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 = 26 and p = 0.39  (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 two answer blanks. A blank is represented by _____.)

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

first blank

n·p does not exceed

n·p exceeds   

both n·p and n·q exceed

n·q does not exceed

n·q exceeds

n·p and n·q do not exceed

second 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 two answer blanks. A blank is represented by _____.)

no, p̂ cannot be approximated by a normal random variable because _____ _____.

first blank

n·p does not exceed

n·p exceeds    

both n·p and n·q exceed

n·q does not exceed

n·q exceeds

n·p and n·q do not exceed

second blank (Enter an exact number.)

(c) Suppose n = 64 and​​​​​​​ p = 0.39. (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 two answer blanks. A blank is represented by _____.)

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

first blank

n·p does not exceed

n·p exceeds    

both n·p and n·q exceed

n·q does not exceed

n·q exceeds

n·p and n·q do not exceed

second 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) Suppose n = 26 and p = 0.39  (For each answer, enter a number. Use 2 decimal places.)
n·p = 26*0.39 = 10.14
n·q = 26*(1-0.39) = 15.86

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

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

both n·p and n·q exceed 10.
What are the values of μ_p̂ and σ_p̂? (For each answer, enter a number. Use 3 decimal places.)
μ_p̂ = 0.39

σ_p̂ = sqrt(0.39*(1-0.39)/26) = 0.096

(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 two answer blanks. A blank is represented by _____.)
n·p = 25*0.15 = 3.75
n·q = 25*(1-0.15) = 21.25

no, p̂ cannot be approximated by a normal random variable because _____ _____.

n·p does not exceed 10.

(c) Suppose n = 64 and p = 0.39. (For each answer, enter a number. Use 2 decimal places.)
n·p = 64*0.39 = 24.96
n·q = 64*(1-0.39) = 39.04

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

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

both n·p and n·q exceed 10.


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 = 0.39

σ_p̂ = sigma sub p hat = sqrt(0.39*(1-0.39)/64) = 0.061