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chapter four

chapter four. Some Important Probability Distributions. Important Probability Distributions. Normal and Standard Normal Distributions Student’s t Distribution Chi-squared ( χ 2 ) Distribution F Distribution

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chapter four

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  1. chapter four Some ImportantProbability Distributions

  2. Important Probability Distributions • Normal and Standard Normal Distributions • Student’s t Distribution • Chi-squared (χ2) Distribution • F Distribution • Statistical Inference: If we know the probability distribution of the sample mean or variance (estimators of the population values) we are able to draw inferences about their true (population) values.

  3. The Normal Distribution • X ~ N(μx, σx2) • X a continuous random variable (-∞, +∞) • ~ distributed as • N Normal distribution with • (population) parameters: mean μx, variance σx2 • f(X) = [1/(σx√2π)]exp{-(1/2)[(X- μx)/σx]2} • Shaped like a bell [Fig. C-1 (4-1)], it is a good model for a continuous r.v. whose value depends on a number of factors, each contributing a small influence.

  4. Characteristics of the Normal Distribution • Symmetrical around its mean μx • The probability of a normally distributed r.v. taking a value far from its mean becomes progressively smaller. • About 68% of the area under the normal curve lies within (μx± σx) • About 95% within (μx± 2σx) • About 99.7% within (μx± 3σx) • Is fully described by its two parameters (μx, σx2). The probability that X lies within any interval can be calculated, or found in Table E-1(A-1), knowing (μx, σx2).

  5. Characteristics of the Normal Distribution • Any linear combination of normally distributed r.v.’s is normally distributed. • Let X ~ N(μx, σx2), Y ~ N(μy, σy2), independent • For W = aX + bY, a and b constants • Then W ~ N(μW, σW2) • μW = aμX + bμY • σW2 = a2σX2 + b2σY2 • For a Normal distribution, skewness is 0, kurtosis is 3

  6. Figure C-1 Areas under the normal curve.

  7. Figure C-2 (a) Different means, same variance; (b) same mean, differentvariances; (c) different means, different variances.

  8. Standard Normal Distribution • Let Z = (X – μx)/σX, • if X ~ N(μx, σx2) • Then Z ~ N(0, 1) is the unit or standard normal variable • Any normally distributed r.v. can be converted to a standard normal variable • This greatly simplifies computing probabilities • See Fig. C-3 (4-3) and Tables E-1(a) [A-1(a)], E-1(b) [A-1(b)]

  9. Figure C-3 (a) PDF and (b) CDF of the standard normal variable.

  10. Example 1 • Suppose X~N(70, 9) is the daily sale of bread in a bakery. What is the probability that X > 75? • Compute Z • Z = (X – μ)/σ = (75 -70)/3 = 1.67 • Find P(Z < 1.67) = 0.9525 in Table E-1(b)[A-1(b)] • Then P(Z > 1.67) = 1 – 0.9525 = 0.0475 • P(Z > 1.67) = P(X > 75) = 0.0475 or 4.75%

  11. Example 2 • What is the probability that 65 < X < 75? • Compute Z’s • Z1 = (65 – 70)/3 = -1.67 • Z2 = (75 – 70)/3 = 1.67 • From Table E-1(b)[A-1(b)] find • P(Z < -1.67) = 0.0475 • P(Z < 1.67) = 0.9525 • Then P(-1.67 < Z < 1.67) = 0.9525-0.0475 = 0.9050 • P(65 < X < 75) = 90.5%

  12. Random Sample • X1, X2, …Xn constitute a random sample of size n if all the Xs are drawn independently from the same probability distribution (each X has the same PDF). • The Xs drawn in this way are independently and identically distributed (i.i.d.) random variables • The term random sample refers to a sample of i.i.d. r.v.’s, sometimes called an i.i.d.sample. • If X ~ N(μx, σx2) and each Xi is drawn independently, then X1,…,Xn are i.i.d. r.v.’s with the normal PDF as their common probability distribution.

