A researcher wants to study the average lifetime of a certain brand of rechargea

Question

A researcher wants to study the average lifetime of a certain brand of rechargeable batteries I in hours). In testing the hypothesis, Ho: =950 hours vs. H1: 950 hours, a random sample of 25 rechargeable batteries is drawn from a normal population whose standard deviation is 200 hours.

a. Calculate , the probability of a Type II error when =1000 and a=0.10.

b. Calculate the power of the test when =1000 and a=0.10.

c. Interpret the meaning of the power of the test.

d. Recalculate if n is increased from 25 to 40.

e. Review the results of the previous questions. What is the effect of increasing the sample size on the value of ?

f. Recalculate if a is lowered from 0.10 to 0.05.

g. Review the results of the previous questions. What is the effect of decreasing the significance level on the value on ?

Can someone please help me with f and g? Thank you in advance!

Explanation / Answer

(f) Here if a is lowered from 0.10 to 0.05.

then we have to calculte the x below which we will not reject the null hypothesis.

So, the x = 950 +- Z0.05 ( /n) = 950 +- 1.96 * (200/ 25) = 950 + 1.96 * 40 = 1028.4

stanadard error of sample mean = /n = 200/25 = 40

so here the probability of type II error = Pr(x < 1028.4) = NORM (x < 1028.4 ; 1000 ; 40)

Z= (1028.4 - 1000)/40 = 0.71

Pr(Type II error) = Pr(Z < 0.71) = 0.7611

and Power of the test = 1 - 0.7611 = 0.2389

so when a = 0.10 then Pr(Type II error) = 0.6536

(g) So, a is lowered form 0.10 to 0.05 the probability of type II error will increase and power of the test will increase. so decreasng the significane value will increse the type II error.

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