C R I T I C A L R E G I O N S

8.2 TESTING THE MEAN

P-Values VS Critical Regions

• The P-Value method is the most popular method of statistical testing• There is another method of statistical testing

called the critical region (also called the traditional method)• For a fixed, preset value of the level of

significance , both methods are logically equivalent.• However, we will only use the critical region for the case

where are testing when is known

The Critical Region Method• The values of for which we reject are called the

critical region of the distribution.• The location of the critical region depends on • The total area in the critical region is the level of significance • Note that this is NOT the P-value previously discussed• The level of significance should be a fixed, preset number assigned

before drawing any samples• The most commonly used levels of significance are and

• The critical value(s) are the z-scores that correspond to the shaded areas

Critical Regions for H0: = kFigure 8-7

• When using the critical region method, we check if the sample test statistic falls in the critical region.• If it does, we reject • If it does not, then we fail to reject

The Critical Region Method

1. Check requirements• Is a random variable?• Is known?• Do you have a simple random sample or can you assume a simple

random sample was used?• Is the distribution normal or is ?

2. State the null and alternate hypotheses, Identify the level of significance

3. Take a simple random sample and compute a sample test statistic• Standardized Sample Test Statistic:

4. Sketch a graph and show the critical region and critical value(s). • The area of the critical region =

5. Conclude the test• If the test statistic falls in the critical region, reject • If the test statistic does not fall in the critical region, fail to reject

6. Interpret the results in the context of the application

How to Test When is KnownCritical Region Method

EXAMPLE 5: Critical Region Method of Testing

Consider Example 3 regarding sunspots. Let x be a random variable representing the number of sunspots observed in a four-week period. A random sample of 40 such periods from Spanish colonial times gave the number of sunspots per period. The raw data are given in Example 3. The sample mean is 47.0. Previous studies indicate that for this period, = 35. It is thought that for thousands of years, the mean number of sunspots per four-week period was about = 41. Do the data indicate that the mean sunspot activity during the Spanish colonial period was higher than 41? Use = 0.05.

EXAMPLE 5: Critical Region Method of Testing SOLUTION

1. Check requirements• Is a random variable? YES• Is known? YES, • Do you have a simple random sample or can you assume

a simple random sample was used? YES• Is the distribution normal or is ? YES,

2. State the null and alternate hypotheses, Identify the level of significance

EXAMPLE 5: Critical Region Method of Testing SOLUTION

3. Take a simple random sample and compute a sample test statistic

4. Sketch a graph and show the critical region and critical value(s).

Critical Value:invNorm(.95)

EXAMPLE 5: Critical Region Method of Testing SOLUTION

5. Conclude the test• The sample test statistic

does not fall in the critical regiontherefore we fail to reject

6. Interpret the results in the context of the application• At the 5% level of significance, the sample evidence is

insufficient to justify rejecting . It seems that the average sunspot activity during the Spanish colonial period was the same as the historical average.