  13. Table C-1 25 Random numbers from N(0, 1) and N(2, 4).

  14. Sampling or Probability Distribution • Sampling or probability distribution of an estimator, such as the sample mean Xbar. • Suppose we obtain 20 random samples of 20 observations each from a normal distribution N(10, 4) and calculate the 20 sample means (Table C-2). • Grouping the 20 means in a frequency distribution (Table C-3) gives us the empirical sampling, or probability, distribution of the sample means (Fig. C-4). • A sampling distribution is a probability distribution where the random variable is an estimator, such as the sample mean or variance.

  15. Table C-2 20 Sample means from N(10, 4).

  16. Table C-3 Frequency distribution of 20 sample means.

  17. Figure C-4 Distribution of 20 sample means from N(10, 4) population.

  18. Sampling Distribution of Xbar • If X1, X2, …, Xn is a random sample from a normal population with mean μx and variance σx2, then the sample mean Xbar also follows the normal distribution with the same mean μx, but with variance σx2/n. • Xbar ~ N(μx, σx2/n) • Where n is the sample size, making the variance of Xbar usually much less than σx2 . • The square root of the variance of Xbar is called the standard error (or se) of Xbar, another name for the standard deviation of an estimator.

  19. Sampling Distribution of Xbar • Theoretically, E(Xbar) = 10 and var(Xbar) = 4/20 =0.2 • In Table C-2, Xbar = 10.052, var(Xbar) = 0.339 • More samples (observations on Xbar) bring us closer to the theoretical values • Note that Z = (Xbar-μx)/(σx/√n) ~ N(0,1) • Knowing that an estimator (Xbar) follows a particular distribution lets us calculate the probability that a given sample mean is above or below the population mean.

  20. Example • Suppose X is miles per gallon for a particular car model with X ~ N(20, 4). What is the probability for a random sample of 25 that • 19 < average mpg < 21? • Xbar ~ N(20, 4/25) • Z = (Xbar – 20)/√(4/25) = (Xbar-20)/0.4 • Z1 = (21-20)/0.4 = 2.5 • Z2 = (19-20)/0.4 = -2.5 • P(-2.5 < Z < 2.5) = 0.9938 – 0.0062 = 0.9876 from Table E-1(b)[A-1(b)].

  21. Central Limit Theorem • If X1, X2, …,Xn is a random sample from any population (probability distribution) with mean μx and variance σx2, the sample mean Xbar tends to be normally distributed with mean μx and variance σx2/n as the sample size, n, increases indefinitely (technically, approaches infinity). • In practice, a sample with n > 30 is usually sufficient for Xbar to be approximately normal. • See Fig. C-5 (4-5).

  22. Figure 4-5 The central limit theorem: (a) Samples drawn from a normalpopulation; (b) samples drawn from a non-normal population.

  23. The t distribution • We know Xbar ~ N(μx, σx2/n) • Then Z = (Xbar – μx)/(σx/√n) ~ N(0, 1) • Suppose we only know μx and estimate σx2 by its sample estimator Sx2= ∑(X – Xbar)2/(n-1) • We can obtain the new variable • t = (Xbar - μx)/(Sx/√n) • Which follows Student’s t distribution with k = n-1 degrees of freedom • The only parameter is the degrees of freedom where tk denotes a t statistic with k degrees of freedom

  24. Properties of the t distribution • Symmetric about its mean • Mean zero and variance k/(k – 2) • Only defined for degrees of freedom k > 2 • Approaches the standard normal distribution as k becomes “large” (variance t approaches 1) • “large” could be 100 or less • See Fig. C-6 (4-6)and Table E-2 (A-2) (t distribution)

  25. Figure 4-6 The t distribution for selected degrees of freedom (d.f.).

  26. Examples • Over 15 days the sale of bread averaged 74 loaves with a sample s.d. of 4. What is the probability of obtaining this average sales if the true daily average is 70 loaves? • Since we estimate the population σ with the sample s.d. S, compute t instead of Z • t = (74 – 70)/(4/√15) = 3.873 with (15 - 1) = 14 d.f. • From Table E-2(A-2): P(t14 > 3.787) = 0.001 • So P(t14 > 3.873) < 0.001 or 0.1%

