Question

Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us.

(a)

• Suppose n = 39 and p = 0.21.

(For each answer, enter a number. Use 2 decimal places.)
n·p = _____________
n·q = _____________

Can we approximate p̂ by a normal distribution? Why? (Fill in the blank. There are four answer blanks. A blank is represented by _____.)

_____, p̂ _____ be approximated by a normal random variable because _____ _____.

first blank

Yes/No    

second blank

can/cannot    

third blank

both n·p and n·q exceed

n·q does not exceed    

n·p exceeds

n·q exceeds

n·p and n·q do not exceed

n·p does not exceed

fourth blank (Enter an exact number.)

______________

What are the values of μ_p̂ and σ_p̂? (For each answer, enter a number. Use 3 decimal places.)
μ_p̂ = ____________

σ_p̂ =_____________

(b)

Suppose

• n = 25 and p = 0.15.

Can we safely approximate p̂ by a normal distribution? Why or why not? (Fill in the blank. There are four answer blanks. A blank is represented by _____.)

_____, p̂ _____ be approximated by a normal random variable because _____ _____.

first blank

Yes/No    

can/cannot    

both n·p and n·q exceed

n·q does not exceed    

n·p exceeds

n·q exceeds

n·p and n·q do not exceed

n·p does not exceed

fourth blank (Enter an exact number.)
=____________________

(c)

Suppose

• n = 61 and p = 0.31.

(For each answer, enter a number. Use 2 decimal places.)
n·p = _____________
n·q = _____________

Can we approximate p̂ by a normal distribution? Why? (Fill in the blank. There are four answer blanks. A blank is represented by _____.)

_____, p̂ _____ be approximated by a normal random variable because _____ _____.

Yes/No    

can/cannot    

both n·p and n·q exceed

n·q does not exceed    

n·p exceeds

n·q exceeds

n·p and n·q do not exceed

n·p does not exceed

fourth blank (Enter an exact number.)
=____________

What are the values of μ_p̂ and σ_p̂? (For each answer, enter a number. Use 3 decimal places.)
μ_p̂ = ________

σ_p̂ = _______

Answer 1

(A) Given that n = 39 and p = 0.21

np = 39*0.21 = 8.19

nq = 39*(1-0.21) = 30.81

Yes, p̂ can be approximated by a normal random variable because both n·p and n·q exceed 5

(B) Given that n = 25 and p = 0.15

np = 25*0.15 = 3.75

nq = 25*(1-0.15) = 21.25

No, p̂ can be approximated by a normal random variable because n·p does not exceed 5

(C) Given that n = 61 and p = 0.31

np = 61*0.31 = 18.91

nq = 61*(1-0.31) = 42.09

Yes, p̂ can be approximated by a normal random variable because both n·p and n·q exceed 5