Steps necessary for calculating the p-value for a test of hypothesis

1. Determine the value of the test statistic corresponding to the result of the sampling experiment.

2.

a) If the test is one- tailed, the p-value is equal to the tail area beyond z in the same direction as the alternative hypothesis. Thus, if the alternative hypothesis is of the form >, the p - value is the area to the right of, or above, the observed z value. Conversely, if the alternative is of the form <, the

p - value is the area to the left of, or below, the observed z value. (Fig.1.6;1.7)

b) If the test is two tailed, the p -value is equal to twice the area beyond the observed z -value in the direction of the sign of z. That is, if z is positive, the p -value is twice the area to the right of, or above, the observed z- value. Conversely, if z is negative, the p- value is twice the area to the left of, or below, the observed z -value. (See Fig.1.8)

Example:

The management of Health club claims that its members lose an average of 10kg or more within the first month after joining the club. A random sample of 36 members of this health club was taken and found that they lost an average of 9.2 kg within the first month of membership with standard deviation of 2.4kg. Find the p- value for this test.

Solution:

Let be the mean weight lost during the first month of membership by all members and be corresponding mean for the sample.

Step 1. State the null and alternative hypothesis

(The mean weight lost is 10kg or more)

(The mean weight lost is less than 10kg)

Step 2. Select the distribution to use

Because the sample size is large we use the normal distribution to make the test and calculate p- value.

Step 3. Calculate the p- value.

The < sign in the alternative hypothesis indicates that test is left tailed. The p- value is given by the area in the left tail of the sampling distribution curve of where is less than 9.2. To find this area, we first find the z value for as follows

The area to the left of under the sampling distribution of is equal to the area under the standard normal curve to the left of . The area to the left of is 0.0228. Consequently,

Thus, based on the p- value of 0.0228 we can state that for any (significance level) greater than 0.0228 we will reject the null hypothesis and for any less than 0.0228 we will accept the null hypothesis.

Suppose we make the test for this example at . Because is less than p- value of 0.0228, we will not reject the null hypothesis. Now suppose we make the test at . Because is greater than the p- value of 0.0228, we will reject the null hypothesis.

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