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Chapter 10

Chapter 10. The Normal and t Distributions. The Normal Distribution. A random variable Z (- ∞ ∞) is said to have a standard normal distribution if its probability distribution is of the form: The area under p(Z) is equal to 1

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Chapter 10

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  1. Chapter 10 The Normal and t Distributions

  2. The Normal Distribution • A random variable Z (-∞ ∞) is said to have a standard normal distribution if its probability distribution is of the form: The area under p(Z) is equal to 1 Z has and , page 210

  3. The Normal Distribution • Pr (Z ≥ 1.5), Figure 10.1 (a), page 210 • Table A.1 give (Z ≥ 1.5), for positive values of Z* • Find ά such that Pr (Z ≥ Z*)= α , ά is the probability If Z*= 1.5 than from the table α = .067 Pr (Z ≤ -Z*)= Pr (Z ≥ Z*) Pr (Z ≤ -1.5)= .067

  4. The Normal Distribution • To determine the probability in two symmetrical tails of the distribution: ІZІ = Z* means Z ≤ -Z* andZ ≥ Z* together • Pr(ІZІ ≥ Z*)= Pr(Z ≤ -Z*)+ Pr(Z ≥ Z*) = = 2Pr(Z ≥ Z*) area in Fig. 10.1b The probability of not being in either tail is unshaded area or: Pr(ІZІ ≤ Z*) = 1 - Pr(ІZІ≥ Z*)

  5. The Normal Distribution • Second type of problems: Find Zc such that Pr (Z ≥ Zc) = α α is a specific amount of probability and Zc is the critical value of Z that bounds α probability on the right-hand tail Table A.1 for a given probability we search for Z value

  6. The Normal Distribution • When we deal with a two-tailed probability Find Zc such that Pr (ІZІ ≥ Zc) = α we solve it by determining Zc such that Pr (ІZІ ≥ Zc) = α/2 For example: a) determine values of Z that bound a total of 5 percent of the probability in both tails, find Zc such that Pr (Z ≥ Zc) = 0.025, using table A.1 we find Zc = 1.96 b) what symmetrical values of Z contain between them 50 percent of the total probability, find Zc and -Zc such that Pr (ІZІ≤ Zc) = 0.5, from table A.1 These are bound by determining Zc such that Pr (Z ≥ Zc) = 0.25 Z values -0.67 and 0.67 contain between them 50 percent of the probability

  7. Other Normal Distributions • Random variable X (-∞ ∞) is said to have a normal distribution if its probability distribution is of the form: where b>0 and a can be any value. and

  8. Other Normal Distributions • Standard normal is one of the members of this family with μ=0 and σ=1 if a=0 and b=1 • Figure 10.4 shows different normal distributions, page 214 • All members of the normal distribution family can be viewed as being linear transformations of each other • Figure 10.5, page 215

  9. Other Normal Distributions • Any transformation can be thought of as a transformation of the standard normal distribution

  10. Other Normal Distributions • α=Pr(X ≥ Xk)= Pr(Z ≥Zk), where • X has a normal distribution with μ=5 and σ=2 Pr(X≥ 6) ? From Table A.1 we find Pr(Z ≥ 0.5)=0.309

  11. Other Normal Distributions • X has a normal distribution with μ=5 and σ=2 Pr(4.38≤X≤ 8) =? Pr(-0.31≤ Z ≤ 1.5) =0.555

  12. The t Distribution • The equation of the probability density function p(t) is quite complex: p(t) = f (t; df), -∞< t <∞ • t has and when df>2 • Probability problems: Find α such that Pr(t ≥ t*) =α Table A.2 can be used to find probability df=5, Pr(t ≥ 1.5) = 0.097 and Pr(t ≥ 2.5) = 0.027

  13. The t Distribution • Table A.3 provides answers to problems: Find tc such that Pr(t ≥ tc) =α Find tc such that Pr(t ≤ -tc) =α Find tc such that Pr(|t |≥ tc) =α df=20 and α=0.05 => tc =1.725 one tail tc =2.086 with two tails Notice that with df=20 and α=0.05(1 tail) and α=0.10(2 tails) => tc =1.725

  14. The Chi-Square Distribution • When we have d independent random variables z1, z2 , z3, . . . Zd , each having a standard normal distribution. • We can define a new random variable χ2 = , df=d Figure 10.6 page 222 χ2 has μ = d and S = Find (χ2 ) such that Pr(χ2 ≥ (χ2)c) =α Table A.4 df =10 and α=0.10 then χ2 ≥ (χ2)c=15.99

  15. The F Distribution • Suppose we have two independent random variables χ2n and χ2dhaving chi-square distributions with n and d degrees of freedom • A new random variable F can be defined as: • This random variable has a distribution with n and d degrees of freedom • 0 ≤ F < ∞

  16. The F Distribution • Find Fc such that Pr(Fn,d ≥ Fc) =α • Table A.5 gives the Fc values for n and d when α = 0.05 • Table A.6 gives the Fc values for n and d when α = 0.01 • For F distribution with 5 and 10 df Fc = 3.33 for α = 0.05

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