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1 Introduction to Estimation Chapter 10

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2 10.1 Introduction Statistical inference is the process by which we acquire information about populations from samples. There are two types of inference: Estimation Hypotheses testing

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3 10.2 Concepts of Estimation The objective of estimation is to determine the value of a population parameter on the basis of a sample statistic. There are two types of estimators: Point Estimator Interval estimator

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4 Point Estimator A point estimator draws inference about a population by estimating the value of an unknown parameter using a single value or point.

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5 Population distribution Point Estimator Parameter ? Sampling distribution A point estimator draws inference about a population by estimating the value of an unknown parameter using a single value or point. Point estimator

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6 An interval estimator draws inferences about a population by estimating the value of an unknown parameter using an interval. Interval estimator Population distribution Sample distribution Parameter Interval Estimator

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7 Selecting the right sample statistic to estimate a parameter value depends on the characteristics of the statistic. Estimator’s Characteristics Estimator’s desirable characteristics: Unbiasedness: An unbiased estimator is one whose expected value is equal to the parameter it estimates. Consistency: An unbiased estimator is said to be consistent if the difference between the estimator and the parameter grows smaller as the sample size increases. Relative efficiency: For two unbiased estimators, the one with a smaller variance is said to be relatively efficient.

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8 10.3 Estimating the Population Mean when the Population Variance is Known How is an interval estimator produced from a sampling distribution? A sample of size n is drawn from the population, and its mean is calculated. By the central limit theorem is normally distributed (or approximately normally distributed.), thus…

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9 We have established before that 10.3 Estimating the Population Mean when the Population Variance is Known

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10 This leads to the following equivalent statement The Confidence Interval for ( is known) The confidence interval

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11 Interpreting the Confidence Interval for 1 – of all the values of obtained in repeated sampling from a given distribution, construct an interval that includes (covers) the expected value of the population. 1 – of all the values of obtained in repeated sampling from a given distribution, construct an interval that includes (covers) the expected value of the population.

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12 Lower confidence limit Upper confidence limit 1 - Confidence level Graphical Demonstration of the Confidence Interval for

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13 The Confidence Interval for ( is known) Four commonly used confidence levels z

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14 Example: Estimate the mean value of the distribution resulting from the throw of a fair die. It is known that = 1.71. Use a 90% confidence level, and 100 repeated throws of the die Solution: The confidence interval is The Confidence Interval for ( is known) The mean values obtained in repeated draws of samples of size 100 result in interval estimators of the form [sample mean -.28, Sample mean +.28], 90% of which cover the real mean of the distribution.

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15 The Confidence Interval for ( is known) Recalculate the confidence interval for 95% confidence level. Solution:.95.90

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16 The Confidence Interval for ( is known) The width of the 90% confidence interval = 2(.28) =.56 The width of the 95% confidence interval = 2(.34) =.68 The width of the 90% confidence interval = 2(.28) =.56 The width of the 95% confidence interval = 2(.34) =.68 Because the 95% confidence interval is wider, it is more likely to include the value of

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17 Example 10.1 Doll Computer Company delivers computers directly to its customers who order via the Internet. To reduce inventory costs in its warehouses Doll employs an inventory model, that requires the estimate of the mean demand during lead time. It is found that lead time demand is normally distributed with a standard deviation of 75 computers per lead time. Estimate the lead time demand with 95% confidence. The Confidence Interval for ( is known)

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18 Example 10.1 – Solution The parameter to be estimated is the mean demand during lead time. We need to compute the interval estimation for From the data provided in file Xm10-01, the sample mean is The Confidence Interval for ( is known) Since 1 - =.95, =.05. Thus /2 =.025. Z.025 = 1.96

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19 Using Excel Tools > Data Analysis Plus > Z Estimate: Mean The Confidence Interval for ( is known)

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20 Wide interval estimator provides little information. Where is ?????????????????????????????? Information and the Width of the Interval

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21 Here is a much narrower interval. If the confidence level remains unchanged, the narrower interval provides more meaningful information. Here is a much narrower interval. If the confidence level remains unchanged, the narrower interval provides more meaningful information. Wide interval estimator provides little information. Where is Ahaaa! Information and the Width of the Interval

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22 The width of the confidence interval is affected by the population standard deviation ( ) the confidence level (1- ) the sample size (n). The Width of the Confidence Interval

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23 90% Confidence level To maintain a certain level of confidence, a larger standard deviation requires a larger confidence interval. /2 =.05 Suppose the standard deviation has increased by 50%. The Affects of on the interval width

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24 /2 = 2.5% /2 = 5% Confidence level 90% 95% Let us increase the confidence level from 90% to 95%. Larger confidence level produces a wider confidence interval The Affects of Changing the Confidence Level

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25 90% Confidence level Increasing the sample size decreases the width of the confidence interval while the confidence level can remain unchanged. The Affects of Changing the Sample Size

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26 10.4 Selecting the Sample size We can control the width of the confidence interval by changing the sample size. Thus, we determine the interval width first, and derive the required sample size. The phrase “estimate the mean to within W units”, translates to an interval estimate of the form

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27 The required sample size to estimate the mean is Click to see how the formula is developed. 10.4 Selecting the Sample size Here

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28 Example 10.2 To estimate the amount of lumber that can be harvested in a tract of land, the mean diameter of trees in the tract must be estimated to within one inch with 99% confidence. What sample size should be taken? Assume that diameters are normally distributed with = 6 inches. Selecting the Sample size

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29 Solution The estimate accuracy is +/-1 inch. That is w = 1. The confidence level 99% leads to =.01, thus z /2 = z.005 = 2.575. We compute If the standard deviation is really 6 inches, the interval resulting from the random sampling will be of the form. If the standard deviation is greater than 6 inches the actual interval will be wider than +/-1. If the standard deviation is really 6 inches, the interval resulting from the random sampling will be of the form. If the standard deviation is greater than 6 inches the actual interval will be wider than +/-1. Selecting the Sample size

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