The drying time of a certain type of paint under specified test conditions is kn

Question

The drying time of a certain type of paint under specified test conditions is known to be normally distributed with mean value 75 min and standard deviation 9 min. Chemists have proposed a new additive designed to decrease average drying time. It is believed that drying times with this additive will remain normally distributed with = 9. Because of the expense associated with the additive, evidence should strongly suggest an improvement in average drying time before such a conclusion is adopted. Let denote the true average drying time when the additive is used. The appropriate hypotheses are H0: = 75 versus Ha: < 75. Consider the alternative value = 74, which in the context of the problem would presumably not be a practically significant departure from H0.

(a) For a level 0.01 test, compute at this alternative for sample sizes n = 64, 1024, and 2500. (Round your answers to four decimal places.)

I need (beta or type II error) for the different sample sizes. Please tell me how to do it if you can please! I really do not understand this. Thank you

Explanation / Answer

This is a left tailed test, One will fail to reject null hypothesis (Type II error) if one gets Z statistic greater than -2.58. This -2.58 Z critical value correspond to some Xbar critical such that

-2.58=Z critical=(Xbar critical-mu)/sigma

-2.58=( Xbar critical-75)/9

Xbar critical=51.78

Now, probbaility of Type II error is

P(Xbar>51.78)=P(Z>[(51.78-74)/9]=P(Z>-2.47)=0.9932

Power=1-0.9932=0.0068

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