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Chapter 5: Sampling Di_strib

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FIGURĘ 5.14 The dimensions of a mechanical assembly, for f&ęrcise 5.48.

5.48    A mechanical assembly (Figurę 5.14) consists of a shaft with a bearing at each end. The total length of the assembly is the sum ! + Y + Z of the shaft length X and the lengths Y and Z of the bearings. These lengths vary from part to part in production, independently of each other and with normal distributions. The shaft length X has mean 11.2 inches and standard deviation 0.002 inch, while each bearing length Y and Z has mean 0.4 inch and standard deviation 0.00L inch.

(a)    According to the 68-95-99.7 rule, about 95% of all shafts have lengths in the rangę 11.2 ± d\ inches. What is the value of di? Similarly, about 95% of the bearing lengths fali in the rangę 0.4 ± dz. What is the vałue of dz?

(b)    It is common practice in industry to State the "natural tolerance" of parts in the form used in (a). An engineer who knows no statistics thinks that tolerances add, so that the natural tolerance for the total length of the assembly (shaft and two bearings) is 12 ± d inches, where d ~'d\ + 2dz. Find the standard deviation of the total length

X + Y + Z. Then find the value d such that about 95% of all assemblies have lengths in the rangę 12 ± d. Was the engineer correct?

5.49    Leona and Fred are friendly competitors in high schóol. Both are about to take the ACT college entrance examination. They agree that if one of them scores 5 or morę points better than the other, the loser will buy the winner a pizza. Suppose that in fact Fred and Leona have eąual ability, so that each score varies normally with mean 24 and standard deviation 2. (The variation is due to luck in guessing and the accident of the specific questions being familiar to the student.) The two scores are independent. What is the probability that the scores differ by 5 or morę points in either direction?

5.50 ACT, Inc„ the producer of the ACT test of readiness for college work, also produces tests for 8th- and 9th-grade students designed to help |hem