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1 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 9 Estimation Using a Single Sample

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2 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. A point estimate of a population characteristic is a single number that is based on sample data and represents a plausible value of the characteristic. Point Estimation

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3 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example A sample of 200 students at a large university is selected to estimate the proportion of students that wear contact lens. In this sample 47 wore contact lens. Let = the true proportion of all students at this university who wear contact lens. Consider “success” being a student who wears contact lens.

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4 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example A sample of weights of 34 male freshman students was obtained. 185161174175202178 202139177170151176 197214283184189168 188170207180167177 166231176184179155 148180194176 If one wanted to estimate the true mean of all male freshman students, you might use the sample mean as a point estimate for the true mean.

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5 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example After looking at a histogram and boxplot of the data (below) you might notice that the data seems reasonably symmetric with a outlier, so you might use either the sample median or a sample trimmed mean as a point estimate. 260220180140 Calculated using Minitab

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6 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Bias A statistic with mean value equal to the value of the population characteristic being estimated is said to be an unbiased statistic. A statistic that is not unbiased is said to be biased. Sampling distribution of a unbiased statistic Sampling distribution of a biased statistic Original distribution

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7 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Criteria Given a choice between several unbiased statistics that could be used for estimating a population characteristic, the best statistic to use is the one with the smallest standard deviation. Unbiased sampling distribution with the smallest standard deviation, the Best choice.

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8 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Large-sample Confidence Interval for a Population Proportion A confidence interval for a population characteristic is an interval of plausible values for the characteristic. It is constructed so that, with a chosen degree of confidence, the value of the characteristic will be captured inside the interval.

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9 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Confidence Level The confidence level associated with a confidence interval estimate is the success rate of the method used to construct the interval.

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10 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Recall * n 10 and n(1- ) 10 Specifically when n is large*, the statistic p has a sampling distribution that is approximately normal with mean and standard deviation. For the sampling distribution of p, p = and for large* n The sampling distribution of p is approximately normal.

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11 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Some considerations

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12 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Some considerations This interval can be used as long as np 10 and n(1-p) 10

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13 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Approximate Confidence Interval This gives as the 95% approximate confidence interval estimate for This interval can be used as long as np 10 and n(1-p) 10

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14 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. The 95% Confidence Interval The endpoints of the interval are often abbreviated by where – gives the lower endpoint and + the upper endpoint. These are called the 95% confidence limits for .

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15 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example For a project, a student randomly sampled 182 other students at a large university to determine if the majority of students were in favor of a proposal to build a field house. He found that 75 were in favor of the proposal. Let = the true proportion of students that favor the proposal.

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16 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example - continued So np = 182(0.4121) = 75 >10 and n(1-p)=182(0.5879) = 107 >10 we can use the formulas given on the previous slide to find a 95% confidence interval for . The 95% confidence interval for is (0.341, 0.484).

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17 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. The General Confidence Interval The general formula for a confidence interval for a population proportion when 1. p is the sample proportion from a random sample, and 2. The sample size n is large (np 10 and np(1-p) 10) is given by

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18 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Finding a z Critical Value Finding a z critical value for a 98% confidence interval. Looking up the cumulative area or 0.9900 in the body of the table we find z = 2.33 2.33

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19 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Some Common Critical Values Confidence level z critical value 80%1.282 90%1.645 95%1.960 98%2.326 99%2.576 99.8%3.090 99.9%3.291

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20 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Terminology The standard error of a statistic is the estimated standard deviation of the statistic.

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21 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Terminology The bound on error of estimation, B, associated with a 95% confidence interval is (1.96)·(standard error of the statistic). The bound on error of estimation, B, associated with a confidence interval is (z critical value)·(standard error of the statistic).

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22 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Sample Size The sample size required to estimate a population proportion to within an amount B with 95% confidence is The value of may be estimated by prior information. If no prior information is available, use = 0.5 in the formula to obtain a conservatively large value for n. Generally one rounds the result up to the nearest integer.

