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:a) The probability that a randomly chosen specimen has an acceptable hardness is 0.7938.b) If the acceptable range of hardness is (70-c, 70+c), then the value of c would 95% of all specimens have an acceptable hardness of 5.88.c) Expected number of acceptable specimens among the ten is 7.938.d) Binomial with n = 10 and p = P(X < 73.84)[tex]p = P(Z <(73.84 - 70) / 3 ) = P(Z < 1.28) = 0.8997\\\\P(X <= 8) = 1 - P(X = 9) - P(X = 10)\\= 0.2650635[/tex]Step-by-step explanation:a ) [tex]P(67 < X< 75) = P( (67 - 70) / 3 < X < (75 - 70) / 3 )\\\\= P( - 1 < Z < 1.67) = 0.9525 - 0.1587 = 0.7938[/tex]b ) [tex]c = 1.96 * 3 = 5.88[/tex]                    { Since Z = 1.96 for 95% CI refer table.}c ) Expected number of acceptable specimens among the ten [tex]= 10 * P(67 < X< 75) \\\\= 10 * 0.7938 = 7.938[/tex]d ) Binomial with n = 10 and p = P(X < 73.84)[tex]p = P(Z <(73.84 - 70) / 3 ) = P(Z < 1.28) = 0.8997\\\\P(X <= 8) = 1 - P(X = 9) - P(X = 10)\\= 0.2650635[/tex]