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Chapter 4: Q45E (page 168)

A machine that produces ball bearings has initially been set so that the true average diameter of the bearings it produces is \({\rm{.500}}\)in. A bearing is acceptable if its diameter is within \({\rm{.004}}\)in. of this target value. Suppose, however, that the setting has changed during the course of production, so that the bearings have normally distributed diameters with mean value \({\rm{.499}}\)in. and standard deviation \({\rm{.002}}\)in. What percentage of the bearings produced will not be acceptable?

Short Answer

Expert verified

The\({\rm{7}}{\rm{.3\% }}\)of total bearings are not acceptable.

Step by step solution

01

Introduction

The standard deviation is a statistic that calculates the square root of the variance and measures the dispersion of a dataset compared to its mean.

02

Explanation

Let the diameter of the bearings be denoted by \({\rm{rv}}\) \({\rm{X}}\), then it's mean and standard deviation are given to us as:

\(\begin{array}{l}{\rm{\mu = 0}}{\rm{.499 inches }}\\{\rm{\sigma = 0}}{\rm{.002 inches }}\end{array}\)

The acceptable diameter of bearing is given to us as \({\rm{0}}{\rm{.496}}\)to \({\rm{0}}{\rm{.504}}\)inches. Hence \({\rm{P(0}}{\rm{.496 < X < 0}}{\rm{.504)}}\)denotes the fraction of the acceptable bearing.

Standardizing gives:

\({\rm{0}}{\rm{.496 < X < 0}}{\rm{.504}}\)if and only if

\(\frac{{{\rm{0}}{\rm{.496 - 0}}{\rm{.499}}}}{{{\rm{0}}{\rm{.002}}}}{\rm{ < }}\frac{{{\rm{X - 0}}{\rm{.499}}}}{{{\rm{0}}{\rm{.002}}}}{\rm{ < }}\frac{{{\rm{0}}{\rm{.504 - 0}}{\rm{.499}}}}{{{\rm{0}}{\rm{.002}}}}\frac{{{\rm{ - 0}}{\rm{.003}}}}{{{\rm{0}}{\rm{.002}}}}{\rm{ < }}\frac{{{\rm{X - 0}}{\rm{.499}}}}{{{\rm{0}}{\rm{.002}}}}{\rm{ < }}\frac{{{\rm{0}}{\rm{.005}}}}{{{\rm{0}}{\rm{.002}}}}{\rm{ - 1}}{\rm{.5 < Z < 2}}{\rm{.5}}\)

Thus

\({\rm{P(0}}{\rm{.496 < X < 0}}{\rm{.504) = P( - 1}}{\rm{.5 < Z < 2}}{\rm{.5)}}\)Here \({\rm{Z}}\)is a standard normal distribution \({\rm{rv}}\) with \(cdf\phi (z)\). Hence

\(P(0.496 < X < 0.504) = \phi (2.5) - \phi ( - 1.5)\)To get \(\phi (2.5)\)and\(\phi ( - 1.5)\), we check Appendix Table \({\rm{A}}{\rm{.3,}}\)from there

\(\phi ( - 1.5) = 0.0668{\rm{ and }}\phi (2.5) = 0.9938\)

Hence

\(\begin{array}{l}{\rm{P(0}}{\rm{.496 < X < 0}}{\rm{.504) = 0}}{\rm{.9938 - 0}}{\rm{.0668}}\\{\rm{P(0}}{\rm{.496 < X < 0}}{\rm{.504) = 0}}{\rm{.927}}\end{array}\)

Since this probability denotes the fraction of acceptable bearings, Hence

\(\begin{array}{c}{\rm{P( bearing not acceptable ) = 1 - P(0}}{\rm{.496 < X < 0}}{\rm{.504)}}\\{\rm{ = 1 - 0}}{\rm{.927P( bearing not acceptable )}}\\{\rm{ = 0}}{\rm{.073}}\end{array}\)

Hence \({\rm{7}}{\rm{.3\% }}\)of total bearings are not acceptable.

Proposition: Let Z be a continuous \({\rm{rv}}\) with \(cdf\phi (z)\). Then for any \({\rm{a}}\) and \({\rm{b}}\)with\({\rm{a < b}}\),

\(P(a < Z < b) = \phi (b) - \phi (a)\)

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Most popular questions from this chapter

Question: Suppose only \({\rm{75\% }}\) of all drivers in a certain state regularly wear a seat belt. A random sample of 500 drivers is selected. What is the probability that

a. Between 360 and 400 (inclusive) of the drivers in the sample regularly wear a seat belt?

b. Fewer than 400 of those in the sample regularly wear a seat belt?

Mopeds (small motorcycles with an engine capacity below\({\rm{50\;c}}{{\rm{m}}^{\rm{3}}}\)) are very popular in Europe because of their mobility, ease of operation, and low cost. The article "Procedure to Verify the Maximum Speed of Automatic Transmission Mopeds in Periodic Motor Vehicle Inspections" (J. of Automobile Engr., \({\rm{2008: 1615 - 1623}}\)) described a rolling bench test for determining maximum vehicle speed. A normal distribution with mean value \({\rm{46}}{\rm{.8\;km/h}}\) and standard deviation \({\rm{1}}{\rm{.75\;km/h}}\)is postulated. Consider randomly selecting a single such moped.

a. What is the probability that maximum speed is at most\({\rm{50\;km/h}}\)?

b. What is the probability that maximum speed is at least\({\rm{48\;km/h}}\)?

c. What is the probability that maximum speed differs from the mean value by at most \({\rm{1}}{\rm{.5}}\)standard deviations?

The temperature reading from a thermocouple placed in a constant-temperature medium is normally distributed with mean m, the actual temperature of the medium, and standard deviation. What would the value of s have to be to ensure that \(95\% \)of all readings are within \(.18\)of \(\mu \)?

Suppose that blood chloride concentration (mmol/L) has a normal distribution with mean\({\rm{104}}\)and standard deviation\({\rm{5}}\)(information in the article "Mathematical Model of Chloride Concentration in Human Blood," \({\rm{J}}\). of Med. Engr. and Tech.,\({\rm{2006: 25 - 30}}\), including a normal probability plot as described in Section\({\rm{4}}{\rm{.6}}\), supports this assumption).

a. What is the probability that chloride concentration equals\({\rm{105}}\)? Is less than\({\rm{105}}\)? Is at most\({\rm{105}}\)?

b. What is the probability that chloride concentration differs from the mean by more than 1 standard deviation? Does this probability depend on the values of\({\rm{\mu }}\)and\({\rm{\sigma }}\)?

c. How would you characterize the most extreme\({\rm{.1\% }}\)of chloride concentration values?

The current in a certain circuit as measured by an ammeter is a continuous random variable \({\rm{X}}\) with the following density function:

\({\rm{f(x) = \{ }}\begin{array}{*{20}{c}}{{\rm{.075x + }}{\rm{.2}}}&{{\rm{3}} \le {\rm{x}} \le {\rm{5}}}\\{\rm{0}}&{{\rm{otherwise}}}\end{array}\)

a. Graph the pdf and verify that the total area under the density curve is indeed \({\rm{1}}\).

b. Calculate \({\rm{P(X}} \le {\rm{4)}}\). How does this probability compare to \({\rm{P(X < 4)}}\)?

c. Calculate \({\rm{P(3}}{\rm{.5}} \le {\rm{X}} \le {\rm{4}}{\rm{.5)}}\) and also \({\rm{P(4}}{\rm{.5 < X)}}\).
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