aluminium sheets used to make beverage cans have thicknesses that are normally d

Question

aluminium sheets used to make beverage cans have thicknesses that are normally distributed with mean 10 and standard deviation 1.3. a particular sheet is 10.8 thousandths of an inch thick.

5- deduce from the previous question, the area under the normal curve to the left of z=-1.38. then check your answer from the z table.
6- find the area under the normal curve between z=-1.38 and z= 1.38. can we get the answer basing on questions 4 and 5? if yes, how?
7- what z score corresponds to the 75th percentile? find the thickness of the sheet in the original units of thousandths of inches.
8- what z score corresponds to the 25th percentile?
9- can we answer question 8 graphically? if yes, explain.
10- what thickness of the sheet (in the original units of thousandths of inches) corresponds to the median (50th percentile)
please answer the rest questions with details and steps clearlly.

Explanation / Answer

Question 5

Here, we have to find P(Z<-1.38)

By using the normal table or excel, we get

P(Z<-1.38) = 0.083793322

Question 6

We have to find P(-1.38<Z<1.38)

P(-1.38<Z<1.38) = P(Z<1.38) – P(Z<-1.38)

P(Z<1.38) = 0.916206678

P(Z<-1.38) = 0.083793322

P(-1.38<Z<1.38) = P(Z<1.38) – P(Z<-1.38)

P(-1.38<Z<1.38) = 0.916206678 - 0.083793322

P(-1.38<Z<1.38) = 0.832413355

Required probability = 0.832413355

Question 7

X = Mean + Z*SD

We are given

Mean = 10

SD = 1.3

For 75th percentile, critical Z value = 0.67448975

X = 10 + 0.67448975*1.3

X = 10.87683668

75th percentile = 10.87683668

Question 8

We have to find P(Z<z) = 0.25

By using z table or excel, we get

Z = -0.67448975

Question 9

Yes, we can answer question 8 graphically as the distribution is symmetric and area at both ends are equal, so we can easily calculate the required values for z or percentiles from the available information.

Question 10

Here, we have to find thickness of the sheet X for the median or 50th percentile.

X = Mean + Z*SD

We are given

Mean = 10

SD = 1.3

For 50th percentile, critical Z value =0.00

X = 10 + 0.00*1.3 = 10 + 0.00 = 10

Required answer: Median = 50th percentile = 10

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