1.2. Tests of the mean of a normal distribution:

Population variance known

In this and following sections we will present specific procedures for developing and implementing hypothesis test procedures with applications to business and economic problems.

We are given a random sample of n observations from a normal population with mean and known variance . If the observed sample mean is , then the test statistic is

and we can use the following tests with significance level .

1. To test either null hypothesis

or against the alternative

the decision rule is

Reject if

2. To test either null hypothesis

or against the alternative

the decision rule is

Reject if

3. To test the null hypothesis

against the two sided alternative

the decision rule is

Reject if or ,

where is the number for which

and is the standard normal distribution.

A statistical test of hypothesis procedure contains the following five steps:

1. State the null and alternative hypothesis

2. Select the distribution to use

3. Determine the rejection and nonrejection regions

4. Calculate the value of the test statistic

5. Make a decision.

Example:

A manufacturer of detergent claims that the content of boxes sold weigh on average at least 160 grams. The distribution of weights is known to be normal, with standard deviation of 14 grams. A random sample of 16 boxes yielded a sample mean weight of 158.9 grams. Test at the 10% significance level the null hypothesis that the population mean is at least 160 grams.

Solution:

Let be the mean average of all boxes and be the corresponding mean for the sample.

; ;

The significance level is is 0.1. That is, the probability of rejecting the null hypothesis when it is actually is true should not exceed 0.1. This is the probability of making a Type I error. We perform the test of hypothesis using the five steps as follows.

Step 1. State the null and alternative hypothesis

We write the null and alternative hypothesis as

grams

grams

Step 2. Select the distribution to use

Since population standard deviation is known we will use .

Step 3. Determine the rejection and nonrejection regions

The significance level is 0.1. The < sign indicates that the test is left tailed. We look for 0.9 from in the standard normal distribution table, (Table 1 of Appendix). The value of z is . (Fig. 1.4).

Step 4. Calculate the value of the test statistic

The decision to reject or not to reject the null hypothesis will depend on whether the evidence from the sample falls in the rejection or nonrejection region. If the value of the sample mean falls in rejection region, we reject . Otherwise we do not reject the null hypothesis. To locate the position of on the sampling distribution curve of in Figure 1.4 we first calculate z value for . This is called the value of the test statistic.

Step 5. Make a decision

In the final step we make a decision based on the value of the test statistic for in previous step. This value of is not less than the critical value of , and it falls in the nonrejection region. Hence we accept and conclude that based on sample information, it appears that the mean weight of all boxes is greater than 160 grams.

By accepting the null hypothesis we are stating that the difference between the sample mean and the hypothesized value of the population mean is not too large and may occurred because of the chance or sampling error. There is a possibility that the mean weight is less than 160 grams, by the luck of the draw, we selected a sample with a mean that is not too far from required mean of 160 grams.