hw10sol_fall03 - AMS 310.01 FALL 2003 Homework#10 Solutions...

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AMS 310.01  FALL 2003   Homework #10  Solutions  7.30 (a)  The probability of Type I error is the probability that  X is less than 2.9960 or  greater than 3.0040 when  3.0000 μ = . Using a normal approximation to the  distribution of the sample mean, this probability is given by  2.9960 3.0000 3.0040 3.0000 ( ) 1 ( ) 0.0250/ 30 0.0250/ 30 F F - - + - ( 0.876) 1 (0.876) 2 ( 0.876) 0.381 F F F = - + - = - = (b)  The probability of type II error is the probability that  2.9960 3.0040 X < <   when  3.0050 μ = . Using a normal approximation, this probability is given by 3.0040 3.0050 2.9960 3.0050 0.0250/ 30 0.0250/ 30 F F - - - ( 0.219) ( 1.97) 0.4133 0.0244 0.3889 F F = - - - = - = 7.32 Using the normal approximation for the distribution of the sample mean,  we need to find  x  such that  100 0.01 12/ 40 x F - = Using the normal table, we see that  100 2.33 12/ 40 x - = - So, we reject the null hypothesis if  x x <  where  95.58 x = 7.38 (a)    The null hypothesis is  50 μ  ohms,       and the alternative hypothesis is  50 μ <  ohms.
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