Question

Suppose 36.5% of households have a dog. Consider a random sample of 14 households. (a.) Find the probability that exactly 4 in the sample have a dog. (b.) Give the mean and variance of X, where X denotes the number out of 14 households that have a dog.

Answer 1

Solution

Given that ,

p = 36.5% = 0.365

1 - p = 1 - 0.365 = 0.635

n = 14

(a)

x = 4

Using binomial probability formula ,

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

P(X = 4) = ((14! / 4! (14 - 4)!) * 0.365⁴ * (0.635)^{14 - 4}

=  ((14! / 4! (10)!) * 0.365⁴ * (0.635)¹⁰

= 0.1894

Probability = 0.1894

(b)

mean = n * p = 14 * 0.365 = 5.11

standard deviation = n * p * q = 14 * 0.365 * 0.635 = 3.245