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 = 34 and p = 0.25. (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·q does not exceed n·p and n·q do not exceed both n·p and n·q exceed n·p exceeds n·p does not exceed n·q exceeds 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·q does not exceed n·p and n·q do not exceed both n·p and n·q exceed n·p exceeds n·p does not exceed n·q exceeds fourth blank (Enter an exact number.) (c) Suppose n = 64 and p = 0.16. (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·q does not exceed n·p and n·q do not exceed both n·p and n·q exceed n·p exceeds n·p does not exceed n·q exceeds 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

Rule :

If np and nq > 5. So,  p̂ is a normal random variable

(a) Suppose n = 34 and p = 0.25

n*p = 34*0.25 = 8.5 > 5

n*q = n*(1-p) = 34 * (1-0.25) = 25.5 > 5

np and nq > 5. So,  p̂ is a normal random variable

μp̂ = np = 8.5

σp̂ = SQRT(npq) = SQRT(34*0.25*0.75) = 2.525

(b) Suppose n = 25 and p = 0.15

n*p = 25*0.15 = 3.75 < 5

n*q = n*(1-p) = 25 * (1-0.15) = 21.25 > 5

np < 5 and nq > 5. So,  p̂ is not a normal random variable

μp̂ = np = 3.75

σp̂ = SQRT(npq) = SQRT(25*0.15*0.85) = 1.785

(c) Suppose n = 64 and p = 0.16

n*p = 64*0.16 = 10.24 > 5

n*q = n*(1-p) = 64 * (1-0.16) = 53.76 > 5

np > 5 and nq > 5. So,  p̂ is a normal random variable

μp̂ = np = 10.24

σp̂ = SQRT(npq) = SQRT(64*0.16*0.84) = 2.933