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Estimation Chapter 7

Estimation Chapter 7. “ Farewell! Thou are too dear for my possessing And like enough thou know’st thy estimate.” William Shakespeare, Sonnet 87. Topics and Goals for Chapter 7. Unbiased point estimators for the population mean: sample mean sample median sample trimmed mean

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Estimation Chapter 7

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  1. EstimationChapter 7 “Farewell! Thou are too dear for my possessing And like enough thou know’st thy estimate.” William Shakespeare, Sonnet 87 MGMT 242

  2. Topics and Goals for Chapter 7 • Unbiased point estimators for the population mean: • sample mean • sample median • sample trimmed mean • Interval estimate of a population mean • Confidence interval for a proportion • Sample size and confidence intervals • What to do when the population variance is unknown • Confidence intervals with the Student’s t-distribution • Assumptions for the Student’s t-distribution MGMT 242

  3. Unbiased Estimators of the Population Mean • An estimator of the mean, µhat (µ with a caret over it) is unbiased if E(uhat) = µ, that is if the long run average of µhat equals the population mean. • The sample mean, xbar, is an unbiased estimator of µ; • The sample median, xm, is an unbiased estimator of µ; • The “trimmed sample mean” (doesn’t take top 10%, bottom 10% of values) is an unbiased estimator of µ. • Any linear combination of the sample values, divided by the number of values, is an unbiased estimator of µ (see Problem 7.8, where the middle of the range is used to estimate µ). MGMT 242

  4. Efficient Estimators of the Population Mean • The most efficient estimator of the population mean is that which will give an estimate with the smallest standard deviation. • Example: Problems 7.1, 7.2, 7.8 (Electronic Reserve). MGMT 242

  5. Confidence Interval for Population Mean-I • Suppose we measure a sample mean; how close is this value to the population mean? • If we know the standard deviation for the population, this is a straightforward problem; if we don’t, it’s more complicated--for now we’ll suppose we know the value of , the population standard deviation • The population distribution for the sample means, sample size N, will approach a normal distribution, mean µ, and standard deviation of the mean, mean =  /N, as N gets large (practically, for N greater than about 10 to 20). MGMT 242

  6. Confidence Interval for Population Mean-II • There is a 95% probability that the measured value of the sample mean, xbar, lies within the range µ-1.96 mean to µ + 1.96 mean (See Board demo) • This corresponds to the inequality µ-1.96 mean  xbar  µ +1.96 mean • With a little manipulation the inequality above can be changed to the one for the confidence interval (CI) xbar-1.96 mean  µ  xbar +1.96 mean where mean = /N. • Interpretation: 95% of the trials (in the long run) will give values of xbar within limits.(Concepts example) MGMT 242

  7. Confidence Interval for Population Mean-III • General Case: Confidence Interval (CI) for level (1-)*100 % (e.g.  = 0.05 corresponds to 95% level) • Then the CI (1-) is given by xbar - z (1-/2)mean  µ  xbar + z (1-/2)mean , where mean =  / N and z (1-/2)is the z-score for the (1-/2) centile (see board diagram): MGMT 242

  8. Interpretation of Confidence Interval The diagram to the left is from the “Concepts” StatPlus add-in. µ, the “true” mean salary, equals $5600; The 95% CI for given  and N runs from MGMT 242

  9. Confidence Interval for Proportion-I • General Case: Confidence Interval (CI) for level (1-)*100 % (e.g.  = 0.05 corresponds to 95% level) • Then the CI (1-) is given by p - z (1-/2)p    p + z (1-/2)p, where z (1-/2)is the z-score for the (1-/2) centile,  is the population proportion (proportion yeses in a yes/no questionnaire, proportion test positive in a medical test, etc.), p = x /N is the sample estimate of  (x is the number of successes in a sample size N) and p, the standard deviation of the proportion, is estimated by p = {p(1-p)/N} 1/2 MGMT 242

  10. Confidence Interval for Proportion-II • The CI (1-) for 1-  = 0.95, a 95% CI, is given by p - 1.96 p    p + 1.96 p, p = {p(1-p)/N} 1/2 • Example (Ex. 7.20, text): 84 out of 125 individuals are aware of a certain product; a 95% CI for this proportion is given by p = 84/125 = 0.672; 1.96 p = 1.96 x [p(1-p)/N] = 0.082, so 95% CI is given by 0.672 - 0.082 to 0.672 + 0.082 or (0.590, 0.754) MGMT 242

  11. Sample Size Required for Given CI width • We know that the CI gets smaller as the sample size, N, increases. Suppose we require (at a certain significance level) a specific width, E, for the CI. Then E = 2 z (1-/2)mean and, since mean = /N, we get N = (2 z (1-/2) /E)2 • Example: Exercise 7.25, text: want 95% CI for insurance claims to $50 wide (=E), with  estimated $400; Then N = (2x1.96x400 /50)2 = 984. MGMT 242

  12. Student’s t-Distribution for Unknown  • Sample Standard Deviation, s, used to estimate population standard deviation, , if  unknown • s = { (xi - xbar)2/(N-1)}(1/2) for sample, size N, with mean xbar • Uncertainty in standard deviation (from sample size estimate) is clearly bigger, the smaller the sample size; • Have to account for this uncertainty by use of a new statistic, the “Student’s-t” variable. • t = (x - ) / s, for individual value of sample, or • t= (xbar - ) / SEM, with SEM = s /N, for sample mean. MGMT 242

  13. Student’s t-Distribution--Continued I • The Student’s t-distribution gives the probability of the t-statistic (see previous slide) occurring by chance • The distribution will clearly depend on sample size, N • The larger the sample size, the more nearly the sample standard deviation, s, should approach the value for population standard deviation,  • The effective sample size is the “degrees of freedom” (abbreviated as “df”); df = N-1 • Probability for large t, small N, is lower than for same value of z (see “Concepts” illustration). • ‘ MGMT 242

  14. Student’s-t Distribution • The graph at left compares a Student’s-t distribution (solid blue line) with the standard normal bell curve (dotted red line) for df=2 (N=3). Note that the probability of the Student’s-t is less than that for the z-curve, for statistic values greater than 2, or less than -2. MGMT 242

  15. Student’s-t Distribution--Example • Ex. 7.34, Text. Comparison shopping at 14 New York area department stores to get refrigerator prices yields the following results: $341,347,319,331,326,298,335,351,316,307,335,320,329,346 Find the 95% Confidence Interval (CI) for the price: (From Xcel) xbar = $328.64 SEM= s /N = 15.49 / 14 = $4.14 df = 14 -1 = 13 t13 = 2.160 (from Table 4, text, or Excel) 95% CI: 328.64- 2.160 x 4.14 to 328.64 + 2.160x 4.14 or $319.70 to $337.58 MGMT 242

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