Question

# (i) A binomial random variable has a mean equal to 187 and a standard deviation of...

(i) A binomial random variable has a mean equal to 187 and a standard deviation of 12.

(a) Find the value of n. (Give your answer correct to the nearest whole number.)

(b) Find the value of p. (Give your answer correct to four decimal places.)

(ii) he probability of success on a single trial of a binomial experiment is known to be 0.3. The random variable x, number of successes, has a mean of 75.

(a) Find the number of trials involved in this experiment. (Give your answer correct to the nearest whole number.)

(b) Find the standard deviation of x. (Give your answer correct to two decimal places.)

i)

For binomial distribution ,

Given, Mean = 187 , Standard deviation = 12

That is

np = 187 and Sqrt( np(1-p) ) = 12

Put the value of np from first equation in second and solve for p

Sqrt( np (1-p) ) = 12

Sqrt ( 187 ( 1 - p) ) = 12

Take square of both sides

187 ( 1 - p) = 144

1 - p = 144 / 187

p = 1 - 144/187

p = 0.2299

So,

np = 187

n * 0.2299= 187

n = 813

a)

n = 813

b)

p = 0.2299

ii)

a)

Given, p = 0.3 and mean = np = 75

So,

n * 0.3 = 75

n = 75 / 0.3

n = 250

b)

Standard deviation = sqrt( np(1-p) )

= sqrt( 250 * 0.3 * 0.7)

= 7.25