STAT 211EXAM 2 – FORM AFALL 2004

Let X be the cost ($) of an appetizer and y be the cost of main course at a certain restaurant for a customer who orders both courses. Suppose that x and y has the following joint distribution:

y
x / 10 / 15 / 20 / Total
5 / 0.20 / 0.15 / 0.05 / 0.40
6 / 0.10 / 0.15 / 0.10 / 0.35
7 / 0.10 / 0.10 / 0.05 / 0.25
Total / 0.40 / 0.40 / 0.20 / 1

Where Var(X)=0.6275, Var(Y)=14, Cov(X,Y)=0.35, and the pmf of Y is

y101520otherwise

p(y)0.400.400.200

Answer questions 1 to 9 using this information.

  1. Are X and Y independent?

(a) Yes(b) No. The covariance is positive and p(x,y)p(x)p(y) for all x and y.

  1. Find the probability that the customer orders appetizer cost $5 and main course cost $15.

(a) 0.05(b) 0.10(c) 0.15(d) 0.20(e) 0.25

  1. Find the probability that total cost for the customer, X+Y being at most $21.

(a) 0(b) 0.15(c) 0.30(d) 0.60(e) 0.70

The possibilities are observed. p(5,10)+p(5,15)+p(6,10)+p(6,15)+p(7,10)=0.70

  1. Find the probability of the appetizer being $6.

(a) 0.20(b) 0.25(c) 0.30(d) 0.35(e) 0.40

  1. Which of the following is the expected cost of the main course?

(a) 5.85(b) 6(c) 6.35(d) 14(e) 15

E(Y)=10(0.40)+15(0.40)+20(0.20)=14

  1. Find the probability that maximum cost for the appetizer or the main course being $15.

(a) 0.10(b) 0.15(c) 0.25(d) 0.35(e) 0.40

The possibilities are observed. p(5,15)+p(6,15)+p(7,15)=0.40

  1. Which of the following is the variance in the total cost, Var(X+Y)?

(a) 0.6275(b) 14(c) 14.6275(d) 14.9775(e) 15.3275

Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)=0.6275+14+2(0.35)=15.3275

  1. Which of the following is the correlation between X and Y?

(a) 0.0398(b) 0.1181(c) 0.3500(d) need more information to compute it

Corr(X,Y)==0.1181

  1. Given that the main course was $20, which of the following is the probability of the appetizer being $7?

(a) 0(b) 0.25(c) 0.50(d) 0.75(e) 1

P(X=7 | Y=20)=p(7,20)/P(Y=20)=0.05/0.20=0.25

  1. If I give you the data on reaction time and ask you if it comes from the pdf that we define, which of the following would you do to answer my question?

(a) ordered data should be divided into equal intervals to set the percentiles and then cdf should be set at the percentiles to find the cut off values for the graph (data versus cutoff values) so that we can check if the 45 line is observed.

(b) unordered data should be divided into equal intervals to set the percentiles and then cdf should be set at the percentiles to find the cut off values for the graph (data versus cutoff values) so that we can check if the 45 line is observed.

(c) ordered data should be divided into equal intervals to set the percentiles and then pdf should be set at the percentiles to find the cut off values for the graph (data versus cutoff values) so that we can check if the 45 line is observed.

(d) unordered data should be divided into equal intervals to set the percentiles and then pdf should be set at the percentiles to find the cut off values for the graph (data versus cutoff values) so that we can check if the 45 line is observed.

  1. Which of the following is incorrect?

(a) If the data is modeled with the binomial distribution, it is discrete

(b) If the data is modeled with the geometric distribution, it is discrete

(c) If the data is modeled with the poisson distribution, it is continuous

(d) If the data is modeled with the exponential distribution, it is continuous

  1. Let X be the number of ticketed airline passengers denied a flight because of overbooking. The pmf of X is defined a p(x)=c(5-x) for x=0,1,2,3,4. Which of the following is the numerical value of c makes p(x) a legitimate pmf of X?

(a) 1/15(b) 1/10(c) 1/5(d) 10(e) 15

15c=1 then c=1/15

Suppose that the reaction time (sec), X to a certain stimulus is a continuous random variable with the following pdf and the cdf,

Answer questions 13 to 16 using this information.(Hint: where ln(.) is the natural log.)

  1. Which of the following is the numerical value of k?

(a) 0.63(b) 0.43(c) 0.23(d) 0.13(e) 0.03

kln(10)=1 then k=1/ln(10)=0.43

  1. Which of the following is the probability thatthe reaction time exceeds 3 sec.?

(a) kln(3)(b) 1-kln(3)(c) k/3(d) 1-k/3(e) none of the listed

P(X>3)=1-P(X3)=1-kln(3)

  1. Which of the following is the expected reaction time?

(a) 10k(b) 9k(c) 8k(d) k(e) none of the listed

E(X)=

  1. Which of the following is the E(X2)?

(a) 50k(b) 49.5k(c) 49k(d) 0.02k(e) none of the listed

E(X2)=

  1. The number of errors per 1000 selected lines of a computer code is the interest. Which of the following is the possible distribution to use?

(a) Poisson Distribution(b) Binomial Distribution(c)Exponential Distribution

If more than one error per lineif either error or not per line

  1. The lifetime of a randomly selected component is the interest. Which of the following is the possible distribution to use?

(a) Poisson Distribution(b) Binomial Distribution(c) Exponential Distribution

  1. The number of customer complaints in a randomly selected week is the interest. Which of the following is the possible distribution to use?

(a) Poisson Distribution(b) Binomial Distribution(c) Exponential Distribution

Suppose that the distribution of slot width on a forging are normally distributed with the mean value of 1 in. and standard deviation of 0.0025 in. Answer questions 20 to 21 using this information.

  1. What percentage of such forgings has a slot width that is between 0.995 and 1.005 in.?

(a) 0.0228(b) 0.0456(c) 0.9544(d) 0.9772(e) 1

P(0.995X1.005)=P(-2z2)=P(z2)-P(z<-2)=0.9772-0.0228=0.9544 where z=(x-1)/0.0025

  1. Which of the following is the median slot width?

(a) 0.0025(b) 0.5(c) 1(d) 1.0025(e) 1.5

P(X>median)=0.5=P(Z>(median-1)/0.0025) then (median-1)/0.0025=0 using the standard normal table will give you median=1 or remember that the mean and the median in symmetric continuous distributions are the same.

  1. If the data are standard normal distributed, what percent of the data is in between -1 and 1?

(a) 0.1587(b) 0.4761(c) 0.5239(d) 0.6826(e) 0.8413

P(-1Z1)=P(Z1)-P(Z<-1)=0.8413-0.1587=0.6826

  1. If the data are standard normal distributed, what percent of the data exceed 5?

(a) 0(b) 0.25(c) 0.50(d) 0.75(e) 1

P(Z>5)=0

  1. Which of the following is? (Hint: (.) is a gamma function)

(a) 20.8584 (b) 4.4292(c) 3.1416d) 2.8584(e) none of the listed

=3! -/2=5.1138

  1. Suppose you did not know the distribution of the height data but you were told it is a skewed distribution with the known mean and the standard deviation. You want to compute the probability of randomly selected height being within two standard deviation of the expected height. Which of the following should you do?

(a)Still use the normal distribution to find the probability

(b)Use Empirical rule to find the probability

(c)Use Chebyshev’s inequality to find the probability

(d)We have not learned anything in class to solve such a problem.