Exam 3 Review with key Math 166

1. Classify each random variable as finite, infinite discrete or continuous.

I   A person is given 30 minutes to complete a test.

a) X is the time he spends taking the test.

b)      Y is the number of times a person takes the test until he passes.

c)      N is the number of correct answers a person gets.

II A die is tossed.

      a) X is the distance the die travels upward.

b) Y is the top number on the die.

c) N is the number of times the die is tossed until a 6 appears.

 

2. Three balls are drawn all at once without replacement from a box containing 4 red and 6 blue balls. X is the number of red balls drawn. Write the distribution of X. Find E(X). Show all work.

 

 

3. Three red balls weighing 2 ounces each and 5 blue balls weighing 4 ounces each are in a box. Three balls are randomly selected all at once. X is the total weight of the three balls chosen. Write the distribution of  X. Find E(X). Show your work.

 

4. a) If the odds you will win a raffle are 5:20, what is the probability you will win?

 

b)      If the chance of rain is 30%, what are the odds it will not rain?

 

5. Stocks I, II, and III have returns of 10%, 12% and 15% respectively. A person invests 20% of his investment money in stock I, 30% in stock II and 50% in stock III.

Find the expected return as a percent.

 

6. Two stocks are watched over a 6 month period. The average share price of each stock for each month is recorded and the results follow.

Stock I   price    $25     $30     $15        Stock II  price    $25     $40    $10

        # months      3          2         1                  #months       1         3        2

Compute the expected value of the price per share for each.

Compute the standard deviations and compare.

 

7. A student takes 12 credits and receives the following grades:

grade        A     B     C      D      F

#credits    3      5      2       2     0

X is the number of points assigned to a letter grade in a 4-point system.

a) Write the distribution of X.

b) Find the expected value of X.

c) If before this semester, the student had 65 credits with a gpa of 3.5, find the student’s new gpa.

 

 

 

8. a) Find the mean, median, mode, and standard deviation of the scores.

                                                quiz score     0      1        2          3          4          5

                                               # students    16      21    14        15        24        10

 

 

In b) and c), find the mean, median and st. dev. of  the price per share.

b) the price per share of a stock:  stock price in $          20       21       22      23     24

                                                           # days               5          3        14        4       4

 

 

c) the price per share of a stock: stock price in $       20        21       22      23     24

                                                             # days             10        6          1        3      10

 

Why is the standard deviation in c higher than in b?

 

9. A student takes 5 tests and receives the following scores.

 

Test #               1         2         3         4         5

Score               80        93        76        65        90

 

Find the mean, median, mode and st.dev. of his scores.

 

10. Which of the following are binomial? If binomial give n, p, E(X), and the standard deviation of  X. If not binomial say why not.

 

a)      A box contains 5 red, 6 blue and 4 yellow balls. A person draws one ball and notes whether or not the ball is red. Then the ball is returned to the box. This is repeated 10 times. X is the number of red balls drawn.

 

 

b)      A person draws 1 card from a deck and notes the suit. The card is  not returned to the deck. This is repeated 15 times. X is the number of hearts.

 

c)      A certain gene occurs in 7% of the general population. Seven people from one family and 6 people from another family are selected. X is the number of people who have the gene.

 

d)      Same as c) but 50 people are selected at random.

 

 

 

 

 

 

 

11. It is estimated that 15% of cars on a certain highway are going over the speed limit. If 500 cars are observed and X is the number over the speed limit, find

 

a)      E(X)  and (X). (The st dev is very close to a whole number and you can use this or round to 4 decimal places, it won’t make much difference.)

 

 

 

 

b)      Find P(60 < X < 90)

 

c)      Find the normal approximation to the probability in b.

 

 

12.  Same as 11 but only 7% of cars are speeding and only 20 cars are observed,

 

a)      Find P(1  

 

 

b)      Find the normal approximation to the probability in a. Why is this estimate not as good as the estimate in problem 11 c?

 

 

 

13.    Lengths of babies born at a certain hospital are normally distributed with mean 20 inches and standard deviation 2.6 inches.

a)      Find the probability that a randomly selected baby is less than 19 inches long. Round to 2 decimal places.

b)      Find the probability that a randomly selected baby is at least 19 inches and at most 21 inches. Round to 4 decimal places.

c)      Find a length L so that 70% of babies are at least as long as L.

