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MN1025 – Business Statistics 15 Lecture 4—Friday 1/2/2008 CONFIDENCE INTERVALS for the MEAN Reference: Lind et al. , Chapters 9, 10. If you are interested in the theory you could read Chapter 8. 4.1 Confidence intervals We are interested in estimating the average of some quantity. The question might be for example: how much do university graduates earn in their first year, on average? A survey of all graduates would pro- vide an exact figure for this average ( the population mean ), say μ = £ 23125. This number is generally unknown. In practice one takes a sample. The sample mean, for example ¯ x = £ 22950, provides an estimate for the unknown population mean, μ . In addition, from the sample one can construct a confidence interval , which is a range of likely values for the unknown pop- ulation mean. In our example, the 95% confidence interval is 22950 ± 550, i.e., £ 22950 plus or minus £ 550. Since 22950 - 550 = 22400 and 22950 + 550 = 23500, this corresponds to a range of values from £ 22400 to £ 23500. Another way of writing this range is (22400, 23500). In general, a confidence interval can be written in the form ¯ x ± d (in our example, ¯ x = 22950 and d = 550), where d is half the width of the interval and depends on the sample size and the sample standard deviation. The alternative form for the confidence interval is ( a, b ), (in our example, a = 22400 and b = 23500), where a is the lower limit, a = ¯ x - d , and b is the upper limit, b = ¯ x + d . 4.2 Sampling For his research project, a graduate student at the University of Poppleton wants to find out the av- erage IQ of Poppleton’s 12050 undergraduates. He tests 10 randomly selected students and obtains: Data Display: sample1 112 104 128 98 123 115 112 130 94 128 Descriptive Statistics: sample1 Variable N Mean StDev Minimum Maximum sample1 10 114.40 12.88 94.00 130.00 These data give him the average IQ of the 10 stu- dents in his sample. He uses the fact that the sample mean, ¯ x , is the best estimate for the unknown popu- lation mean, μ , and concludes that the average IQ of Poppleton undergraduates is approximately 114.4. Of course, if he had taken a different sample, he would have obtained a different estimate. Here are the descriptive statistics for 10 randomly selected samples of 10 undergraduates each. Descriptive Statistics: sample1, sample2, ... Variable N Mean StDev Minimum Maximum sample1 10 114.40 12.88 94.00 130.00 sample2 10 112.00 19.70 79.00 136.00 sample3 10 122.40 15.62 97.00 151.00 sample4 10 111.60 18.09 82.00 138.00 sample5 10 106.90 12.08 87.00 122.00 sample6 10 120.00 22.83 88.00 158.00 sample7 10 112.60 10.75 89.00 123.00 sample8 10 107.70 14.47 84.00 132.00 sample9 10 118.70 22.25 82.00 157.00 sample10 10 115.10 19.10 88.00 139.00 The sample means, ¯ x , and thus the estimates for the population mean, μ , vary widely, as can also be seen in the histogram of the 10 sample means below.

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