Mathematics, 21.11.2019 20:31 camilacarrielh15

The finished inside diameter of a piston ring is normally distributed with a mean of 10 centimeters and a standard deviation of 0.03 centimeter. (a) what proportion of rings will have inside diameters exceeding 10.075 centimeters ? (b) what is the probability that a piston ring will have an inside diameter between 9.97 and 10.03 centimeters? (c) below what value of inside diameter will 15% of the piston rings fall?

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Answer from: tracitalbot

a) 0.62% rings will have inside diameters exceeding 10.075 centimeters.

b) There is a 68% probability that a piston ring will have an inside diameter between 9.97 and 10.03 centimeters

c) Below 9.97cm.

Step-by-step explanation:

Problems of normally distributed samples can be solved using 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.

In this problem, we have that:

The finished inside diameter of a piston ring is normally distributed with a mean of 10 centimeters and a standard deviation of 0.03 centimeter. This means that $\mu = 10, \sigma = 0.03$

(a) What proportion of rings will have inside diameters exceeding 10.075 centimeters ?

This proportion is 1 subtracted by the pvalue of Z when X = 10.075 .

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

$Z = \frac{10.075 - 10}{0.03}$

Z = 2.5

Z = 2.5 has a pvalue of 0.9938.

This means that 1-0.9938 = 0.0062 = 0.62% rings will have inside diameters exceeding 10.075 centimeters.

(b) What is the probability that a piston ring will have an inside diameter between 9.97 and 10.03 centimeters?

Those are the values that are within 1 standard deviation of the mean.

The 68-95-99.7 states that 68% of the measures of a normally distributed sample are within 1 standard deviation of the mean.

So, there is a 68% probability that a piston ring will have an inside diameter between 9.97 and 10.03 centimeters

This is the value of X of which Zscore has a value on the 15th percentile. This is Z = -1.035 .

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

$-1.035 = \frac{X - 10}{0.03}$

X - 10 = 0.03*(-1.035)

X = 9.97

Below 9.97cm.

Answer from: Quest

The answer is 1532.75

Answer from: Quest

d. you cannot tell because there are no values shown.

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