider than a 95% confidence interval, and a confidence interval with 200 observations is narrower than one with 100 observations at the same confidence level. These properties apply to all confidence intervals, not just the one for the population proportion.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 27 Interpretation of the Confidence Level So what does it mean to say that we have “95% confidence”? The meaning refers to a long-run interpretation—how the method performs when used over and over with many different random samples. If we used the 95% confidence interval method over time to estimate many population proportions, then in the long run about 95% of those intervals would give correct results, containing the population proportion.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 8 Statistical Inference: Confidence Intervals Section 8.3 Constructing a Confidence Interval to Estimate a Population Mean

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 29 The confidence interval again has the form Point estimate margin of error The sample mean is the point estimate of the population mean. The exact standard error of the sample mean is In practice, we estimate σ by the sample standard deviation, s, so How to Construct a Confidence Interval for a Population Mean

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 30 For large n… from any population and also For small n from an underlying population that is normal… The confidence interval for the population mean is: How to Construct a Confidence Interval for a Population Mean

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 31 In practice, we don’t know the population standard deviation  Substituting the sample standard deviation s for to get introduces extra error. To account for this increased error, we must replace the z-score by a slightly larger score, called a t –score. The confidence interval is then a bit wider. This distribution is called the t distribution. How to Construct a Confidence Interval for a Population Mean

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 32 Summary: Properties of the t-Distribution  The t-distribution is bell shaped and symmetric about 0.  The probabilities depend on the degrees of freedom,.  The t-distribution has thicker tails than the standard normal distribution, i.e., it is more spread out.  A t -score multiplied by the standard error gives the margin of error for a confidence interval for the mean.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 33 t-Distribution Figure 8.7 The t Distribution Relative to the Standard Normal Distribution. The t distribution gets closer to the standard normal as the degrees of freedom ( df ) increase. The two are practically identical when. Question: Can you find z -scores (such as 1.96) for a normal distribution on the t table (Table B)?

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 34 Table 8.3 Part of Table B Displaying t-Scores. The scores have right-tail probabilities of 0.100, 0.050, 0.025, 0.010, 0.005, and When, and is the t -score with right-tail probability = and two-tail probability = It is used in a 95% confidence interval,. t-Distribution

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 35 Figure 8.8 The t Distribution with df = 6. 95% of the distribution falls between and These t -scores are used with a 95% confidence interval when n = 7. Question: Which t -scores with df = 6 contain the middle 99% of a t distribution (for a 99% confidence interval)? t-Distribution

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 36 Using the t Distribution to Construct a Confidence Interval for a Mean Summary: 95% Confidence Interval for a Population Mean When the standard deviation of the population is unknown, a 95% confidence interval for the population mean is: To use this method, you need:  Data obtained by randomization  An approximately normal population distribution

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 37 Example: Stock Market Activity on Different Days of the Week Companies often finance their daily operations by selling shares of common stock. One statistic monitored closely is the numbers of shares, or trading volume, that are bought and sold each day. In this example, we compare the number of shares traded of General Electric stock on Mondays and Fridays during February through April of 2011.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 38 Figure 8.9 MINITAB Dot Plot for General Electric Trading Volume. Data are shown for 11 Mondays and 12 Fridays during February, March, and April Question: What does this plot tell you about how the trading volumes compare for these two days of the week? Example: Stock Market Activity on Different Days of the Week

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 39 Let denote the population mean for Monday volume. The estimate of is the sample mean of 45, 43, 43, 66, 91, 53, 35, 45, 29, 64, and 56, which is. The sample standard deviation of. The standard error of the sample mean is Example: Stock Market Activity on Different Days of the Week

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 40 Since, the degrees of freedom are. For a 95% confidence interval, from Table 8.3 we use. The 95% confidence interval is With 95% confidence, the range of believable values for the mean Monday volume is million shares to million shares. Example: Stock Market Activity on Different Days of the Week

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 41 Finding a t Confidence Interval for Other Confidence Levels The 95% confidence interval uses since 95% of the probability falls between and. For 99% confidence, the error probability is 0.01 with in each tail and the appropriate t-score is. To get other confidence intervals use the appropriate t-value from Table B.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 42 How Do We Find a t- Confidence Interval for Other Confidence Levels? Table 8.5 Part of Table B Displaying t -Scores for Large df Values. The z -score of 1.96 is the t -score with right-tail probability of and.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 43 If the Population is Not Normal, is the Method “Robust”? A basic assumption of the confidence interval using the t-distribution is that the population distribution is normal. Many variables have distributions that are far from normal. We say the t-distribution is a robust method in terms of the normality assumption.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 44 How problematic is it if we use the t- confidence interval even if the population distribution is not normal? For large random samples, it’s not problematic because of the Central Limit Theorem. What if n is small? Confidence intervals using t-scores usually work quite well except for when extreme outliers are present. If the Population is Not Normal, is the Method “Robust”?

