1.4. Hypothesis testing using the p –value approaches

In previous section, the value of the significance level was selected before the test performed. Sometimes we may prefer not to predetermine . Instead, we may want to find a value such that a given null hypothesis will be rejected for any greater than this value and it will not be rejected for any smaller than this value. In this approach, we calculate the p-value for the test, which is defined as the smallest level of significance at which the given null hypothesis is rejected.

Definition:

The p-value is the smallest significance level at which the null hypothesis is rejected.

Using the p-value approach, we reject the null hypothesis if

p- value

and we do not reject the null hypothesis if

p- value

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.

Exercises

1. Find the p-value for each of the following hypothesis tests

a) ; ; ; ; ;

b) ; ; ; ; ;

c) ; ; ; ;

2. Consider ; against the alternative .

A random sample of 60 observations taken from this population produced a sample mean of 31.4 and a standard deviation of 8.

a) Calculate the p-value.

b) Considering the p-value of part a), would you reject the null hypothesis if the test were made at the significance level of 0.05?

c) Considering the p-value of part a), would you reject the null hypothesis if the test were made at the significance level of 0.01?

3. In a given situation, suppose was rejected at . Answer the following questions as “yes”, “no”, or “can’t tell” as the case may be.

a) Would also be rejected at ?

b) Would also be rejected at

c) Is the p-value smaller than 0.05?

4. In a problem of testing against , the following sample quantities are recorded.

a) State the test statistic and find the rejection region with .

b) Calculate the test statistic and draw a conclusion with .

c) Find the p-value and interpret the results.

Answers

1. a) 0.0046; b) 0.0017 ;c) 0.0162; 2. a) 0.0204; b) yes, reject ; c) no, do not reject ;3. a) can’t tell; b) yes; c) no; 4. a) ; ; b) , is rejected at ; c) 0.0125;