A certain mathematics professor lets students out of class at X minutes after th

Question

A certain mathematics professor lets students out of class at X minutes after the hour where X has a continuous uniform distribution with minimum alpha = 50 minutes, and maximum beta unknown. A student in this class believes that beta is at least 59 minutes. To test this null hypothesis, she measures the length of one class period and decides to reject the null hypothesis if it is less than 56.5 minutes long. a) Assuming that the student is right and beta = 59, what is the probability of Type I Error? b) If in actuality beta = 57.5, what is the probability of making a Type II Error? c) If in reality beta = 53 and the length of the measured class is X = 52, what type of error will be made? A. No Error B. Type I Error C. Type II Error

Explanation / Answer

= 50 minutes and = 59 minutes

Null Hypothesis : H0 : >= 59 mins

ALternative Hypothesis : Ha : < 59 mins

(a) Here = 59 minutes then type I error is to reject the null hypothesis when it is true.

then the null hypothesis is rejected when length of one class is less than 56.5 minutes long.

Pr (t < 56.5) = (56.5 - )/ ( - ) = (56.5 - 50)/ ( 59 - 50) = 0.722

(b) = 57.5 minutes then type II error is to failed to reject a false null hypothesis even if it false.

so type II error will occur when we will measure the length of that one class more than 56.5 minutes long .

so Type II error = Pr ( t > 56.5) = 1 - ( 56.5 -50)/ (57.5 - 50) = 0.133

(c) if real = 53 minutes and measured class is 52 minutes so there is no chance of having type I error as length of class is less than 56.5 minutes so there is no error in this part.

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