Question

1. A large supermarket produces its own store brand hand cream. Last year the machines used to produce the cream were refurbished. Since then the supermarket has been receiving a lot of complaints from unhappy customers that there is not enough cream in the container. The machine is calibrated for 100ml with a standard deviation of 5ml. The label on the cream says there is minimum of 98ml.

1. What is the probability that there will be less cream in the container than specified on the label? Considering your calculations, is there cause for concern? (7 points, no more than 50 words)
2. The management has decided to change the labels on the hand cream and requires no more than 5% of containers to be under the advertised contents. What should be printed on the label as minimum amount? (3 points)
3. If you were advising the supermarket, what would you advise them to do? What should the company consider? What data would they need? (4 points, no more than 150 words)

2. A medical researcher has found that in a group of 60 employees who use the same canteen (cafeteria) in their offices, only 20 are healthy and every other person has either high blood pressure, high level of cholesterol or both. If 20 people have high blood pressure and 35 have high level of cholesterol.

1. How many people have only high blood pressure? (2 point)
2. How many people have only high level of cholesterol? (2 point)
3. How many people with high blood pressure have high level of cholesterol too? (2 point)

If a person is selected randomly from this group, what is the probability that he/she

4. Is healthy? (2 point)
5. Has high blood pressure, given that they have high level of cholesterol? (3 point)
6. Has high level of cholesterol given that they do not have high level of blood pressure? (3 point)
7. If you were the medical researcher, what would you advise the company in regard to the food served at the cafeteria? Does it seem to be health? (5 points, no more than 200 words)

Answer 1

Note: Final answers are highlighted in colour.

1.

a) = 100, = 5

By converting the question into Standard normal distribution,

Z=

P(X<98)= P(Z<)= P(Z<-0.4)=0.3446 (From the below Z table)

This means about 34.46% of the packets will contain less than specified on the label. This is a real concern.

b) Probability should be 0.05

That means P(X<a)= 0.05

We need to identify such a

Z value for probability of 0.05 from the Z table= -1.645

= -1.645

X= (-1.645)*(5)+ 100= 91.775

So they can change the label specifying the minimum value as 91.7

c)

I would advise the company to set its machine again in such a way that either mean increases from 100 or standard deviation reduces from 5. This way we can be more reliably promiss the minimum amount.

Also they will need to collect the number of complaints from customers and need to collect the amount of cream they received. In this way we can know the amount at which customers are getting frustrated and we can adjust the cream level to a satisfactory level.

Profit vs Customer satisfaction analysis should also be done.

2.

Total People(T)= 60

Healthy(H)= 20, Either High Blood Pressure or Cholestrol(BP U Ch)= 40

High Blood Pressure(BP)= 20, High Cholestrol(Ch)= 35.

a) People with Both High Blood Pressure and High Cholestrol(BP Ch)= (BP + Ch) - (BP U Ch)= 20+35-40=15

Therefore People with only high blood pressure()= BP - (BP Ch)= 20-15= 5

b)People that have only high level of cholesterol ()= Ch- (BP Ch)= 35- 15= 20

c)

people with high blood pressure that have high level of cholesterol too= (BP Ch) = 15

3.

a) If a person is selected randomly from this group, probability that he/she is healthy= H/T= 20/60= 1/3

b)Has high blood pressure, given that they have high level of cholesterol= P(BP|Ch)=

= (BP Ch)/ T= 15/60= 1/4

P(Ch)= Ch/T= 35/60

Therefore P(BP|Ch)= = 15/35= 0.42857

c)Has high level of cholesterol given that they do not have high level of blood pressure==

= 1-P(BP)= 1- BP/T= 1-20/60= 2/3

= = 20/60= 1/3

Therefore = = 1/2= 0.5

d) As the 2/3rd of the people eating at the cafetaria are unhealthy, I feel there is some issue with the food being served there.

If I were the medical researcher, I would urge the company to establish frequent quality checks and revoke the vendorship of some of the vendors that are producing food with bad quality.

I would ask the company to take timely customer feedbacks and assess them time to time.