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chapter five

chapter five. Statistical Inference: Estimation and Hypothesis Testing. Statistical Inference. Drawing conclusions about a population based on a random sample from that population

linda-hicks
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chapter five

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  1. chapter five Statistical Inference: Estimation and Hypothesis Testing

  2. Statistical Inference • Drawing conclusions about a population based on a random sample from that population • Consider Table D-1(5-1): Can we use the average P/E ratio of the 28 companies shown as an estimate of the average P/E ratio of the 3000 or so stocks on the NYSE? • If X = P/E ratio of a stock and Xbar the average P/E of the 28 stocks, can we tell what the expected P/E ratio, E(X), is for the whole NYSE?

  3. Table D-1 (5-1) Price to Earnings (P/E) ratios of 28 companies on the New York Stock Exchange (NYSE).

  4. Estimation Is the First Step • The average P/E from a random sample of stocks, Xbar, is an estimator (or sample statistic) of the population average P/E, E(X), called the population parameter. • The mean and variance are parameters of the normal distribution • A particular value of an estimator is called an estimate, say Xbar = 23. • Estimation is the first step in statistical inference.

  5. How good is the estimate? • If we compute Xbar for each of two or more random samples, the estimates likely will not be the same. • The variation in estimates from sample to sample is called sampling variation or sampling error. • The error is not deliberate, but inherent in a random sample as the elements included in the sample will vary from sample to sample. • What are the characteristics of good estimators?

  6. Hypothesis Testing • Suppose expert opinion tells us that the expected P/E of the NYSE is 20, even though our sample Xbar is 23. • Is 23 close to the hypothesized value of 20? • Is 23 statistically different from 20? • Statistically, could 23 be not that different from 20? • Hypothesis testing is the method by which we can answer such questions as these.

  7. Estimation of Parameters • Point estimate • Xbar = 23.25 from Table D-1 (5-1) is a point estimate of μX the population parameter • The formula Xbar = ∑Xi/n is the point estimator or statistic, a r.v. whose value varies from sample to sample • Interval estimate • Is it better to say that the interval from 19 to 24 most likely includes the true μX, even though Xbar = 23.25 is our best guess of the value of μX?

  8. Interval Estimates • If X ~ N(μX, σ2X), then the sample mean • Xbar ~ N(μX, σ2X/n ) for a random sample • Or Z = (Xbar- μX)/(σX/√n) ~ N(0, 1) • And for unknown σX2, t = (Xbar-μX)/(SX/√n) ~ t(n-1) • Even if X is not normal, Xbar will be for large n • We can construct an interval for μX using the t distribution with n-1 = 27 d.f. from Table E-2 (A-2) • P(-2.052 < t < 2.052) = 0.95

  9. Interval Estimates • The t values defining this interval (-2.052, 2.052) are the critical t values • t = -2.052 is the lower critical t value • t = 2.052 is the upper critical t value • See Fig. D-1 (5-1). By substitution, we can get • P(-2.052 <(Xbar-μX)/(SX/√n) < 2.052), OR • P(Xbar-2.052(SX/√n) <μX< Xbar+2.052(SX/√n)) = 0.95 • An interval estimator of μX for a confidence interval of 95% or confidence coefficient of 0.95 • 0.95 is the probability that the random interval contains the true μX

  10. Figure D-1 (5-1) The t distribution for 27 d.f.

  11. Interval Estimator • The interval is random because Xbar and SX/√n vary from sample to sample • The true but unknown μX is some fixed number and is not random • DO NOT SAY: that μX lies in this interval with probability 0.95 • SAY: there is a 0.95 probability that the (random) interval contains the true μX

  12. Example • For the P/E example • 23.25 – 2.052(9.49/√28) < μX < 23.25 + 2.052(9.49/√28) • Or 19.57 < μX < 26.93 (approx.) as the 95% confidence interval for μX • This says, if we were to construct such intervals 100 times, then 95 out of 100 intervals would contain the true μX

