Answer to Question #107335 in Statistics and Probability for NKULULEKO HAROLD LOUW

Question

In a factory where light bulbs are produced, the mean lifespan of a certain type of bulb was 1 000 hours^2 with variance 10^6 hours. However, a new and faster production technique was introduced recently and there is some fear that this has reduced the mean lifespan. To investigate this concern, the plan is to measure the lifespan of 200 light bulbs produced with the new technique. If the mean lifespan of these bulbs is 980 hours or less it will be decided that the overall mean lifespan µ of new production technique is smaller than 1 000 and the new technique will be revised; but if the mean lifespan of the 200 bulbs is more than 980, nothing will be done. When answering the questions below, assume that the variance for the new technique is unchanged since the old technique.d)i) Calculate the probability that this statistical procedure will not reject H0 while this is incorrect since the true µ is 990 hours.ii) What is the name of the accompanying error?

Answer 1

$H_0: \mu=a_0=1000, H_1:\mu<a_0=1000\text{ (left-tailed)}.\\ \mu\text{ is the mean lifespan after applying the new technique}.$

$\sigma^2=10^6, n=200.\\ 1-\beta=P\{X\leq980|\mu=990\}=P\{Z\leq\frac{980-\mu}{\sigma/\sqrt{n}}\}=P\{Z\leq\frac{980-990}{50\sqrt{2}}\}=P\{Z\leq-0.1414\}=0.4438\text{ --- Power of the test}.\\ \text{(probability that we will reject the null hypothesis while the null}\\ \text{hypothesis is false in fact)}.\\ i) \beta=1-0.4438=0.5562\\ ii)\text{Type II Error}.$