Final test contents :

problem 1 : confidence interval for an canknowpopulation proportion ; sample size calculation

problem 2 : CI for population mean ; sample sizecalculation

problem 3 : Bivariate discrete distribution• Marginal distribution. Marginal distribution mean. Marginal distribution variance

a covariancea correlation

. Independence

problem 4 : Expected value of ax, taxiVariance of Cixi taxi

problem 5 : Distribution of I (random sample mean)using central limit theoremELI )Varix)

percentiles of Normal distributionproblem 6 : Maximum likelihood estimator derivation

Bias. Mean squared error (Bonus) )central limit theorem.

Exercise set for chapter 5-8

1 Chapter 5Problem 1. Two random variables have a joint probability mass function in Table 1. Show

that the correlation between X and Y is zero but X and Y are not independent.

Table 1: Joint probability mass function

x -1 0 0 1

y 0 -1 1 0

fX,Y (x, y) 1/4 1/4 1/4 1/4

Problem 2. A random variable X has the probability distribution fX(x) = x/18, 0 x 6.Determine the following:

(a) The expected value of X.

(b) The variance of X.

(c) The cumulative distribution of X.

1

-

① Marginal distribution of X-

O text -4 E Eo -

① Marginal distribution of y& o #

⑦ Eun -- that to. I + l -¥=o ¥lI÷±ELY) = O -

Varun = E (x'] - 02 = C- s ) ? I to? I + l? # = I -

Var(Y) = I⑦ E" " -- c-no . # + o.e.is#+o...*+..o..itoJIIiEE,④ COUCH Y) = E ( XY) - ECx) . Ely) = o

TY" ' LT

⑤ CowCx, y) = cWC f-x.ylhl) # fxcclfyu,TUVaea#Independ ene of x and y⇐ fx,ycx , y) =f*x, figs for all x

= So:c . # a. = #t.EE#.. :"#are

= Efx'] - ⇐ Ix) )! Joba? da - 62 not independent

x4 6

""=/ miss. ÷ !÷l ." . ÷: *.Ho - 16 - is -16=2

Problem 3. X and Y are independent, normal random variables with E(X) = 2, V ar(X) =5, E(Y ) = 6, V ar(Y ) = 8.

(a) E(3X + 2Y ).

(b) V ar(3X + 2Y ).

(c) Note that the sum of two normal random variables is still a normal random variable.

What is the distribution of 3X + 2Y ? Please specify its mean and variance.

2 Chapter 6Problem 4. Consider the following two samples: Sample 1: 10, 9, 8, 7, 8, 6, 10, 6; Sample

2: 10, 6, 10, 6, 8, 10, 8, 6.

2

① E ( GX, + Cz Xz) = Ci . Elk ) + GECK)

② Var ( C, Xi taxa ) = Cf - VarKil t CE Varcxz) +2 . C, .cz Couch, Kj"

g

-

=

-

=3. E (x) t 2. ELY) = 3.2+2.6 = 18

= 32 . Varix) +2? VarCY) = B? 5 t 22.8=45 -132=77

-

3×-124 n N ( I 8,77 )

range for samplel : 4

Sdi : I- 6

range for soupLe 2 : 4 -Sdz : l - 85-

(a) Calculate the sample range for both samples. Would you conclude that both samples

exhibit the same variability? Explain.

(b) Calculate the sample standard deviations for both samples. Do these quantities indicate

that both samples have the same variability? Explain.

3 Chapter 7Problem 5. Describe the population/distribution parameter that the following numerical

summaries trying to estimate.

(a) Sample mean

(b) Sample variance

(c) Sample proportion

Problem 6. A random sample of size n1 = 16 is selected from a normal population with a

mean of 75 and a standard deviation of 8. A second random sample of size n2 = 9 is taken

from another normal population with mean 70 and standard deviation 12. Let X̄1 and X̄2 be

the two sample means. Note the difference of two normal random variables is still a normal

random variable. Find:

(a) The probability that X̄1 � X̄2 exceeds 4.

(b) The 95th percentile of the distribution for X̄1 � X̄2.

3

-

Estimate→ population mean

Festinate population variance

Estimate population proportion

I = I a

-

Problem 7. Suppose we have a random sample of size 2n from a population denoted by X,

and E(X) = µ, V ar(X) = �2. Let X̄1 =

12n

P2ni=1 Xi and X̄2 =

12

Pni=1 Xi be two estimators

of µ. Which is the better estimator of µ? Hint, compare the bias and the MSE of the two

estimators.

4 Chapter 8An article in the Journal of the American Statistical Association [“Illustration of Bayesian

Inference in Normal Data Models Using Gibbs Sampling” (1990, Vol. 85, pp. 972?985)]

measured the weight of 30 rats under experiment controls. Suppose that 12 were underweight

rats.

(a) Calculate a 95% two-sided confidence interval on the true proportion of rats that would

show underweight from the experiment.

4

XT follows a Normal distribution N( 75, ÷ )

XI follows a Normal distribution N(yo , if )

(Assume theF- ( I - XT ) = F- ( XT ) - ECI ) = g

two samples Var (Xi -XT) = 4+16=20ware drawn

independently) XT -XT -N ( 5 , 20)-

a) PCI -I > 4) = I - PII -I a. 41=1 -

b) put Zo,Ello .95)t

-

Eon (III)-→ 95th percentile of standard

= I - Teo, I ( 0,224)Normal distribution-

= 5th. 64 .Fo I l - 0.4-12= 12.37 = 0.588

--need

Bias for I : ECXT ) - µ = µ - gu = o

Bias for Ii : EUI ) - µ = µ - fu =o

⇐ ( In III. Xi ) = In - Titi ELY = In - END 1- IT END + ' ' ' fan#Aka)

= IT . 2n - µ = fuO' XT is better

MSE for xi : ECI -µ5= VarCI) = In- than XI in

MSE for Iz : ELXT -pep = Varix) = I terms of MSE .c N

&

( 'p' -196171¥ , pathos )"411.961

.

→ [ 0.225,0.575 ]

I = Yo = 0.4 , n > 30 Tag

(b) Using the point estimate of p obtained from the preliminary sample, what sample size is

needed to be 95% confident that the error in estimating the true value of p is less than

0.02?

(c) How large must the sample be if you wish to be at least 95% confident that the error in

estimating p is less than 0.02, regardless of the true value of p?

5

no, (£45' Edict-E,

E) e perp,b) F - 0.4

← no, I"o9÷)? o .4. o:b+Margin of error

y, 2304.96

pup) reaches maximum 4 no, I:?÷z)'

. as - as

at p -- o - 57 2401