  13. Figure D-2 (5-2) (a) 95% and (b) 99% confidence intervals for μx for 27 d.f.

  14. In General • From a random sample of n values X1, X2,…, Xn, compute the estimators L and U such that • P(L<μX<U) = 1 – α • The probability is (1 – α) that the random interval from L to U contains the true μX • 1-α is the confidence coefficient and α is the level of significance or the probability of committing a type I error • Both may be multiplied by 100 and expressed as a percent • If α = 0.05 or 5%, 1 – α = 0.95 or 95%

  15. Properties of Point Estimators • The properties of Xbar, compared to the sample median or mode, make it the preferred estimator of the population mean, μX: • Linearity • Unbiasedness • Minimum variance • Efficiency • Best Linear Unbiased Estimator (BLUE) • Consistency

  16. Properties of Point Estimators • Linearity • A linear estimator is a linear function of the sample observations • Xbar = ∑(Xi/n) = (1/n)(X1 + X2 +…Xn) • The Xs appear with an index or power of 1 only • Unbiasedness: E(Xbar) = μX (Fig. D-3,5-3) • In repeated applications of a method, if the mean value of an estimator equals the true parameter (population) value, the estimator is unbiased. • With repeated sampling, the sample mean and sample median are unbiased estimators of the population mean.

  17. Figure D-3 (5-3) Biased (X*) and unbiased (X) estimators of populationmean value, μx.

  18. Properties of Point Estimators • Minimum Variance • a minimum-variance estimator has smaller variance than any other estimator of a parameter • In Fig. D-4 (5-4), the minimum-variance estimator of μX is also biased • Efficiency (Fig. D-5, 5-5) • Among unbiased estimators, the one with the smallest variance is the best or efficient estimator

  19. Figure D-4 (5-4) Distribution of three estimators of μx.

  20. Figure D-5 (5-5) An example of an efficient estimator (sample mean).

  21. Properties of Point Estimators • Efficiency example • Xbar ~ N(μX, σ2/n) sample mean • Xmed ~ N(μX, (π/2)(σ2/n)) sample median • (var Xmed)/(var Xbar) = π/2 ≈ 1.571 • Xbar is a more precise estimator of μX. • Best Linear Unbiased Estimator (BLUE) • An estimator that is linear, unbiased, and has the minimum variance among all linear and unbiased estimators of a parameter

  22. Properties of Point Estimators • Consistency (Fig. D-6, 5-6) • A consistent estimator approaches the true value of the parameter as the sample size becomes large. • Consider Xbar = ∑Xi/n and X* = ∑Xi/(n + 1) • E(Xbar) = μX but E(X*) = [n/(n + 1)] μX. • X* is biased. • As n gets large, n/(n + 1) → 1, E(X*) → μX . • X* is a biased, but consistent estimator of μX .

  23. Figure D-6 (5-6) The property of consistency. The behavior of the estimatorX* of population mean μx as the sample size increases.

  24. Hypothesis Testing • Suppose we hypothesize that the true mean P/E ratio for the NYSE is 18.5 • Null hypothesis H0: μX = 18.5 • Alternative hypothesis H1 • H1: μX > 18.5 one-sided or one-tailed • H1: μX < 18.5 one-sided or one-tailed • H1: μX ≠ 18.5 composite, two-sided or two-tailed • Use the sample data (Table D-1 (5-1), average P/E = 23.25) to accept or reject H0 and/or accept H1

  25. Confidence Interval Approach • H0: μX = 18.5, H1: μX ≠ 18.5 (two-tailed) • We know t = (Xbar - μX)/(SX/√n) ~ tn-1. • Use Table (E-2) A-2 to construct the 95% interval • Critical t values (-2.052, 2.052) for 95% or 0.95 • P(Xbar-2.052(SX/√n) <μX< Xbar+2.052(SX/√n)) = 0.95 • 23.25 – 2.052(9.49/√28) < μX < 23.25 + 2.052(9.49/√28) • Or 19.57 < μX < 26.93 • H0: μX = 18.5 < 19.57, outside the interval • Reject H0 with 95% confidence

  26. Confidence Interval Approach • Acceptance region • 19.57 < H0:μX < 26.93 interval for 95% • Critical region or region of rejection • H0:μX < 19.57 and 26.93 < H0:μX . • Accept H0 if value within acceptance region • Reject H0 if value outside the acceptance region • Critical values are the dividing line between acceptance and rejection of H0

