Question

If np greater than or equals 5 and nq greater than or equals 5, estimate Upper P (more than 9 ) with n = 13 and p = 0.3 by using the normal distribution as an approximation to the binomial distribution; if np less than 5 or nq less than 5, then state that the normal approximation is not suitable.

Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A. P(more than 9) = ________

B. The normal distribution cant be used

Answer 1

Solution

Given that ,

p = 0.3

1 - p = 1 - 0.3 = 0.7

n = 13

np = 0.3 * 13 = 3.9 < 5

nq = 0.7 * 13 = 9.1 > 5

the normal approximation is not suitable .

Using binomial probability formula ,

P(X = x) = ((n! / x! (n - x)!) * p^x * (1 - p)^{n - x}

P(X > 9) = P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13

= ((13! / 10! (3)!) * 0.3¹⁰ * (0.7)³ + ((13! / 11! (2)!) * 0.3¹¹ * (0.7)² +

((13! / 12! (1)!) * 0.3¹¹ * (0.7)¹ + ((13! / 13! (0)!) * 0.3¹³ * (0.7)⁰

=  0.00065

P(X > 9) = 0.00065