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 = 0.40. (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·q exceeds    

• n·p and n·q do not exceed

• n·p exceeds

• n·p does not exceed

• both n·p and n·q 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·q does not exceed

• n·q exceeds    

• n·p and n·q do not exceed

• n·p exceeds

• n·p does not exceed

• both n·p and n·q exceed

fourth blank (Enter an exact number.)

(c) Suppose n = 47 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 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·q exceeds    

• n·p and n·q do not exceed

• n·p exceeds

• n·p does not exceed

• both n·p and n·q 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 = 42

p = 0.40

q = 1 - p = 0.60

np = 42 X 0.40 = 16.80

nq = 42 X 0.60 = 25.20

Yes, can be approximated by a normal random variable because both np and nq exceed 10.

= 0.490

(b)

n = 25

p = 0.15

q = 1 - p = 0.85

np = 25 X 0.15 = 3.75

nq = 25 X 0.85 = 21.25

No, cannot be approximated by a normal random variable because nq exceed 10 . np does not 10

(c)

n = 47

p = 0.39

q = 1 - p = 0.61

np = 47 X 0.39 = 18.33

nq = 47 X 0.61 = 28.67

Yes, can be approximated by a normal random variable because both np and nq exceed 10.

= 0.390