Answer to Question #88387 in Statistics and Probability for Randi Carpenter

Question

suppose that 1 out of 10 plasma televisions shipped with a defective speaker. out of a shipment of n=400 plasma televisions. find the probability that there are
a) at most 40 with defective speakers (Hint: Use the dishonest- coin principle with P= 1/10=0.1 to the find the mean and standard deviation.
B) Most than 52 with defective speakers

Answer 1

Dishonest-Coin Principle

Let X denote the number of heads in n tosses of a coin (assume $n\geq30)$. Let p denote the probability of heads on each toss.Then, X has an approximately normal distribution with mean $\mu=n\cdot p$ and standard deviation $\sigma =\sqrt{n\cdot p\cdot (1-p)}$.

We have that $p=0.1, n=400$. Then $\mu=400\cdot 0.1=40, \sigma =\sqrt{400\cdot 0.1\cdot (1-0.1)}=6.$

a) at most 40 with defective speakers

$$z={x-\mu \over \sigma}={40.5-40 \over 6}=0.083333$$

$$P(X\leq40)=P(X<40.5)=P(Z<0.083333)=0.533207$$

b) Most than 52 with defective speakers

$$z={x-\mu \over \sigma}={52.5-40 \over 6}=2.083333$$

$$P(X>52)=P(X>52.5)=0.018610$$