Answer:
Check the explanation
Step-by-step explanation:
(a) P(Type I error) = P( is rejected |
Here we have : µ = 250 and : µ ≠ 250
Here it is given that if 240 < X < 260 then the machine is considered operating satisfactory i.e. is not rejected.
So rejection region is X <= 240 and X >= 260.
Also we know that X follows N( µ , (400/12) ) as population follows N( µ, 400) ; So ( X - µ) / (20 / (12)1/2) follows std. normal distribution.
So, P( is rejected | is true) = P(X <= 240 | µ = 250) + P(X >= 260 | µ = 250)
= P(( X - µ) / (20 / (12)1/2) <=(240 -250)/ (20 / (12)1/2| µ = 250) +
P(( X - µ) / (20 / (12)1/2) >=(260 -250)/ (20 / (12)1/2| µ = 250)
= P(( X - µ) / (20 / (12)1/2) <= -1.7320508076 | µ = 250) + P(( X - µ) / (20 / (12)1/2) >= 1.7320508076 | µ = 250)
And since ( X - µ) / (20 / (12)1/2) follows std. normal distribution. This can be found from Z-tables.
so, = 0.0416 + 0.0416 = 0.0832
(b) P(Type II error) = P( is not rejected |
is false) = P(240 < X < 260 | µ ≠ 250)
We have to calculate probability of type II error specifically at µ = 242 so,
= P(240 < X < 260 | µ =242) = P((240 -242)/ (20 / (12)1/2 <= ( X - µ) / (20 / (12)1/2) <= (260 -242)/ (20 / (12)1/2)| µ = 242)
= P(-0.3464101615 <= ( X - µ) / (20 / (12)1/2) <= 3.1176914535 | µ = 242)
And since ( X - µ) / (20 / (12)1/2) follows std. normal distribution. This can be found from Z-tables.
= 0.6346