The Normal and t Distributions

1
Chapter 10

• The Normal and t Distributions

2
The Normal Distribution

• A random variable Z (-8 8) 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 ? such that Pr (Z Z) a ,
• ? is the probability
• If Z 1.5 than from the table a .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
• ?Z? Z means Z -Z and Z Z together
• Pr(?Z? 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(?Z? Z) 1 - Pr(?Z? Z)

5
The Normal Distribution

• Second type of problems
• Find Zc such that Pr (Z Zc) a
• a is a specific amount of probability and Zc is
  the critical value of Z that bounds a 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 (?Z? Zc) a
• we solve it by determining Zc such that
• Pr (?Z? Zc) a/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 (?Z? 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 (-8 8) is said
• to have a normal distribution if its probability
  distribution is of the form
• where bgt0 and a can be any value.
• and

8
Other Normal Distributions

• Standard normal is one of the members of this
  family with µ0 and s1 if a0 and b1
• 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

• aPr(X Xk) Pr(Z Zk), where
• X has a normal distribution with µ5 and s2
• 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 µ5 and s2
• Pr(4.38X 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), -8lt t lt8
• t has and when
  dfgt2
• Probability problems
• Find a such that Pr(t t) a
• Table A.2 can be used to find probability
• df5, 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) a
• Find tc such that Pr(t -tc) a
• Find tc such that Pr(t tc) a
• df20 and a0.05 gt tc 1.725 one tail
• tc 2.086 with two tails
• Notice that with df20 and a0.05(1 tail) and
  a0.10(2 tails) gt 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
• ?2 , dfd
• Figure 10.6 page 222
• ?2 has µ d and S
• Find (?2 ) such that Pr(?2 (?2)c) a
• Table A.4 df 10 and a0.10 then ?2 (?2)c15.99

15
The F Distribution

• Suppose we have two independent random variables
  ?2n and ?2d having 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 lt 8

16
The F Distribution

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