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?

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 and standard deviation , the zscore of a measure X is given by:

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

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

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

So

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

(c) below what value of inside diameter will 15% of the piston rings fall?

This is the value of X of which Zscore has a value on the 15th percentile. This is .

So

Below 9.97cm.

f(x)=2/3x-8, which give g(x) the greatest value.

step-by-step explanation:

< xcy ≅ < xby

step-by-step explanation:

in δcax and δbax,

so, δcax ≅ δbax (side-angle-side or sas)

since δcax ≅ δbax, we can conclude < acx ≅ < abx

in δacy and δaby,

so, δacy ≅ δaby (side-angle-side or sas)

since δacy ≅ δaby, we can conclude < acy ≅ < aby

now,

< acy ≅ < aby

=> < acx + < xcy ≅ < abx + < xby

=> < acx + < xcy ≅ < acx + < xby   (since < acx ≅ < abx already proved above)

subtracting < acx from both sides, we get

< acx + < xcy -< acx ≅ < acx + < xby -< acx

cancelling out < acx and -< acx from both sides, we get

< xcy ≅ < xby (proved)