Elementary Statistics

Chapter 6 Test Review Key

1. (10 points) Given the following probability distribution:

Number of Books

(x)

P(x)

0 0.07

1 0.20

2 0.10

3 0.13

4 0.40

5 0.10

Find the following (round to 2 decimal places):

a) Mean 89.2=µ

b) Standard Deviation 52.1=σ

c) Variance 32.22 =σ

Elementary Statistics

Chapter 6 Test Review Key

2. (5 points) Using the formula for Binomial, setup the following problems (no need to plug into

the calculator)

a) n = 8, p = 0.7, find P(x = 3)

383)7.01()7.0(

!3)!38(

!8)3(

−−⋅−

=P

b) n = 12, p = 0.31, find P(x < 4)

3123

212211210120

)31.01()31.0(!3)!312(

!12

)31.01()31.0(!2)!212(

!12)31.01()31.0(

!1)!112(

!12)31.01()31.0(

!0)!012(

!12

)3()2()1()0(

)4(

−

−−−

−⋅−

+−⋅−

+−⋅−

+−⋅−

=

=+=+=+==

<

xPxPxPxP

xP

Elementary Statistics

Chapter 6 Test Review Key

3. (10 points) Using the attached tables, find the following binomial probabilities (circle the values

on the attached table along with your calculators)

a. n = 9, p = 0.25, find P(x < 3)

P(x < 3) = 0.0751 + 0.2253 + 0.3003 = 0.6007

* This is the normal binomial distribution table (not cumulative)

Elementary Statistics

Chapter 6 Test Review Key

b. n = 8, p = 0.35, find P(x<5)

P(x < 5) = P(x=0,1,2,3,4) = 0.8389

* This is the cumulative binomial distribution table

Elementary Statistics

Chapter 6 Test Review Key

4. (35 points) Various problem that use the same techniques demonstrated with this example:

If 30% of the bolts produced by a machine are defective, if 10 bolts are chosen at random, find

the following probabilities assuming it follows a binomial distribution:

a) P(less than 7 are defective)

P(x=0,1,2,3,4,5,6)

P(x < 6)

binomcdf(10,0.30,6)

0.9894

b) P(exactly 3 are defective)

P(x=3)

binompdf(10,0.30,3)

0.2668

c) P(at most 2 are defective)

P(x=0,1,2)

P(x < 2)

binomcdf(10,0.30,2)

0.3828

d) P(more than 3 are defective)

P(x=4,5,6,7,8,9,10)

P(x > 4)

1-binomcdf(10,0.30,3)

0.3504

e) P(at least 7 are defective)

P(x=7,8,9,10)

P(x > 7)

1-binomcdf(10,0.30,6)

0.0106

f) P(between 2 and 5 are defective, inclusive)

P(x=2,3,4,5)

P(2 < x < 5)

binomcdf(10,0.30,5)-binomcdf(10,0.30,1)

0.8033

Elementary Statistics

Chapter 6 Test Review Key

5. (5 points) 15% of Cowley Students believe in Bigfoot. Compute the mean, standard deviation,

and variance of the random variable X, the number of Cowley students that believe in Bigfoot

based on a random sample of 200 students.

n = 200, p= 0.15

5.25)15.01)(15.0(200)1(

05.5)15.01)(15.0(200)1(

30)15.0(200

2 =−=−=

=−=−=

===

pnp

pnp

np

X

X

X

σ

σ

µ

Elementary Statistics

Chapter 6 Test Review Key

6. (30 points) Various problems that use the same techniques demonstrated with this example:

Between 1 and 5 P.M., calls to a help desk follow a Poisson distribution with an average of 5

calls per minute. Over 10 minutes, find the following probabilities:

50)10(5

10,5

===

==

t

t

Xλµ

λ

a) P(Exactly 45 calls come in)

P(x = 45)

poissonpdf(50,45)

0.0458

b) P(Less than 30 calls come in)

P(x = 0, 1, 2, …, 29)

P(x < 29)

poissoncdf(50,29)

9.168E-4

0.0009168

c) P(At most 40 calls come in)

P(x = 0, 1, 2, …, 40)

P(x < 40)

poissoncdf(50,40)

0.0861

d) P(More than 52 calls come in)

P(x = 53, 54, …)

P(x > 53)

1-poissoncdf(50,52)

0.3542

e) P(At least 60 calls come in)

P(x = 60, 61, …)

P(x > 60)

1-poissoncdf(50,59)

0.0923

f) P(Between 20 and 40 calls come in, inclusive)

P(x = 20, 21, …, 40)

P(20 < x < 40)

poissoncdf(50,40)-poissoncdf(50,19)

0.0861

Elementary Statistics

Chapter 6 Test Review Key

7. (5 points) Using the formula for Poisson, setup the following problems (no need to plug into the

calculator)

a) λ = 3, t = 5, find P(x = 10)

!10

15)10(

!)(

15)5(3

1510 −

−

⋅=

⋅=

===

eP

x

exP

t

x µµ

λµ

b) λ = 3, t = 1, find P(x < 3)

!2

3

!1

3

!0

3

)2()1()0(

)3(

!)(

3)1(3

323130 −−−

−

⋅+

⋅+

⋅

=+=+=

<

⋅=

===

eee

xPxPxP

xP

x

exP

t

x µµ

λµ