Question

Making handcrafted pottery usually takes two major steps:wheel throwing and firing. The time of wheel throwing and thetime of firing are normally distributed random variables with meansof 40 min and 60 min and standard deviations of 2 min. and 3 min,respectively.

(a) What is the probability that a piece of pottery will befinished within 95 minutes?


(b) What is the probability that it will take longer than 110minutes?

Answer 1

a)
`P(R<95)=P((R-\mu)/(\sigma)<(95-\mu)/(\sigma))=P(Z<(95-100)/(3.606))=P(Z<-1.387)`

And we can find this probability using the normal standard table or excel and we got:

`P(z<-1.387)=0.0827`

b)
`P(R>110)=P((R-\mu)/(\sigma)>(110-\mu)/(\sigma))=P(Z>(110-100)/(3.606))=P(Z>2.774)`

And we can find this probability using the complement rule and the normal standard table or excel and we got:

`P(z>2.774)=1-P(Z<2.774) = 1-0.9972 = 0.0028`

Explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

Part a

Let X the random variable that represent the time for the step 1 and Y the time for the step 2, we define the random variable R= X+Y for the total time and the distribution for R assuming independence between X and Y is:

`R \sim N(40+60 = 100,√(2^2 +3^2)= 3.606 s)`

Where
`\mu=65.5` and
`\sigma=2.6`

We are interested on this probability

`P(R<95)`

And the best way to solve this problem is using the normal standard distribution and the z score given by:

`z=(R-\mu)/(\sigma)`

If we apply this formula to our probability we got this:

`P(R<95)=P((R-\mu)/(\sigma)<(95-\mu)/(\sigma))=P(Z<(95-100)/(3.606))=P(Z<-1.387)`

And we can find this probability using the normal standard table or excel and we got:

`P(z<-1.387)=0.0827`

Part b

`P(R>110)=P((R-\mu)/(\sigma)>(110-\mu)/(\sigma))=P(Z>(110-100)/(3.606))=P(Z>2.774)`

And we can find this probability using the complement rule and the normal standard table or excel and we got:

`P(z>2.774)=1-P(Z<2.774) = 1-0.9972 = 0.0028`

Answer 2

Given Information:

Mean = μ = 40 + 60 = 100 minutes

Standard deviation = σ = 2² + 3² = 13 minutes

Required Information:

a. P(X < 95) = ?

b. P(X > 110) = ?

Answer:

a. P(X < 95) = 0.0823

b. P(X > 110) = 0 .0028

Explanation:

a)

Let random variable X represents the time in minutes of wheel throwing and firing.

The probability that a piece of pottery will be finished within 95 minutes means,

P(X < 95) = P(Z < (x - μ)/√σ)

P(X < 95) = P(Z < (95 - 100)/√13)

P(X < 95) = P(Z < -1.39)

The z-score corresponding to -1.39 is 0.0823

P(X < 95) = 0.0823

Therefore, there is 8.23% probability that a piece of pottery will be finished within 95 minutes.

b)

P(X > 110) = 1 - P(X < 110)

P(X > 110) = 1 - P(X < (x - μ)/√σ)

P(X > 110) = 1 - P(X < (110 - 100)/√13)

P(X > 110) = 1 - P(X < 2.77)

The z-score corresponding to 2.77 is 0.9972

P(X > 110) = 1 - 0.9972

P(X > 110) = 0 .0028

Therefore, there is 0.28% probability that a piece of pottery will take longer than 110 minutes.