  27. Examples • What is the probability that the 15 day average bread sale is less than 68 or greater than 72? • Compute t values • t1 = (68 – 70)/(4/√15) = -1.936 with 14 d.f. • t2 = (72 – 70)/(4/√15) = 1.936 with 14 d.f. • P(t < -1.936 or t > 1.936) = P(|t14| > 1.936) • P (|t14| > 1.936) between 0.05 and 0.10 in a two tailed test

  28. Chi-square Distribution • Sampling distribution of the sample variance • S2 = ∑(X – Xbar)2/(n - 1) • Statistical theory shows that the square of a standard normal variable is distributed as a chi-square (χ2) with one degree of freedom. • Z2 = χ2(1) • For Z1, Z2,…,Zkindependent std. normal variables: • ∑Zi2 = Z12 + Z22 +…+Zk2 ~ χ2(k) • Degrees of freedom k is the parameter of the χ2

  29. Figure 4-7 Density function of the X2 variable.

  30. Properties of the Chi-square Distribution • Takes only positive values from 0 to +∞ • Skewed, but as d.f. increase approaches symmetrical normal distribution • The mean of a χ2 variable is k with variance 2k • Or the variance is twice the mean • For two independentχ2 variables, Z1 and Z2, with k1 and k2 d.f., then (Z1 + Z2) ~ χ2(k1 + k2) • Table E-4 (A-4) gives probabilities that a particular χ2 value with known d.f. exceeds a given number.

  31. Example • Suppose S2 is the sample variance of a random sample of size n drawn from a normal population with variance σ2, then statistical theory holds that • (n – 1)(S2/σ2) ~ χ2(n-1) • Or d.f. times the ratio of sample to population variance ~ χ2 with (n – 1) d.f. • If n = 20, S2 = 16 and σ2 = 8, what is the probability of obtaining this sample variance? • (19)(16/8) = 19(2) = 38 is χ2 with 19 d.f. • From Table E-4 (A-4), P(χ2(19) > 38) ≈ 0.005 or 0.5%

  32. The F Distribution • X1, X2,…,Xm random sample of size m drawn from normal pop., mean μX and variance σ2X • Y1, Y2,…, Yn random sample of size n drawn from normal pop., mean μY and variance σ2Y • If X’s and Y’s are independent, what is the probability that σ2X = σ2Y? • Estimators S2X = ∑(X – Xbar)2/(m - 1) and S2Y = ∑(Y – Ybar)2/(n – 1) • F = S2X/S2Y ≈ 1 if variances are equal

  33. The F Distribution • F = S2X/S2Y ~ F(m-1),(n-1) • Or, the F ratio follows the F distribution with m – 1 (numerator) and n – 1 (denominator) d.f. • Also known as the variance ratio distribution • Designated Fk1,k2, k1 = m-1, k2 = n-1 or the numerator and denominator d.f. respectively.

  34. Properties of the F Distribution • Skewed, ranges from 0 to +∞ • As d.f. (k1, k2) become large, F approaches N • The square of a t-distributed r.v. with k d.f. is distributed as an F with 1 and k d.f. • t2 ~ F(1,k) • And F(m,n) = χ2/m, or mF(m,n) = χ2m as n → ∞ • In very large samples, the χ2 can be used instead of the F and vice versa.

  35. Figure 4-8 The F distribution for various d.f.

  36. Example • Consider the means and variances for SAT scores for men and women in Table 4.12 (p.92, 3rd ed.), as samples from much larger normal populations. Based on the sample variances, are the population variances the same? • For the Verbal scores, the F-ratio is (23 d.f. given) • F = 303.9/142.08 = 2.1353 ~ F23,23 • From Table A-3, 0.05 > P(F24,24≈ 2.14) > 0.01 • Or, there is a 1% to 5% probability of getting an F-ratio of 2.14 when the true F-ratio is 1. If this probability is too low, then the variances are different.

  37. Example 2 • An instructor gives the same econometric exam to two classes of 100 and 150 students. He draws a random sample of 25 from the first class and 31 from the second, observing the sample variances in grade-point averages of 100 and 132, respectively. If gpa’s are normally distributed, are the variances the same? • F = 132/100 = 1.32 • P(F30,24 > 1.32) > 0.25 • If 25% probability is too high, then the variances are the same.

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