Question

Suppose that Stephen is the quality control supervisor for a food distribution company. A shipment containing many thousands of apples has just arrived. Unknown to Stephen, 13% of the apples are damaged due to bruising, worms, or other defects. If Stephen samples 10 apples from the shipment, use the binomial distribution to estimate the probability that his sample will contain at least one damaged apple. Select the true statement.

A. Stephen can use a sample of size 10 to reliably determine if the truck load contains damaged apples.
B. Stephen could use a sample of size less than 10 to reliably determine if the truck load contains damaged apples.
C. Stephen could have used a normal approximation to determine this probability.
D. A sample of size 10 is too small to reliably determine if the truck load contains damaged apples.
E. Stephen should sample with replacement so that the probability is exactly binomial.

Answer 1

D. A sample of size 10 is too small to reliably determine if the truck load contains damaged apples.

Explanation:

We have probability of having damaged apple = 13%

= 13/100

= 0.13

Sample size n = 10

Using the binomial distribution

P(X=x) = nCx * P^x(1-P)^n-x

Probability sample will have at least one damaged apple

= P(X>=1) = 1-p(x<1)

= 1 - P(X=0)

= 1-(10C0 *P⁰(1-0.13)¹⁰

= 1 - 1(1-0.13)¹⁰

= 1 - 0.87¹⁰

= 1 - 0.2484

= 0.7516

The answer is option D. A sample of size 10 is too small to reliably determine if the truck load contains damaged apples.