Question

Assume a binomial probability distribution has p = 0.80 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. No, because np ≥ 5 and n(1 − p) ≥ 5. Yes, because np ≥ 5 and n(1 − p) ≥ 5. Yes, because np < 5 and n(1 − p) < 5. No, because np < 5 and n(1 − p) < 5. Yes, because n ≥ 30. (c) What is the probability of 300 to 310 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 330 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 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 more accurate. The advantage would be that using the normal probability distribution to approximate the binomial probabilities makes the calculations less accurate. How would you calculate the probability in part (d) using the binomial distribution. (Use f(x) to denote the binomial probability function.) P(x ≥ 330) = f(0) + f(1) + + f(329) + f(330) P(x ≥ 330) = f(331) + f(332) + f(333) + f(334) + + f(400) P(x ≥ 330) = 1 − f(329) − f(330) − f(331) − f(332) − − f(400) P(x ≥ 330) = f(330) + f(331) + f(332) + f(333) + + f(400) P(x ≥ 330) = f(0) + f(1) + + f(328) + f(329)

Answer 1

(a)

Mean, =np =400*0.80 =320

Standard deviation, = =8

(b)

Yes, because np ≥ 5 and n(1 − p) ≥ 5

(np =320 > 5 and n(1-p) =400*0.20 =80 > 5).

(c)

Formula: f(X =r) =C(n, r).p^r.(1-p)^n-r

The probability of 300 to 310 successes =f(300 X 310) =f(300)+P(301)+..........+f(310) =0.1122

[f(300 X 310) =1 - f(X < 300) - f(X > 310) =1 - 0.0062 - 0.8816 =0.1122].

Using normal approximation of the binomial distribution:

Z =(x - ​​​​​​)/

P(300 X 310) =P[(299.5 - 320)/8 ​​​​​Z (310.5 - 320)/8] =P(-2.5625 Z -1.1875) =0.1123

(Continuity correction is made).

(d)

The probability of 330 or more successes =P(X 330) =P[(Z (329.5 - 320)/8] =P(Z 1.1875) =0.1175

(e)

The advantage would be that using the the normal probability distribution to approximate the binomial probabilities reduces the number of calculations.

f(X 330) =f(330)+f(331)+..........+f(400) =0.1164