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Ira Lisetskai [31]
3 years ago
7

Making handcrafted pottery generally takes two major steps: wheel throwing and firing. The time of wheel throwing and the time o

f 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?
Mathematics
1 answer:
9966 [12]3 years ago
5 0

Answer:

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.

Step-by-step 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 = \frac{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}} = \sqrt{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 = \frac{X - \mu}{\sigma}

Z = \frac{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 = \frac{X - \mu}{\sigma}

Z = \frac{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.

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