Hypothesis Tests

Hypothesis Tests

The steps in the classical hypothesis test process are as follows:

  1. Define the null and alternative hypotheses
  2. Establish the level of significance
  3. Select the appropriate test statistic
  4. Use the level of significance and the test statistic to build your decision rule.
  5. Using the sample data calculate the value of the test statistic.
  6. Reach a conclusion and fail to reject or reject the null hypothesis

Examples - Large Samples, Single Populations, Test of Mean and Test of Proportion

Example 1 - One population, large sample mean

  1. The null hypothesis is that the mean is equal to 12. The Alternative is that the mean is not equal to 12.
  2. The level of significance is set at alpha = 0.02
  3. n>30 so we have a large sample and can use the z distribution.
  4. Now we need to build a decision rule. We have a two tailed test so there is 0.01 in each tail. Using 0.5000 - 0.01 = 0.4900 we go to table A-2 and find that z is equal to plus and minu 2.33. If we calculate a value of z between -2.33 and +2.33 we will not reject the null hypothesis. If we calculate a value of z that is greater than 2.33 or less than -2.33 we will reject the null hypothesis and accept the alternative hypothesis.
  5. Using the given data we have z equal to (12.19 - 12)/(0.11/Square root(36)) = 10.36.
  6. The sample statistic, 10.36, is well past our critical z value of 2.33 so we will reject the null hypothesis and accept the alternative hypothesis.

Example 2, One population, proportions

  1. The null hypothesis is that the population proportion is greater than or equal to 0.10. The alternative is that the population proportion is less than 0.01
  2. The level of significance is set at alpha = 0.05
  3. Both np and nq are greater than 5 so we can use the z distribution.
  4. Now we need to build a decision rule. We have a one tailed test so there is 0.05 in the tail. Using 0.5000 - 0.05 = 0.4500 we go to table A-2 and find that z is equal to 1.645. In this case we have a left-tailed test. Small values of the sample proportion would lead us to reject the null hypothesis. If we calculate a value of z less -1.645 we will reject the null hypothesis and accept the alternative hypothesis. Otherwise we will not reject the null hypothesis.
  5. Using our data we calculate the value of z as (0.09 - 0.10)/Square Root(0.10 * 0.90/1233) = 1.17.
  6. The value that we have calculated falls in our do-not-reject the null hyopthesis region.

Examples - Large Samples, Two Populations, Test of Mean - Independent and Dependent Samples

Example 3 - Two populations, large sample, mean

  1. The null hypothesis is that the mean number of days in ICU for children not wearing a seat belt (Group 2)is less than the mean time spent in ICU for children who were wearing a seat belt (Group 1). The alternative hypothesis is that children not wearing seat belts spent more time in the ICU.
  2. The level of significance is set at alpha = 0.01 and we have a one-tailed test so all of alpha goes into the left tail.
  3. n or both groups is >30 so we have a large samples and can use the z distribution.
  4. Now we need to build a decision rule. We have a one-tailed test so there is 0.01 in the left tail (see figure). Using 0.5000 - 0.01 = 0.4900 we go to table A-2 and find that z is equal to -2.33. If we calculate a value of z that is greater than -2.33 we will not reject the null hypothesis. If we calculate a value of z that is less than -2.33 we will reject the null hypothesis and accept the alternative.
  5. Using the given data, which is repeated in the figure on the left we calculate z as -2.3301. We just barely reject the null hypothesis and accept the alternative. If we calculate the p value we get 0.0099, which is very close to our alpha (0.01).