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 45 The Standard Normal Distribution is the t-Distribution with Table 8.5 Part of Table B Displaying t-Scores for Large df Values. The z-score of 1.96 is the t-score with right-tail probability of and.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 8 Statistical Inference: Confidence Intervals Section 8.4 Choosing the Sample Size for a Study

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 47 Sample Size for Estimating a Population Proportion To determine the sample size, 1.first, we must decide on the desired margin of error. 2.second, we must choose the confidence level for achieving that margin of error. In practice, 95% confidence intervals are most common.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 48 The random sample size n for which a confidence interval for a population proportion p has margin of error m (such as m = 0.04) is The z-score is based on the confidence level, such as z=1.96 for 95% confidence. You either guess the value you’d get for the sample proportion based on other information or take the safe approach of setting. Summary: Sample Size for Estimating a Population Proportion

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 49 Example: Sample Size For Exit Poll An election is expected to be close. Pollsters planning an exit poll decide that a margin of error of 0.04 is too large. How large should the sample size be for the margin of error of a 95% confidence interval to equal 0.02?

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 50 The z-score is based on the confidence level, such as for 95% confidence. The 95% confidence interval for a population proportion p is:. The required sample size is: Example: Sample Size For Exit Poll

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 51 Choosing the Sample Size for Estimating a Population Mean The random sample size n for which a confidence interval for a population mean has margin of error approximately equal to m is where the z-score is based on the confidence level, such as z=1.96 for 95% confidence.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 52 In practice, you don’t know the value of the standard deviation,  You must substitute an educated guess for  Sometimes you can use the sample standard deviation from a similar study. When no prior information is known, a crude estimate that can be used is to divide the estimated range of the data by 6 since for a bell-shaped distribution we expect almost all of the data to fall within 3 standard deviations of the mean. Choosing the Sample Size for Estimating a Population Mean

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 53 A social scientist plans a study of adult South Africans to investigate educational attainment in the black community. How large a sample size is needed so that a 95% confidence interval for the mean number of years of education has margin of error equal to 1 year? Assume that the education values will fall within a range of 0 to 18 years. Example: Estimating Mean Education in South Africa

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 54 Other Factors That Affect the Choice of the Sample Size  The first is the desired precision, as measured by the margin of error, m.  The second is the confidence level.  The third factor is the variability in the data.  The fourth factor is cost.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 55 What if You Have to Use a Small n? The t methods for a mean are valid for any n. However, you need to be extra cautious to look for extreme outliers or great departures from the normal population assumption. In the case of the confidence interval for a population proportion, the method works poorly for small samples because the CLT no longer holds.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 56 Confidence Interval for a Proportion with Small Samples If a random sample does not have at least 15 successes and 15 failures, the confidence interval formula is still valid if we use it after adding 2 to the original number of successes and 2 to the original number of failures. This results in adding 4 to the sample size n.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 8 Statistical Inference: Confidence Intervals Section 8.5 Using Computers to Make New Estimation Methods Possible

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 58 The Bootstrap: Using Simulation to Construct a Confidence Interval When it is difficult to derive a standard error or a confidence interval formula that works well you can use simulation. The bootstrap is a simulation method that resamples from the observed data. It treats the data distribution as if it were the population distribution.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 59 To use the bootstrap method:  You resample, with replacement, n observations from the data distribution.  For the new sample of size n, construct the point estimate of the parameter of interest  Repeat process a very large number of times (e.g., selecting 10,000 separate samples of size n and calculating the 10,000 corresponding parameter estimates). The Bootstrap: Using Simulation to Construct a Confidence Interval

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 60 To investigate how much the weight readings tend to vary, he weighed himself ten times, taking a 30-second break after each trial to allow the scale to reset. Suppose your data set includes the following: 160.2, 160.8, 161.4, 162.0, , 162.0, 161.8, 161.6, This data has a mean of and standard deviation of Use the bootstrap method to find a 95% confidence interval for the population standard deviation. Example: How Variable Are Your Weight Readings on a Scale?

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 61 Re-sample with replacement from this sample of size 10 and compute the standard deviation of the new sample. The Bootstrap: Using Simulation to Construct a Confidence Interval Figure 8.11 A Bootstrap Frequency Distribution of Standard Deviation Values. These were obtained by taking 100,000 samples of size 10 each from the sample data distribution. Question: What is the practical reason for using the bootstrap method?

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 62 Now, identify the middle 95% of these 100,000 sample standard deviations (take the 2.5th and 97.5th percentiles). For this example, these percentiles are 0.26 and This is our 95% bootstrap confidence interval for the population standard deviation. In Summary: A typical deviation of a weight reading from the mean might be rather large, nearly a pound. The Bootstrap: Using Simulation to Construct a Confidence Interval