Please Validate the answers: Normal Distribution and Probability

#1
Question 1
a) Sales records over several years show that the number of Daily Telegraph newspapers sold by a Delhi newsagency on Saturdays follows a normal distribution with a mean of 218 copies with a standard deviation of 12.5 copies.
Calculate the probability that the newsagency sells fewer than 180 copies of the Daily Telegraph on a particular Saturday.

Ans:
We have to find z value and from that we will find probability.
μ=Mean, σ=Standard Deviation,
z=z-value,X=Normal Random Variable
Z=(X-μ)/σ
=(180-218)/12.5
=-3.04 (Negative sign indicates that value lies on left side of mean in normal distribution curve)
Corresponding to z = -3.04, from z table we find that area under curve will be 0.00118, hence probability will be 0.118%.

b) Calculate the probability (to two decimal places) that on each of three successive Saturdays there will be at least 200 copies of the Daily Telegraph sold by the newsagency. State any assumption you are making.

Ans:
z=(200-218)/12.5
=-1.44
Corresponding to z = -1.44, from z table we find that area under the curve will be 0.07493.
Probability for 3 consecutive Saturday = (0.07493)^3*100 %= 0.04 %




c) The newsagency wants to order an amount of Daily Telegraph newspapers for a particular Saturday so that there is at least a 99% chance that the entire order will all be sold. What is the maximum number of newspapers that should be ordered?
Ans:
Area under curve for 1% = 0.01017
Z-value corresponding to area under curve of 0.01017= -2.32
-2.32=(X-218)/12.5
X= 189

Question 2
A quality control inspector is concerned about the amount of petrol that is dispensed at the petrol pumps of a particular brand of service station across Delhi. There have been claims by a number of motorists that these pumps are giving misleading readings. The brand manager claims that, while there may be some variation in the amount of petrol actually dispensed, the overall mean across all pumps at all their stations is 20 litres when the pump display reads 20 litres.
To test the accuracy of the claim, the inspector selects a random sample of one pump from each of a random sample of 64 service stations and fills a measuring can with petrol, each time until 20 litres is showing on the pump display. After allowing for evaporation, the actual amounts of petrol that were in the 64 cans had a mean volume of 19.60 litres with a standard deviation of 800 ml.

a) Construct a 95% confidence interval for the population mean amount of petrol and clearly interpret your findings. (Include the value of the parameters and the formula you use. Do not use Minitab.)

Ans:
For constructing 95% confidence interval:
We need to find the z value corresponding to area under curve of 0.025, so z=1.96
Here mean, μ = 19.6, S.D, σ = 0.8
LSL=( μ-z*σ)= (19.6-1.96*0.8)=17.432
USL=( μ+z*σ)= (19.6+1.96*0.8)=20.568

b)Use your interval in (a) to determine if you believe the manager’s claim. Clearly explain your reasoning and interpret your findings.

Ans:
We need to find the percentage of people receiving less than 20 ltrs of petrol.

Z= (20-19.6)/0.8=0.5
For z=0.5,Area under curve=0.30854 (right side area of the curve)
People receiving less than 20 ltrs = (1-0.30854)*100%=69.146 %

This is a high percentage of people receiving petrol less than 20 ltrs due to high variation in measurement and loss due to evaporation.
This needs to be fixed by evaporation compensation in measurement and calibration of dispensing meters.


c) What is the probability that you have made an error in your conclusion in (b)?
 
#2
For part b) of problem 1, it seems as though you found the probability that the agency sells less than the stated number of copies. The question asks for at least that number of copies.