Normal distribution 1

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Ruth Hand

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Christian Lawson-Perfect 2 years, 6 months ago

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Question statement

A machine mass-produces steel bolts with a target diameter of {size}mm, but records show that the diameters are normally distributed with mean {size}mm and standard deviation {sd}mm.

The bolts can only be sold if they have a diameter in the range {x1}mm and {x2}mm.

What proportion of the bolts produced are unacceptable for sale?

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			Name	Type	Generated Value

size	integer	30
sd	number	0.05
x1	number	29.95
x2	number	30.05
z1	number	-1
z2	number	1
ans	number	31.73

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Let $X$ be the diameter of a bolt. Since $X$ is normally distributed, $X\sim N(\var{size},\var{sd}^2).$

If we want to make a transformation to the standard normal distribution $Z \sim N(0,1^2)$ , we can use the transformation

$Z = \frac{X-\mu}{\sigma} = \frac{X-\var{size}}{\var{sd}}$

The lower level of acceptability, $x_1=\var{x1}$ , such that $z_1=\tfrac{\var{x1}-\var{size}}{\var{sd}}=\var{z1}$ .

The upper level of acceptability, $x_2=\var{x2}$ , such that $z_2=\tfrac{\var{x2}-\var{size}}{\var{sd}}=\var{z2}$ .

The range of acceptable $Z$ values is therefore $-1$ and $1$ .

Using the symmetrical property of the normal distribution,

$\begin{split} P(-1<Z<1) &\,= 2 P(0<Z<1) \\ &\,= 2 \times 0.34134 \\ &\,=0.68268. \end{split}$

Therefore, the proportion of bolts produced which are unacceptable for sale are

$1 - 0.68268 = 0.31732, \quad \text{or} \quad 31.73 \% \text{ (2 d.p.)} .$

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Christian Lawson-Perfect 2 years, 6 months ago

Ruth Hand 3 years, 8 months ago

Ben McGovern 4 years, 2 months ago

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Normal distribution 1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

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Comment

Checkpoint description

Christian Lawson-Perfect 2 years, 6 months ago

Ruth Hand 3 years, 8 months ago

Ben McGovern 4 years, 2 months ago

Error in variable testing condition

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Generated value: integer

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Gap-fill

Generated value: `integer`