Question

Q1. From a reputed store’s record it was found that weight of sugar bags sold by it is normally distributed

       with mean 1.0 lb. and standard deviation of 40 g.

a. What is the probability/chances/likelihood that

i. a bag selected randomly will be lighter than 400 g (0.0901)

ii. a bag selected randomly will be heavier than 600 g (0.00013)

         iii. the weight will be more than 463.5 g. if a customer bought one bag,? (0.4013)

       b What proportion of customers are expected to buy heavier than 470 g,

           if one customer can buy one bag? (0.3409)

       c. How many bags are expected to be heavier than 470 g,

          if it has a stock of 1000 bags in the store? (341 bags)

       d. What proportion of bags will be heavier than 1.0 lb.? (0.5)

       e. A random sample of 64 bags was selected.

         i. mean weight of the sample will be more than 461.8 g.? (0.0505)

         ii. the mean weight will between 443.8 g. to 463.4 g.(0.95)

       f. From what weight

           i. 25% bags will be heavier? (480.6 g.)    ii. 25% bags will be lighter? (426.6 g.)

Answer 1

1 pound = 453.592 grams ( you make take up-to as many decimal places as you like.. However there will be some very little variations in the answer but that would be okay )

So, Mean = 1 lb = 453.592 grams , standard deviation = 40 grams

Step1 find z value.

Z = (X - population mean) / standard deviation

Step 2

Compare z value with standard normal distribution z table to find the cumulative probability.

Part a

i) X =400 grams

Z = (400 - 453.592 )/ 40 = -1.3398 = - 1.34

Probability that a random bag weighs less than 400 grams = P(X< 400) = P( z ‹ -1.34)

= 0.0901 or 9.01 %

ii) X =600

Z = (600- 453.592)/40 = 3.6602 =3.66

Probability that a bag weighs more than 600 grams = P(X>600) = P(z >600) = 1 - P(z< 3.6602)

= 1 - 0.9999 = 0.0001

Note in z table probabillities are given upto a maximum of 4 decimal places only

For more accuracy, you may use online calculator

III)

Z =(463.5-453.592)/40 = 0.2477 = 0.25 ( taking the closest z value available on z table)

P( X > 463.5 ) = 1- P ( z < 0.25)

= 1- 0.5987

= 0.4013

b) Z = (470 - 453.592) / 40 = 0.4102 = 0.41

P(z > 0.41) = 1- P(z < 0.41) = 1 - 0.6591 = 0.3409

c) Number of bags expected to be heavier than 470 grams

= probabillity of a bag being heavier than 470 grams × number of bags

= 0.3409 × 1000 = 340.9 = 341 bags

D)

Z = (453.592-453.592)/40 =0

And proabability of a bag heavier than 1 lb or 453.592 grams = P(z > 0) = 0.5.

e) i)  sample size =64 ,

Sample standard deviation = 40 / 641/2 = 40/8 =5

Z = (461.8 - 453.592)/5 = 1.6416 = 1.64

P( z > 1.64) = 1 - P( z< 1.64) = 1- 0.9495 = 0.0505

e)ii)

So for sample mean = 463.4,

Z = (463.4 - 453.592)/ 5 = 1.9616 =1.96

For sample mean = 443.8

Z = (443.8 - 453.592)/5 = - 1.9584 = -1.96

P( - 1.96 < z < 1.96 ) = P(z<1.96) - P( z< -1.96)

= 0.975 - 0.025 = 0.95

F)

For 25% heavier, we have to find z score such that P(X > some value) is 0.25

From z table, such z value is 1.96

So, weight = mean + z × standard deviation  

= 453.592 + 1.96 × 40 = 531.992 = 552 grams

And for 25 % lighter bag weight,

Z score such that probability of getting a lighter bag than this is 25%

Such a z value = - 1.96

So, weight = 453.592 - 1.96× 40 = 375.192 = 375.2 grams