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 69 and standard deviation 3. 

(a) If a specimen is acceptable only if its hardness is between 68 and 73, what is the probability that a randomly chosen specimen has an acceptable hardness? (Round your answer to four decimal places.) 

(b) If the acceptable range of hardness is (69 - C, 69 + c), for what value of c would 95% of all specimens have acceptable hardness? (Round your answer to two decimal places.)

 (c) If the acceptable range is as in part (a) and the hardness of each of ten randomly selected specimens is independently determined, what is the expected number of acceptable specimens among the ten? (Round your answer to two decimal places.) specimens 

(d) What is the probability that at most eight of ten independently selected specimens have a hardness of less than 71.527 [Hint: Y-the number among the ten specimens with hardness less than 71.52 is a binomial variable; what is p?] (Round your answer to four decimal places.) 

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Answer 1

Let .X: Rockwell hardness of is normally distributed with standard deviation 3. a particular alloy mean 69 and .X N NC 69, 32) 22 X-69 ANCO, L) 3 a) probability that the a randomly choosen specimen has an acceptable hardness P= PC 68EX & 73) = P( 68-69 exoe 73-89) = p. ( - 13 azt 4) - P ( - 0:3333 € Z = 1.3333) -0.3333) P(Z < 1.3333) - P(Z = = 0.90878 - 0.36945 = 0.53933 as Scanned with 0.5393 CamScanner

b) The value of c is. PC 69-C EXE 69+c) = 095 P( 5 < x-69 ) = 0.95 pc - E24 & ) = 0.95 Plze ) - Plz 4 0.95 p(24) - El-ptz - 2 p( z = 2 ) = 0.95 2 p(24-Ş) = 0.05 Plze-C) = 0.025 - = Fil (0.025) who = -logo c = 5.88 C = 5.88 cs Scanned with C= 5.88 CamScanner

from ③ we have p= 0.5393 speciman randomly Now, we selected 10 hc 10 of acceptable Therefore the expected specimens among the npr 10 (0.5393) = number ten is 50393 d) probability that at most eight of ten independently selected specimens have a hundre hardness of less than 71.52 is yn B CA-10, p= 0.5393) PLY £8) = § (g) (05393)*(04057) y=0 = 0.98014 = 0.9801 Scanned with CamScanner