Question

The Rockwell hardness of a metal is determined by impressing a hardened point into the surface of the metal and then measuring the depth of penetration of the point. Suppose the Rockwell hardness of a particular alloy is normally distributed with mean 68 and standard deviation 3. (a) If a specimen is acceptable only if its hardness is between 65 and 71, what is the probability that a randomly chosen specimen has an acceptable hardness? (Round your answer to four decimal places.) 0.6826 Correct: Your answer is correct.

Answer 1

Answer: 0.6826

Explanation:

Since it's a normal distribution

z = (x - m)/s where

m = mean = 68

s = standard deviation = 3

z = standard normal variable

x = values of hardness of the specimen

a) we are looking for P( 65 lesser than/equal to x lesser than/ equal to 71

For x= 65,

z = (65-68)/3 = -3/3 = -1

For x= 71

z = (71 -68)/3 = 3/3 =1

Looking at the normal distribution table,

For area covered by

(z = -1 to z= 1) = 0.1587

For area covered by

(z = 0 to z=1) = 0.8413

P( 65 lesser than/equal to x lesser than/ equal to 71) = 0.8413 -0.1587

= 0.6826

Answer 2

0.6826

Explanation:

It is stated that specimen is acceptable if its hardness is between 65 and 71. Since the hardness is normally distributed with mean 68 and standard deviation 3, we can say that acceptable interval is 1 standard deviation far from the mean.

We can state our interval as μ±σ where μ is the mean of the sample and σ is the standard deviation. In normal distribution, 1 standard deviation far to the mean covers 68.26% of the distribution.