Question

 

Assume a binomial probability distribution has

p = 0.70

and

n = 400.

(a)

What are the mean and standard deviation? (Round your answers to two decimal places.)

mean=

standard deviation =

(b)

Is this situation one in which binomial probabilities can be approximated by the normal probability distribution? Explain.

Yes, because np ≥ 5 and n(1 − p) ≥ 5.

Yes, because n ≥ 30.     

Yes, because np < 5 and n(1 − p) < 5.

No, because np ≥ 5 and n(1 − p) ≥ 5.

No, because np < 5 and n(1 − p) < 5.

(c)

What is the probability of 260 to 270 successes? Use the normal approximation of the binomial distribution to answer this question. (Round your answer to four decimal places.)

(d)

What is the probability of 290 or more successes? Use the normal approximation of the binomial distribution to answer this question. (Round your answer to four decimal places.)

(e)

What is the advantage of using the normal probability distribution to approximate the binomial probabilities?

The advantage would be that using the normal probability distribution to approximate the binomial probabilities makes the calculations more accurate.The advantage would be that using the normal probability distribution to approximate the binomial probabilities increases the number of calculations.     The advantage would be that using the the normal probability distribution to approximate the binomial probabilities reduces the number of calculations.The advantage would be that using the normal probability distribution to approximate the binomial probabilities makes the calculations less accurate.

Answer 1

Solution:

Given that,

P = 0.70

1 - P = 1 - 0.70 = 0.30

n = 400

Here, BIN ( n , P ) that is , BIN (400 , 0.70)

then,

n*p = 400*0.70 = 280 ≥ 5

n(1- P) = 400*0.30 = 120 ≥ 5

According to normal approximation binomial,

X Normal

a)

Mean = = n*P = 400*0.70 = 280

Standard deviation = =n*p*(1-p)= 400*0.70*0.30 = 84

b)

Is this situation one in which binomial probabilities can be approximated by the normal probability distribution

Yes, because np ≥ 5 and n(1 − p) ≥ 5.

Using countinuity correction factor

c)

P(a < X< a) = P( a - 0.5 < X < a + 0.5)

P(259.5 < x < 270.5) = P((259.5 - 280) / 84) < (x - ) /  < (270.5 - 280) / 84) )

= P(-2.237 < z < -1.037)

= P(z < -1.037) - P(z < -2.237)

= 0.1499 - 0.0126

Probability = 0.1373

d)

P(X a ) = P(X > a - 0.5)

P(x > 289.5) = 1 - P(x < 289-5)

= 1 - P((x - ) / < (289.5 - 280) / 84)

= 1 - P(z < 1.037)

= 1 - 0.8501

= 0.1499

Probability = 0.1499

e)

The advantage would be that using the normal probability distribution to approximate the binomial probabilities makes the

calculations more accurate.