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23 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Sample Size Calculation Example If a TV executive would like to find a 95% confidence interval estimate within 0.03 for the proportion of all households that watch NYPD Blue regularly. How large a sample is needed if a prior estimate for was 0.15. A sample of 545 or more would be needed. We have B = 0.03 and the prior estimate of = 0.15

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24 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Sample Size Calculation Example revisited Suppose a TV executive would like to find a 95% confidence interval estimate within 0.03 for the proportion of all households that watch NYPD Blue regularly. How large a sample is needed if we have no reasonable prior estimate for . The required sample size is now 1068. We have B = 0.03 and should use = 0.5 in the formula. Notice, a reasonable ball park estimate for can lower the needed sample size.

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25 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Another Example A college professor wants to estimate the proportion of students at a large university who favor building a field house with a 99% confidence interval accurate to 0.02. If one of his students performed a preliminary study and estimated to be 0.412, how large a sample should he take. The required sample size is 4032. We have B = 0.02, a prior estimate = 0.412 and we should use the z critical value 2.58 (for a 99% confidence interval)

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26 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. One-Sample z Confidence Interval for 2. The sample size n is large (generally n 30), and 3. , the population standard deviation, is known then the general formula for a confidence interval for a population mean is given by

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27 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. One-Sample z Confidence Interval for Notice that this formula works when is known and either 1.n is large (generally n 30) or 2.The population distribution is normal (any sample size. If n is small (generally n < 30) but it is reasonable to believe that the distribution of values in the population is normal, a confidence interval for (when is known) is

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28 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Find a 90% confidence interval estimate for the true mean fills of catsup from this machine. Example A certain filling machine has a true population standard deviation = 0.228 ounces when used to fill catsup bottles. A random sample of 36 “6 ounce” bottles of catsup was selected from the output from this machine and the sample mean was 6.018 ounces.

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29 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example I (continued) The z critical value is 1.645 90% Confidence Interval (5.955, 6.081)

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30 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Unknown - Small Size Samples [All Size Samples] An Irish mathematician/statistician, W. S. Gosset developed the techniques and derived the Student’s t distributions that describe the behavior of.

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31 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. t Distributions If X is a normally distributed random variable, the statistic follows a t distribution with df = n-1 (degrees of freedom).

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32 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. t Distributions This statistic is fairly robust and the results are reasonable for moderate sample sizes (15 and up) if x is just reasonable centrally weighted. It is also quite reasonable for large sample sizes for distributional patterns (of x) that are not extremely skewed.

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33 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. t Distributions

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34 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Notice: As df increase, t distributions approach the standard normal distribution. Since each t distribution would require a table similar to the standard normal table, we usually only create a table of critical values for the t distributions. t Distributions

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35 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.

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36 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. One-Sample t Procedures Suppose that a SRS of size n is drawn from a population having unknown mean . The general confidence limits are and the general confidence interval for is

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37 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Confidence Interval Example Ten randomly selected shut-ins were each asked to list how many hours of television they watched per week. The results are 8266908475 88809411091 Find a 90% confidence interval estimate for the true mean number of hours of television watched per week by shut-ins.

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38 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. We find the critical t value of 1.833 by looking on the t table in the row corresponding to df = 9, in the column with bottom label 90%. Computing the confidence interval for is Confidence Interval Example Calculating the sample mean and standard deviation we have n = 10, = 86, and s = 11.842

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39 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. To calculate the confidence interval, we had to make the assumption that the distribution of weekly viewing times was normally distributed. Consider the normal plot of the 10 data points produced with Minitab provided on the next slide. Confidence Interval Example

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40 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Notice that the normal plot looks reasonably linear so it is reasonable to assume that the number of hours of television watched per week by shut-ins is normally distributed. P-Value: 0.753 A-Squared: 0.226 Anderson-Darling Normality Test Typically if the p-value is more than 0.05 we assume that the distribution is normal Confidence Interval Example

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41 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Note on Robustness of t procedures Generally, the t intervals give reasonably good results for sample sizes larger than 15 if the underlying distribution is reasonably symmetric and centrally weighted. When the sample size exceeds 30, the results of the t interval are a bit more conservative than the corresponding z interval where s is used in place of so the t interval is the one that is used.

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