Question

If np >= 5 and nq >= 5​, estimate Upper P( fewer than 6) with n =13 and p= 0.5 by using the normal distribution as an approximation to the binomial​ distribution; if np < 5 or nq < 5, then state that the normal approximation is not suitable.

P(fewer than 6) = ?

Answer 1

Solution:

Given ,

n = 13

p = 0.5

X follows Binomial(13 , 0.5)

q = 1 - p = 1 - 0.5 = 0.5

n p = 13 * 0.5 = 6.5 > 5

n q = 13 * 0.5 = 6.5 > 5

So , np >= 5 and nq >= 5​

We can use normal distribution as an approximation to the binomial​ distribution.

According to normal approximation binomial,

X Normal

Mean = = n*p = 6.5

Standard deviation = =n*p*(1-p) = [13 * 0.5 * 0.5] = 1.80277563773

Now

P(fewer than 6)

= P(X < 6)

Using continuity correction

= P(X < 6 - 0.5)

= P(X < 5.5)

= P[(X - )/ <  (5.5 - )/]

=  P[Z <  (5.5 - 6.5)/1.80277563773]

= P[Z < -0.55]

= 0.2912 ... ( use z table)

P(fewer than 6) = 0.2912