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 = 0.23. (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

n·p exceeds

n·p does not exceed

n·q exceeds

n·q does not exceed

both n·p and n·q exceed

n·p and n·q do not exceed fourth blank (Enter an exact number.)

What are the values of μp̂ and σp̂? (For each answer, enter a number. Use 3 decimal places.) μp̂ = mu sub p hat = σp̂ = sigma sub p hat =

(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

second blank

can

cannot

third blank

n·p exceeds

n·p does not exceed

n·q exceeds

n·q does not exceed

both n·p and n·q exceed

n·p and n·q do not exceed fourth blank (Enter an exact number.)

(c) Suppose n = 61 and p = 0.10. (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

n·p exceeds

n·p does not exceed

n·q exceeds

n·q does not exceed

both n·p and n·q exceed

n·p and n·q do not exceed

fourth blank (Enter an exact number.)

What are the values of μp̂ and σp̂? (For each answer, enter a number. Use 3 decimal places.)

μp̂ = mu sub p hat = σp̂ = sigma sub p hat =

Answer 1

a)

n= 39

p= .23

q= 1-.23 = .77

n.p = 8.97

n.q= 39* .77 = 30.03

yes , p̂ __can ___ be approximated by a normal random variable because _both n·p and n·q exceed 5

μp̂ =0.23

and σp̂ = √( p(1-p)/n ) = sqrt(0.23*0.77/39) =   0.067

b)

n= 25

p= .15

q= 1-.15 = .85

n.p = 3.75

n.q= 39* .77 = 21.25

no, p̂ __can _not__ be approximated by a normal random variable because _both n·p does not exceed 5

c)

n= 61

p= .10

q= 1-.10 = .90

n.p = 6.1

n.q= 54.9

yes , p̂ __can ___ be approximated by a normal random variable because _both n·p and n·q exceed 5

μp̂ =.10

and σp̂ = √( p(1-p)/n ) = sqrt(0.10*0.9/61) = 0.038