Data Analysis with Microsoft Excel: Updated for Office 2007

234 Fundamentals of Statistics

An Example of Hypothesis Testing

Let’s put these abstract ideas into a concrete example. You work at a plant

that manufactures resistors. Previous studies have shown that the number of

defective resistors in a batch follows a normal distribution with a mean of 50

and a standard deviation of 15. A new process has been proposed that will

reduce the number of defective resistors, saving the plant money. You put

the process in place and create a sample of 25 batches. The average number

of defects in a batch is 45. Does this prove that the new process reduces the

number of defective resistors, or is it possible that the process makes no dif-

ference at all, and the 45 is simply a random aberration?

Here are our hypotheses.

H 0 : There is no change in the mean number of defective resistors under

the new process.

Ha: The mean number of defective resistors has changed.

Or, equivalently,

H 0 : The mean number of defective resistors in the new process is 50.

Ha: The mean number of defective resistors is not 50.

Acceptance and Rejection Regions

To decide between these two hypotheses, we assume that the null hypoth-

esis is true. Let m 0 be the mean under the null hypothesis. This means that

under the null hypothesis,

P a 2 z 12 a/ 2 ,

x2m 0

s/!n

,z 12 a/ 2 b 512 a

Multiplying by the standard error and adding m 0 to each term in the inequal-

ity, we get

P am 02 z 12 a/ 2

s

!n

,x,m 01 z 12 a/ 2

s

!n

b 512 a

This means that the sample average should be in the range m 06 z 12 a/ 2 s/!n

with probability 12 a, if the null hypothesis is true. Now let a be our sig-

nifi cance level, so that if the sample average lies outside this range, we’ll

reject the null hypothesis and accept the alternative. These outside values

would constitute the rejection region, mentioned earlier. The values within

the range constitute the acceptance region, under which we’ll accept the

null hypothesis. The upper and lower boundaries of the acceptance region

are known as critical values, because they are critical in deciding whether

to accept or to reject the null hypothesis.