Answer to Question #149303 in Statistics and Probability for JAMES

Question

Weights of potatoes are normally distributed with a mean of 60 Kilograms and a standard deviation of 12 kilograms. A manager of XYZ supermarket receives a consignment of 10,000 bags of these potatoes.

Required:

i. What is the probability that a random bag will weigh more than 72 kilograms? (3 marks)

ii. What is the probability that a random bag will weigh between 54 kilograms and 78 kilograms? (4 marks)

iii. If any bag, weighing less than 54 kilograms is considered underweight. What are the number of bags that are likely to be declared underweight?

Answer 1

Let $X=$ the weight of the bag: $X\sim N(\mu, \sigma^2).$

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

Given $\mu=60kg, \sigma=12kg$

1.

$$P(X>72)=1-P(X\leq72)=1-P(Z\leq\dfrac{72-60}{12})$$

$$1-P(Z\leq1)\approx0.1587$$

2.

$$P(54<X<78)=P(X<78)-P(X\leq54)$$

$$=P(Z<\dfrac{78-60}{12})-P(Z\leq\dfrac{54-60}{12})$$

$$= P(Z\leq1.5)-P(Z<-0.5)$$

$$\approx0.9332-0.3085=0.6247$$

3.

$$P(X<54)=P(Z<\dfrac{54-60}{12})=P(Z<-0.5)$$

$$\approx0.30853754$$

$$0.30853754\cdot10000=3085$$

3085 bags.