  27. Type I and Type II Errors • We rejected H0: μX = 18.5 at a 95% level of confidence, not 100% • Type I Error: reject H0 when it is true • If we hypothesized H0: μX = 21 above, we would not have rejected it with 95% confidence • Type II Error: accept H0 when it is false • For any given sample size, one cannot minimize the probability of both types of error

  28. Type I and Type II Errors • Level of Significance, α • Type I error = α = P(reject H0|H0 is true) • Power of the test, (1 – β) • Type II error = β = P(accept H0|H0 is false) • Trade-off: min α vs. max (1 – β) • In practice: set α fairly low (0.05 or 0.01) and don’t worry too much about (1 – β)

  29. Example • H0: μX = 18.5 and α = 0.01 (99% confidence) • Critical t values (-2.771, 2.771) with 27 d.f. • 18.28 < μX < 28.22 is 99% conf. interval • Do not reject H0 • See Fig. D-2 (5-2) • Decreasing α, P(Type I error), increases β, P(Type II error)

  30. Test of Significance Approach • For one-sided or one-tailed tests • Recall t = (Xbar - μX)/(SX/√n) • We know Xbar, SX, and n; we hypothesize μX . • We can just calculate the value of t for our sample and μX hypothesis • Then look up its probability in Table E -2 (A-2). • Compare that probability to the level of significance, α, you choose, to see if you reject H0

  31. Example • P/E example • Xbar = 23.25, SX = 9.49, n = 28 • H0: μX = 18.5, H1: μX ≠ 18.5 • t = (23.25 – 18.5)/(9.49/√28) = 2.6486 • Set α = 0.05 in a two-tailed test (why?) • Critical t values are (-2.052, 2.052) for 27 d.f. • 2.65 is outside of the acceptance region • Reject H0 at 5% level of significance • Reject null: test is statistically significant • Do not reject: test is statistically insignificant • The difference between observed (estimated) and hypothesized values of a parameter is or is not statistically significant.

  32. One tail or Two? • H0: μX = 18.5, H1: μX ≠ 18.5 two-tailed test • H0: μX< 18.5, H1: μX > 18.5 one-tailed test • Testing procedure is exactly the same • Choose α = 0.05 • Critical t value = 1.703 for 27 d.f. • 2.6 > 1.703, reject H0 at 5% level of significance • The test (statistic) is statistically significant • See Fig. D-7 (5-7).

  33. Figure D-7 (5-7) The t test of significance: (a) Two-tailed;(b) right-tailed; (c) left-tailed.

  34. Table 5-2 A summary of the t test.

  35. The Level of Significance and the p-Value • Choice of α is arbitrary in classical approach • 1%, 5%, 10% commonly used • Calculate the p-value instead • A.k.a.: exact significance level of the test statistic • For P/E example with H0:μX< 18.5, t ≈ 2.6 • P(t27> 2.6) < 0.01, p-value < 0.01 or 1% • Statistically significant at the 1% level • In econometric studies, the p-values are commonly reported (or indicated) for all statistical tests

  36. The χ2 Test of Significance • (n-1)(S2/σ2) ~ χ2(n-1) . • We know n, S2, and hypothesize σ2 • Calculate χ2 value directly and test its significance • Example: n = 31, S2 = 12 • H0: σ2 = 9, H1: σ2 ≠ 9, use α = 5% • χ2(30) = 30(12/9) = 40 • P(χ2(30)> 40) ≈ 10% > 5% = α • Do not reject H0: σ2 = 9

  37. Table 5-3 A summary of the x2 test.

  38. F Test of Significance • F = SX2/SY2 • Or [(∑X-Xbar)2/(m-1)]/∑(Y-Ybar)2/(n-1)] follows the F distribution with (m-1, n-1) d.f. • IFσX2 = σY2, so H0: σX2 = σY2 . • Example: SAT Scores in Ex. 4.15 • varmale = 46.1, varfemale = 83.88, n = 24 for both • F = 83.88/46.1 ≈ 1.80 with 23, 23 d.f. • Critical F value for 24 d.f. each at 1% is 2.66 • 1.8 < 2.66, not statistically significant, do not reject H0

  39. Table 5-4 A summary of the F statistic.

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