1. Let X and Y be jointly distributed with ρ(X,Y ) = 0, σX = 1, andσY = 2. Find Var(X − Y ).

2. A blood test is 99 percent effective in detecting a certain disease whenthe disease is present. However, the test also yield a false-positive resultfor 2 percent of the healthy patient tested. (That is, if a healthy person istested, then with probability 0.02 the test will say that this person has thedisease.) Suppose .5 percent of the population has the disease. Find theconditional probability that a random tested individual actually has thedisease given that his or her test result is positive.

3. If the joint probability distribution X and Y is given by

f(x, y) =3x + cy

18for x = 0, 1; y = 1, 2.

1. Determine c that renders f(x, y) a valid joint probability density func-tion of X and Y .

2. Find the marginal distributions of X and Y , respectively;

3. Determine whether or not X and Y are independent;

4. Find P (X + Y = 2).

5. Find P (X ≤ 0|Y = 1)

6. Find E(X + Y ).

7. Find E(XY ).

8. Find Cov(2X + 1, 3Y ).

4. The mean score for freshmen on an aptitude test at a certain collegeis 540, with a standard deviation of 30. What is the probability that twogroups of students selected at random, consisting of 50 and 50 students,respectively, will differ in their mean scores by an amount between 6 and12 points?

5. The following data, recorded in days, represent the length of time torecovery for patients randomly treated with one of two medications to clearup severe bladder infections

Medication 1 Medication 2

n1 = 16 n2 = 36x̄1 = 17 x̄2 = 19s21 = 3 s2

2 = 2

Let µ1 and µ2, and σ21 and σ2

2 be the means and variances of length of timeto recovery for patients for medication 1 and medication 2, respectively.Assuming normal populations.

1. Test if H0 : σ21 = σ2

2 versus H0 : σ21 6= σ2

2 at α = 0.05.

2. Find a 95% confidence interval for the difference µ1 − µ2 in the meanrecovery time for the two medications.

3. Test H0 : µ1 = µ2 versus H1 : µ1 6= µ2 at α = 0.05.

6. A random sample of 100 recorded deaths in the United States during thepast year showed an average life span of 72 years. Assuming a populationstandard deviation of 8 years, does this seem to indicate that the mean lifespan today is greater than 70 years? Use a 0.05 level of significance.

7. The average grade for an exam is 85, and the standard deviation is10. If 25% of the class is given A’s, and the grades are curved to follow anormal distribution what is the lowest score to have an A.

8. A pair of fair 4-sided dice is rolled 192 times. Let T be the numberthat a total of 5 occurs.

1. Find the probability function of T?

2. What are the mean (expected value) and variance of T?

3. What is the probability that a total of 5 occurs at most 49 times?

Critical Values: Let Z be a standard normal random variable: P (Z > zα) =α

α 0.5398 0.5596 0.5793 0.5987 0.6179

zα -0.1 -0.15 -0.20 -0.25 -0.3

9. An insurance company offers its policyholders a number of differentpremium payment options. For a randomly selected policyholder, let Xbe the number of months between successive payments. The cumulativedistribution function X is

F (x) =

0, if x < 1;0.4, if 1 ≤ x < 3;0.6, if 3 ≤ x < 5;0.8, if 5 ≤ x < 7;1.0, if x ≥ 7.

1. What is the probability mass function of X?

2. Compute P (4 < X ≤ 7).

3. Find the mean, E(X) and the variance, Var(X) of X.

10. The waiting time, in hours, between successive speeders spotted by aradar unit is continuous random variable, X, with cumulative distributionfunction

F (x) =

{0, x < 0;1− e−8x, x ≥ 0.

1. Find the probability density function, f(x), of X and use it to com-pute the probability of waiting less than 6 minutes between successivespeeders.

2. Find the expected value and the variance of the waiting time, X, inhours.

11. A six-sided dice is rolled until the first time T that a six turns up.

1. What is the probability distribution for T?

2. Find P (T > 3).

3. Find P (T > 6|T > 3).

12. If a worn machine tool produces 1% defective parts and the partsproduced are independent, find the mean number of defective parts out of25, and the variance of the number of defective part.

13. Find the probability that a random sample of 25 observations, from anormal population with variance σ2 = 6, will have a variance s2

1. greater than 9.1;

2. between 3.462 and 10.745.

14. If S21 and S2

2 represent the variances of independent random samplesof size n1 = 25 and n2 = 31, taken from normal populations with equalvariances, find

P (S2

1

S22

> 1.26).

