Answer to Question #107503 in Statistics and Probability for yoyo

Question

Restaurant “King Steak” offers a set lunch of a 150 grams steak for $100. The steaks are cut by a machine
and the weight of a steak can be assumed to be normally distributed with mean 152 grams and standard
deviation 1.7 grams.
(a) What is the probability that a customer gets a steak that weighs less than 150 grams?
(b) There is a 4% chance that a customer gets a steak that weighs less than W grams. What is the value of
W?
(c) There is a table with 8 customers, everyone orders this set lunch. What is the probability that there are
exactly 2 customers whose steaks weigh less than 150 grams?
(d) Sometimes there are customers ordering two sets of lunch. Use T to denote the total weight of two
steaks. Find the mean, variance, and standard deviation of T.

Answer 1

Let $X=$ the weight of a steak:$X\sim (N, \sigma^2)$

Then $Z=\dfrac{X-\mu}{\sigma}\sim N(0,1)$

Given that $\mu=152\ g, \sigma=1.7\ g$

(a) What is the probability that a customer gets a steak that weighs less than 150 grams? 

$$P(X<150)=P(Z<{150-152\over 1.7})\approx$$ $$\approx P(Z<-1.17647)\approx0.1197$$

(b) There is a 4% chance that a customer gets a steak that weighs less than W grams. What is the value of W? 

$$P(X<W)=P(Z<{W-152\over 1.7})=0.04$$

$${W-152\over 1.7}\approx-1.7507$$

$$W\approx149\ g$$

(c) There is a table with 8 customers, everyone orders this set lunch. What is the probability that there are exactly 2 customers whose steaks weigh less than 150 grams? 

Let $Y=$ the number of customers whose steaks weigh less than 150 grams: $Y\sim Bin (n,p)$

$$P(Y=y)=\binom{n}{y}p^y(1-p)^{n-y}$$

Given that $n=8, p=0.1197$

$$P(Y=2)=\binom{8}{2}(0.1197)^2(1-0.1197)^{8-2}\approx0.1867$$

(d) Sometimes there are customers ordering two sets of lunch. Use T to denote the total weight of two

steaks. Find the mean, variance, and standard deviation of T.

$$T\sim N(\mu+\mu,\sigma^2+\sigma^2)$$

Then

$$\mu_T=150\ g+150\ g=300\ g$$

$$Var(T)=\sigma_T^2=1.7^2+1.7^2=5.78$$

$$\sigma_T=\sqrt{5.78}\ g=1.7\sqrt{2}\ g\approx2.4\ g$$