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In: Statistics and Probability

Chapter 12 of Froister & Blessing (pp. 156) discusses one-tailed and two-tailed t-tests. Statisticians characteristically include...

Chapter 12 of Froister & Blessing (pp. 156) discusses one-tailed and two-tailed t-tests. Statisticians characteristically include the equal-sign component in the null Hypothesis (H0). This means that the null hypothesis of a two-tailed test of two means would appear as:

     H0: (Mean)1 = (Mean)2 with alternative hypothesis H1: (Mean)1 NOT-= (Mean)2.

Had we created a one-tailed test of means, the null and alternative hypotheses would appear as:

     H0: (Mean)1 <= (Mean)2   with H1: (Mean)1  > (Mean)2, or
     H0: (Mean)1  >= (Mean)2 with H1: (Mean)1 < (Mean)2.

1) What is the graphical relationship of the signs in the alternative hypothesis to the area of rejection of the null hypothesis?

2) What are Type I and Type II errors? Why should we care?

Solutions

Expert Solution

1. The following plot depicts the graphical relationship between the signs in the alternative hypothesis to the area of rejection of null hypothesis:

2. Type 1 errors are errors that occur when the null hypothesis is incorrectly rejected, i.e. the null hypothesis is actually true but it is incorrectly rejected. On the other hand, type 2 errors occur when the null hypothesis is incorrectly accepted, i.e. the null hypothesis should actually be rejected but it is accepted.

Finding out Type 1 and Type 2 errors are crucial depending on the applications of the hypothesis testing domain. Type 1 errors are usually known as false alarm errors and certain industries prefer minimal false alarm scenarios. Type 2 errors are errors that usually result in failure to raise an alarm and are therefore crucial to industries like the medical industry.

orchestra answered 2 years ago
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