Question

The target diameter of bolts from a production line is 10mm. Historically the bolt diameters are normally distributed with a standard deviation of 0.15mm. To monitor this process periodically an engineer takes a random sample of 6 measurements. Let µ be the true average bolt diameter.

Ho:u=10 versus Ha:+10

The rejection region is:   
Find the minimum value of c that yields a test with significance 0.01?

zα	1.282	1.645	1.960	2.326	2.576	3.090
α(tail area)	0.1	0.05	0.025	0.01	0.005	0.001
%ile	90	95	97.5	99	99.5	99.9

Your answer can be rounded to four decimal digit accuracy when entered.

Answer 1

At alpha = 0.01, the critical values are +/- z_0.005 = +/- 2.576

z > 2.576

or, ($\bar x - \mu$)/($\sigma/\sqrt n$) > 2.576

or, ($\bar x$ - 10)/(0.15/$\sqrt 6$) > 2.576

or, $\bar x$> 2.576 * 0.15/$\sqrt 6$ + 10

or, $\bar x$> 10.1577

$\bar x$> 10 + c

or, 10.1577 = 10 + c

or, 0.1577 = c

z < -2.576

or, ($\bar x - \mu$)/($\sigma/\sqrt n$) < -2.576

or, ($\bar x$ - 10)/(0.15/$\sqrt 6$) < -2.576

or, $\bar x$< -2.576 * 0.15/$\sqrt 6$ + 10

or, $\bar x$< 9.8423

$\bar x$< 10 - c

or, 9.8423 = 10 - c

or, c = 0.1577

So the minimum value of c is 0.1577.