Question

QUESTION 8 A machine is supposed to produce widgets with a diameter of 20mm or more. The standard deviation of the population of widgets is known to be 5mm based on machine specifications. Quality Control takes a random sample of 25 widgets from the day's production and tests them. Unknown to the company, the machine is not functioning correctly and produces widgets with an actual average diameter of 18mm. Assuming an alpha value of .05 is used, what is the probability of a Type I error? That is, how likely is it that the company will assume that the machine is stil producing widgets with an average diameter of 20mm or more? Use three decimal places in your calculations.

Answer 1

Formula:

Type Il error Hypothesis Ho Ho Η. μ#μο Suppose that H, is false and that the true value of the mean isp= ot 8, say 8>0 T-(,+8) Test statistic, 2 = + 77 =- Therefore, the distribution of z when H, is true: z B=P(Accept H P(Accept H P(- s H, is false) H, is true) H, is true) Under E Under R g o 2-NO11 =area under the curve z N between - and z SVn -Fat r 74 distribution of the test statistic z, under H, and H, CDF of standard normal distribution

As per the formula, here

$\delta$ = -2

n = 25

$\sigma$ = 5

Since alpha = 0.05

Z(alpha/2) = 1.96

$\delta$
n
/$\sigma$ = -2

Thus,

$\beta$ = $\phi$(3.96) - $\phi$(0.04)

where, $\phi$ is CDF of std. normal distribution.

= 0.99996- 0.516

= 0.484

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