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 = 33 and
• p = 0.29.

(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

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

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

fourth blank (Enter an exact number.)

(c)

Suppose

• n = 65 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

YesNo    

second blank

cancannot    

third blank

n·q does not exceedn·p exceeds    both n·p and n·q exceedn·q exceedsn·p and n·q do not exceedn·p 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 =

11.

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

In a random sample of 63 professional actors, it was found that 44 were extroverts.

(a)

Let p represent the proportion of all actors who are extroverts. Find a point estimate for p. (Round your answer to four decimal places.)

(b)

Find a 95% confidence interval for p. (Round your answers to two decimal places.)
lower limit         
upper limit         

Give a brief interpretation of the meaning of the confidence interval you have found.

We are 95% confident that the true proportion of actors who are extroverts falls outside this interval.We are 95% confident that the true proportion of actors who are extroverts falls within this interval.    We are 5% confident that the true proportion of actors who are extroverts falls within this interval.We are 5% confident that the true proportion of actors who are extroverts falls above this interval.

(c)

Do you think the conditions n·p > 5 and n·q > 5 are satisfied in this problem? Explain why this would be an important consideration.

Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately normal.No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately binomial.    No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately normal.Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately binomial.

Answer 1

Solution:

a) n = 33 and p = 0.29

np = 33 × 0.29 = 9.57

nq = 33 × (1 - 0.29) = 23.43

If np > 5 and nq > 5 then we can approximate p̂ by a normal distribution.

Yes p̂ can be approximated by a normal random variable because both np and nq exceeds 5.

μ_p̂ = p = 0.29

b) n = 25 and p = 0.15

np = 25 × 0.15 = 3.75

nq = 25 × (1 - 0.15) = 21.25

If np > 5 and nq > 5 then we can approximate p̂ by a normal distribution.

No p̂ can not be approximated by a normal random variable because np does not exceeds 5.

μ_p̂ = p = 0.15

c) n = 65 and p = 0.23

np = 65 × 0.23 = 14.95

nq = 65 × (1 - 0.23) = 50.05

If np > 5 and nq > 5 then we can approximate p̂ by a normal distribution.

Yes p̂ can be approximated by a normal random variable because both np and nq exceeds 5.

μ_p̂ = p = 0.23