 

 

14. X is a normal random variable with mean 50 and unknown standard deviation.

Given that P(X<75)=0.62, use symmetry to find

a)      P(50<X<75)

 

 

b)      P(X>25)

 

 

 

 

 

 

 

Reminders for the tvm-solver

Deposits are negative, withdrawals are positive

N=mt=the total number of interest periods in t years  

I%=the annual (nominal) interest rate as a percent, not a decimal

PMT=payment

PV=amount you borrow or negative of the amount you initially invest

FV=your balance at the end of the time period.

P/Y=C/Y=m=the number of times per year that interest is compounded

 

The effective rate of compound interest is the interest earned by $1 in one year.

 

15. a) A couple can get a simple interest rate of 5.5% per year. How much interest will they earn on $10,000 in 7 years?

 

b) How much interest will they get if instead they invest the $10,000 at 5% compounded monthly for 7 years?

 

c) How much interest will they earn if the $10,000 is invested at 5% compounded daily for 7 years?

 

16. What is the effective rate of interest for

a)      4% compounded quarterly?

b)      6% compounded monthly?

 

17. A person borrows $100,000 at annual interest rate 4.8% compounded monthly. He will pay off the loan in monthly installments over 30 years.

a)      Find their monthly payment.

b)      How interest much will they pay in 30 years?

c)      How much will they still owe after 10 years?

 

  1. A couple anticipates the need for $50,000 in 18 years to pay for college for their newborn. If interest is 6% compounded monthly and they make monthly payments to accumulate this amount, what should be their monthly payment?

 

  1. A person anticipates the need for $650 per month for living expenses for the next 5 years. Interest is 4.8% compounded monthly. How much should they put into an account at the beginning of the 5 years to be able to withdraw $650 per month?

 

Suggested problems in Tan 8th edition               7th edition

Section 5.2   24, 26, 30                                                24, 26, see below

Section 5.3   30, 32, 36                                                29, see below, 34

 

8th ed #30 sec. 5.2

Lauren plans to deposit $5000 into a bank account at the beginning of next month and $200 per month into the same account at the end of that month and at the end of each subsequent month for the next 5 years. If her bank pays interest at the rate of 6% per year compounded monthly, how much will she have at the end of 5 years?

 

8th ed # 32 sec. 5.3

Joe secured a loan of $12,000 three years ago from a bank for use toward his college expenses. The bank charges interest at the rate of 4% per year compounded monthly on this loan. He wants to repay this loan through monthly payments over 10 years. If the interest rate is the same, what will his monthly payments be?

 

 

Key:

 

1 a) continuous b) infinite discrete          c) finite

   a) continuous b) finite             c) infinite discrete

 

2.         x           0         1          2          3

            p          1/6       ½         3/10     1/30

E(X)=0(1/6) + 1(1/2) + 2(3/10) + 3(1/30)=1.2

 

3.         x          12        10        8          6

            p          5/28     15/28   15/56   1/56

E(X)=9.75      

 

4. a) 0.2           b) 7:3

 

5. 13.1%

 

6.  mean=25  st.dev.=5       mean=27.5   st.dev.=13.46

The 2nd stock has higher st.dev. because values away from the mean occur with

hgher probabilities.

 

7. a)  x             4          3          2          1          0

         p             3/12     5/12     2/12     2/12     0

 

b) 2.75

 

c) 3.383

 

8. a) mean=2.4  median=2  mode=4 st.dev.=1.6432

 

b) mean=21.9667   median=22  mode=22   stdev=1.196755

 

c) mean=21.9  median=21  modes 20, 24    stdev=1.7195

because prices far from the mean have higher probabilities, the data is more spread away from the mean

 

9. mean=80.8  median=80 all the scores are modes  stdev=10.07

 

10. a) yes, n=10  p=1/3 E(X)=10/3  stdev=1.4907

 

b) no, trials are not identical or independent because selection is made without replacement.

c)      no, the 7 people from one family will have a different probability than the 6 from the other family. Trials are not identical.

 

d)      yes, n=50  p=.07  E(X)=3.5  stdev=1.8704162

 

11. a) E(X)=75  stdev=7.9844   b) 0.94812     c) 0.94778

 

12. a) 0.71863   b) 0.7520155   because np and nq are not both greater than 5. sample is too small

 

13. a) 0.35   b) o.2995   c) 18.636 inches

 

14. a) 0.12   b) 0.62

 

15. a) $3850   b) $4180   c) $4190

 

16. a) 4.060401   b)  6.1677812

 

17.  a) $524.67            b) $88,881.20              c) $80,847.42

 

18. $129.08

 

19. $34,611.76