Name:_Answer Key____________ Date:___________________ Probability and Statistics Chapter 5 Review Packet Determine whether each statement is true or false. If the statement is false, explain why:

1. The number of courses a student is taking this semester is an example of a continuous variable

False (you can count the number of classes a student is taking)

2. When the binomial distribution is used the outcomes must be dependent

False (all events must be mutually exclusive) 3. A binomial experiment has a fixed number of trials

True 4. A person’s weight is discrete

False (a person’s weight fluctuates all day) 5. In a Poisson distribution the variable can take on an infinite amount of values

True Select the best answer:

6. What is the sum of the probabilities of all outcomes in a probability distribution? a. 0 b. ½

c. 1 d. It cannot be determined

7. How many outcomes are there in a binomial experiment? a. 0 b. 1

c. 2 d. It varies

8. The number of trials for a binomial experiment a. Can be infinite b. Is unchanged c. Is unlimited

d. Must be fixed Determine if the distribution represents a probability distribution. If not, state way.

9. Determine if the distribution is a probability distribution X 1 2 3 4 5 P(X) 1/7 2/7 2/7 3/7 2/7

No, the probabilities add up to more than 1 10. Determine if the distribution is a probability distribution

X 50 75 100 P(X) 0.5 0.2 0.3

Yes Open Ended:

11. Calls for a Fire Company: The number of fire calls the Conestoga Valley Fire Company receives per day is distributed as followed: Find the mean, variance, and standard deviation, and construct a graph for the data

Number X 5 6 7 8 9 Probability P(X) 0.28 0.32 0.09 0.21 0.10

Mean: 5(0.28) + 6(0.32) + 7(0.09) +8(0.21) + 9(0.10) = 6.53 Variance: 5! 0.28 + 6! 0.32 + 7! 0.09 +8! 0.21 + 9! 0.10 − 6.53 ! = 44.47 −42.6409 = 1.8291 Standard Deviation: 1.8291 = 1.352

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

ProbabilityP(X)

5

6

7

8

9

12. Selecting a Card: There are 6 playing cards placed face down in a box. They are the 4 of diamonds, the 5 of hearts, the 2 of clubs, the 10 of spades, the 3 of diamonds, and the 7 of hearts. A person selects a card. Find the expected value of the draw.

𝜇 = 4 !!+ 5 !

!+ 2 !

!+ 10 !

!+ 3 !

!+

7 !!= 5.17 𝑜𝑟 5.2

13. Carpooling: If 40% of all commuters ride to work in carpools, find the probability that if 8 workers are selected, 5 will ride in carpools. (Binomial)

𝑛 = 8, 𝑝 = 0.40,𝑋 = 5

𝑃 𝑋 =8!

8 − 5 ! 5!∙ 0.40 ! ∙ 0.60 !!!

𝑃 𝑋 = 0.124 14. Income Tax Errors: The probability that a person will make 0, 1, 2, or 3 errors on his or

her income tax return is 0.50, 0.30, 0.15, and 0.05, respectively. If 30 claims are selected, find the probability that 15 will contain 0, 8 will contain 1 error, 5 will contain 2 errors, and 2 will contain 3 errors. (Multinomial)

𝑁 = 30,𝑋! = 15,𝑋! = 8,𝑋! = 5,𝑋! = 2,𝑃! =0.5,𝑃! = 0.3,𝑃! = 0.15,𝑃! = 0.05

𝑃 𝑀 =30!

15! 8! 5! 2!0.5 !" 0.3 ! 0.15 ! 0.05 !

𝑃 𝑀 = 0.0003 15. Sports Score Hot Line Calls: Sports Scores Hot Line receives, on the average, 8 calls

per hour requesting the latest sports scores. The distribution is Poisson in nature. For any randomly selected hour, find the probability that the company will receive

𝜆 = 8 a. At least 8 calls

𝑃 𝑋 ≥ 8; 8 = 0.1396 +⋯+ 0.0001 =0.5471

b. 3 or more calls

𝑃 𝑋 ≥ 3; 8 = 1 − 𝑃 0,1, 𝑜𝑟 2 𝑐𝑎𝑙𝑙𝑠 =1 − 0.0003 + 0.0027 + 0.0107 = 1 − 0.0137 =0.9863

c. At most 7 calls

𝑃 𝑋 ≤ 7; 8 = 0.0003 +⋯+ 0.1396 =0.4529

16. Youth Group Officers: A youth group has 8 boys and 6 girls. If a slate of 4 officers is selected, find the probability that exactly (Hypergeometric)

a. 3 are girls

[(6C3)(8C1)]/(14C4) = 0.16 b. 2 are girls

[(6C2)(8C2)]/(14C4)=0.42 c. 4 are boys

[(6C0)(8C4)]/(14C4)=0.07