7. -/4 pointsDevoreStat9 4.E.046. The Rockwell hardness of a metal is determined

Question

7. -/4 pointsDevoreStat9 4.E.046. 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 6 specimen has an acceptable hardness? (Round your answer to four decimal places.) 3 and 75, what is the probability that a randomly chosen (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 determined, what is the expected number of acceptable specimens among the ten? (Round your answer to two decimal places.) the hardness of each of ten randomly selected specimens is independently 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 p2] (Round your answer to four decimal places.) You may need to use the appropriate table in the Appendix of Tables to answer this question.

Explanation / Answer

Mean 69 Std Dev 3 1) Calculating Z For 63 (63-69)/3 -2 For 75 (75-69)/3 2 P value for Z = -2 0.4772 Z = 2 0.4772 Therefore probability that a specimen is between 63 and 75 is 0.9544 2) 95% confidence level Range 69-c 69+c Finding c Z value for 95 % confidence level is 1.96 Therefore c is 1.96*std dev 5.88 CI for 95% is (69-5.88,69+5.88) (63.12,74.88) c 5.88 3) Expected number of acceptable specimen out of 10. Therefor probability *10 gives expected number of acceptable specimen 9.54 Expected number of acceptable specimen out of 10 is 9.54

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