15. An interior automotive supplier places several electrical wires in a har-ness. A pull test measures the force required to pull spliced wires apart. Acustomer requires that each wire that is spliced into the harness withstanda pull force of 20 pounds. Let X equal the pull force required to pull 20gauge wires apart. Assume that the distribution of X is N(µ, σ2). Thefollowing data give 20 observations of X

29 25 30 26 26 24 22 23 28 2821 28 24 25 25 26 28 22 26 24

1. Find point estimates for µ and σ2.

2. Find a 95% confidence interval for µ that provides a lower bound forµ.

3. Find a 95% confidence interval for σ2.

4. Test H0 : µ = 26 versus H1 : µ 6= 26 at α = 0.05.

5. Test H0 : σ2 = 4 versus H1 : σ2 6= 4 at α = 0.05.

16. A unfair six-sided dice is tossed 120 with the following results

x 1 2 3 4 5 6

Frequencies 12 29 17 15 26 21

Test the hypothesis of 0.05 level of significance that the recorded data maybe fitted by the following distribution

P (X = x) =

112 , if x = 1;28 , if x = 2;18 , if x = 3;18 , if x = 4;28 , if x = 4;212 , if x = 4.

17. To test whether a golf ball of brand A can be hit a greater distanceoff the tee than a golf ball of brand B, each of 17 golfers hit a ball of eachbrand, eight hitting ball A before ball B and nine hitting ball B beforeball A. Assume that the differences of the paired A distance and B dis-tance are approximately normally distributed and test the null hypothesisH0 : µD = 0 against the alternative hypothesis H1 : µD > 0 using a pairedt-test with the 17 differences. Use a 0.05 level of significance.

Golfer Distance for Ball A Distance for Ball B

1 265 2522 272 2763 246 2434 260 2465 274 2756 263 2467 255 2448 258 2459 276 25910 274 26011 274 26712 269 26713 244 25114 212 22215 235 23516 254 25517 224 231

18. The contingency table shows a random sample of patients with chronicfatigue syndrome treated with a drug or with a placebo. At α = 0.10,can you conclude that the variables treatment and result are dependent?Based on these results, would you recommend using the drug as part of atreatment for chronic fatigue syndrome?

Treatment

Result Drug Placebo

Improvement 20 20No change 10 15

19. In a poll, 1000 males and 1000 females were asked, ”If you could haveonly one of the following, which would you pick: money, health, or love?”Their responses are presented in the table below.

Money Health Love

Men 100 500 400Women 60 620 320

Test the claim that gender and response are independent at 0.05 level ofsignificance.

20. A magazine reported the proportions of adult American who favor”stricter gun-control laws.” A telephone poll of 2000 adult Americans, ofwhom 750 were gun owners and 1250 did not own guns, showed that 250gun owners and 250 non-gun owners favor stricter gun-control laws. Letp1 and p2 be the respective proportions of gun owners and non-gun ownerswho favor stricter gun-control laws.

1. With α = 0.05, test H0 : p1 = 13 against H1 : p1 6= 1

3 .

2. With α = 0.05, test H0 : p1 = p2 against H1 : p1 6= p2.

Critical Values:

• Let Z be a standard normal random variable: P (Z > zα) = α

α 0.16 0.1 0.05 0.025 0.02

zα 1.00 1.28 1.645 1.96 2.00

• Let T (ν) have a t-distribution with ν degrees of freedom: P (T (ν) > Tα(ν)) =α

α

Tα(ν) 0.025 0.05 0.1

Tα(19) 2.903 1.729 1.328Tα(39) 2.023 1.685 1.304Tα(40) 2.021 1.684 1.303Tα(50) 2.001 1.676 1.299

• Let χ2(ν) have a chi-squared distribution with ν degrees of freedom:P

(χ2(ν) > χ2

α(ν))

= α

α

χ2α(ν) 0.025 0.05 0.1 0.9 0.95 0.975

χ2α(2) 7.378 5.991 4.605 0.211 0.103 0.0506

χ2α(3) 9.348 7.815 6.251 0.584 0.352 0.216

χ2α(6) 14.449 12.592 10.645 2.204 1.635 1.237

• Let F (ν1, ν2) have a F -distribution with ν1 and ν2 degrees of freedom:P (F (ν1, ν2) > Fα(ν1, ν2)) = α

α

Fα(ν1, ν2) 0.025 0.05 0.1 0.9 0.95 0.975

Fα(19, 19) 2.53 2.16 1.82 0.54 0.46 0.39Fα(20, 20) 2.46 2.12 1.79 0.56 0.47 0.41