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 = 42 and
• p = 0.18.

(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

YesNo    

second blank

cancannot    

third blank

both n·p and n·q exceedn·p and n·q do not exceed    n·p does not exceedn·q exceedsn·p exceedsn·q 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 = mu sub p hat =

σ_p̂ = 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

YesNo    

second blank

cancannot    

third blank

both n·p and n·q exceedn·p and n·q do not exceed    n·p does not exceedn·q exceedsn·p exceedsn·q does not exceed

fourth blank (Enter an exact number.)

(c)

Suppose

• n = 56 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

YesNo    

second blank

cancannot    

third blank

both n·p and n·q exceedn·p and n·q do not exceed    n·p does not exceedn·q exceedsn·p exceedsn·q 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 = mu sub p hat =

σ_p̂ = sigma sub p hat =

Answer 1

a)

(a)

• Suppose n = 42 and
• p = 0.18.

n·p =42   *   0.180   =           7.560

n·q = 34.44


___Yes,__, p̂ __can___ be approximated by a normal random variable because ___ both n·p and n·q exceed__ __5___.



What are the values of μ_p̂ and σ_p̂? (For each answer, enter a number. Use 3 decimal places.)
μ_p̂ = mu sub p hat =0.18


σ_p̂ = sigma sub p hat = √( p(1-p)/n ) =    0.059

b)

• Suppose n = 25 and
• p = 0.15.

n·p = 3.75

n·q = 21.25


___NO,__, p̂ __canmot___ be approximated by a normal random variable because ___   n·p does not exceedn·q exceeds__ __5___.



What are the values of μ_p̂ and σ_p̂? (For each answer, enter a number. Use 3 decimal places.)
μ_p̂ = mu sub p hat =0.15


σ_p̂ = sigma sub p hat = √( p(1-p)/n ) = 0.071

c)

• Suppose n = 56 and
• p = 0.21.

n·p = 11.76

n·q = 44.24


___YES,__, p̂ __can___ be approximated by a normal random variable because ___  both n·p and n·q exceed __ __5___.



What are the values of μ_p̂ and σ_p̂? (For each answer, enter a number. Use 3 decimal places.)
μ_p̂ = mu sub p hat =0.21


σ_p̂ = sigma sub p hat = √( p(1-p)/n ) = 0.054