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Chapter 4: Q87E (page 192)

The accompanying normal probability plot was constructed from a sample of \({\rm{30}}\) readings on tension for mesh screens behind the surface of video display tubes used in computer monitors. Does it appear plausible that the tension distribution is normal?

Short Answer

Expert verified

Yes, itappears plausible that the tension distribution is normal.

Step by step solution

01

Definition of Probability

The normal probability plot is a graphical tool for detecting significant deviations from normality. Outliers, skewness, kurtosis, the necessity for transformations, and mixes are all examples of this. Raw data, residuals from model fits, and estimated parameters are used to create normal probability charts. A graph of normal probability.

02

Step 2: Explanation of the tension distribution is normal

The pattern in the following normal probability plot is relatively straight, implying that the stress distribution is likely to be normal.

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Most popular questions from this chapter

When a dart is thrown at a circular target, consider the location of the landing point relative to the bull鈥檚 eye. Let \({\rm{X}}\) be the angle in degrees measured from the horizontal, and assume that \({\rm{X}}\) is uniformly distributed on \(\left( {{\rm{0, 360}}} \right){\rm{.}}\)Define\({\rm{Y}}\)to be the transformed variable \({\rm{Y = h(X) = (2\pi /360)X - \pi ,}}\) so \({\rm{Y}}\) is the angle measured in radians and\({\rm{Y}}\)is between \({\rm{ - \pi and \pi }}\). Obtain \({\rm{E(Y)}}\)and\({{\rm{\sigma }}_{\rm{y}}}\)by first obtaining E(X) and \({{\rm{\sigma }}_{\rm{X}}}\), and then using the fact that \({\rm{h(X)}}\) is a linear function of \({\rm{X}}\).

Let \({{\rm{I}}_{\rm{i}}}\) be the input current to a transistor and \({{\rm{I}}_{\rm{0}}}\) be the output current. Then the current gain is proportional to\({\rm{ln}}\left( {{{\rm{I}}_{\rm{0}}}{\rm{/}}{{\rm{I}}_{\rm{i}}}} \right)\). Suppose the constant of proportionality is \({\rm{1}}\) (which amounts to choosing a particular unit of measurement), so that current gain\({\rm{ = X = ln}}\left( {{{\rm{I}}_{\rm{0}}}{\rm{/}}{{\rm{I}}_{\rm{i}}}} \right)\). Assume \({\rm{X}}\) is normally distributed with \({\rm{\mu = 1}}\) and\({\rm{\sigma = }}{\rm{.05}}\).

a. What type of distribution does the ratio \({{\rm{I}}_{\rm{0}}}{\rm{/}}{{\rm{I}}_{\rm{i}}}\) have?

b. What is the probability that the output current is more than twice the input current?

c. What are the expected value and variance of the ratio of output to input current?

The error involved in making a certain measurement is a continuous rv \({\rm{X}}\) with pdf

\({\rm{f(x) = \{ }}\begin{array}{*{20}{c}}{{\rm{.09375(4 - }}{{\rm{x}}^2})}&{{\rm{ - 2}} \le {\rm{x}} \le {\rm{2}}}\\{\rm{0}}&{{\rm{otherwise}}}\end{array}\)

a. Sketch the graph of \({\rm{f(x)}}\).

b. Compute \({\rm{P(X > 0)}}\).

c. Compute \({\rm{P( - 1 < X < 1)}}\).

d. Compute \({\rm{P(X < - }}{\rm{.5 or X > }}{\rm{.5)}}\).

The article "'Response of \({\rm{Si}}{{\rm{C}}_{\rm{i}}}{\rm{/S}}{{\rm{i}}_{\rm{3}}}{\rm{\;}}{{\rm{N}}_{\rm{4}}}\) Composites Under Static and Cyclic Loading-An Experimental and Statistical Analysis" (J. of Engr. Materials and Technology, \({\rm{1997: 186 - 193}}\)) suggests that tensile strength (MPa) of composites under specified conditions can be modeled by a Weibull distribution with \({\rm{\alpha = 9}}\) and\({\rm{\beta = 180}}\).

a. Sketch a graph of the density function.

b. What is the probability that the strength of a randomly selected specimen will exceed \({\rm{175}}\)? Will be between \({\rm{150}}\) and \({\rm{175}}\)?

c. If two randomly selected specimens are chosen and their strengths are independent of one another, what is the probability that at least one has a strength between \({\rm{150}}\) and\({\rm{175}}\)?

d. What strength value separates the weakest \({\rm{10\% }}\) of all specimens from the remaining\({\rm{90\% }}\)?

The article "The Load-Life Relationship for M50 Bearings with Silicon Nitride Ceramic Balls" (Lubrication Engr., \({\rm{1984: 153 - 159}}\)) reports the accompanying data on bearing load life (million revs.) for bearings tested at a \({\rm{6}}{\rm{.45kN}}\) load.

\(\begin{array}{*{20}{c}}{{\rm{47}}{\rm{.1}}}&{{\rm{68}}{\rm{.1}}}&{{\rm{68}}{\rm{.1}}}&{{\rm{90}}{\rm{.8}}}&{{\rm{103}}{\rm{.6}}}&{{\rm{106}}{\rm{.0}}}&{{\rm{115}}{\rm{.0}}}\\{{\rm{126}}{\rm{.0}}}&{{\rm{146}}{\rm{.6}}}&{{\rm{229}}{\rm{.0}}}&{{\rm{240}}{\rm{.0}}}&{{\rm{240}}{\rm{.0}}}&{{\rm{278}}{\rm{.0}}}&{{\rm{278}}{\rm{.0}}}\\{{\rm{289}}{\rm{.0}}}&{{\rm{289}}{\rm{.0}}}&{{\rm{367}}{\rm{.0}}}&{{\rm{385}}{\rm{.9}}}&{{\rm{392}}{\rm{.0}}}&{{\rm{505}}{\rm{.0}}}&{}\end{array}\)

a. Construct a normal probability plot. Is normality plausible?

b. Construct a Weibull probability plot. Is the Weibull distribution family plausible?
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