Question

Making handcrafted pottery generally takes two major steps: wheel throwing and firing. The time of wheel throwing and the time of firing are normally distributed random variables with means of 40 minutes and 60 minutes and standard deviations of 2 minutes and 3 minutes, respectively. Assume the time of wheel throwing and time of firing are independent random variables.

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) 8.23% probability that a piece of pottery will be finished within 95 minutes

b) 0.28% probability that it will take longer than 110 minutes.

Explanation:

Normal distribution:

When the distribution is normal, we use the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Two variables:

Means
`\mu_(a), \mu_(b)`

Standard deviations
`\sigma_(a), \sigma_(b)`

Sum:

`\mu = \mu_(a) + \mu_(b)`

`\sigma = \sqrt{\sigma_(a)^(2) + \sigma_(b)^(2)}`

In this question:

`\mu_(a) = 40, \mu_(b) = 60, \sigma_(a) = 2, \sigma_(b) = 3`

So

`\mu = \mu_(a) + \mu_(b) = 40 + 60 = 100`

`\sigma = \sqrt{\sigma_(a)^(2) + \sigma_(b)^(2)} = √(4 + 9) = 3.61`

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

This is the pvalue of Z when X = 95.

`Z = (X - \mu)/(\sigma)`

`Z = (95 - 100)/(3.61)`

`Z = -1.39`

`Z = -1.39` has a pvalue of 0.0823

8.23% probability that a piece of pottery will befinished within 95 minutes.

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

This is 1 subtracted by the pvalue of Z when X = 110.

`Z = (X - \mu)/(\sigma)`

`Z = (110 - 100)/(3.61)`

`Z = 2.77`

`Z = 2.77` has a pvalue of 0.9972

1 - 0.9972 = 0.0028

0.28% probability that it will take longer than 110 minutes.