HYPOTHESIS TESTING1

Hypothesis◦A hypothesis test is a statistical test that is used to determine whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. ◦A hypothesis test examines two opposing hypotheses about a population: the null hypothesis (H0)and the alternative hypothesis (H1). ◦The null hypothesis is the statement being tested. The null hypothesis is either rejected or fails to be rejected. The null hypothesis is assumed to be true unless we have statistically overwhelming evidence to the contrary. 2

Hypothesis (cont.)◦The alternative hypothesis is the statement you want to be able to conclude is true.◦The alternative hypothesis, H1, is an assertion that holds if the null hypothesis is false. 3

Example◦There are three possible choices for the set of null and alternative hypotheses to be used for a given test. Described in terms of an (unknown) population mean (m), they might be listed as shown below. Notice that each null hypothesis has an equality termin its statement (i.e. =, ≤ , or ≥)4Null HypothesisAlternative HypothesisH0: m= $10 H1:m≠ $10 (mis $10, or it isn’t) H0: m≥ $10 H1: m< $10 (m at least $10, or it is less)H0: m≤ $10 H1: m> $10 (mis no more than $10, or it is more)

Directional and Nondirectional Testing◦A directional claim or assertion holds that a population parameter is greater than (>), at least (≥), no more than (≤), or less than (<) some quantity. ◦A nondirectional claim or assertion states that a parameter is equal to some quantity. For example, ABC claims that 35% of its customers are senior citizens.◦Directional assertions lead to what is called one-tail test, where a null hypothesis can be rejected by an extreme result in one direction only. ◦A nondirectional assertion involves a two-tail test, in which a null hypothesis can be rejected by an extreme result occurring in either direction.5

Example 11.ABC’s assertion: “35% of its customers are senior citizens.”2.Null hypothesis: H0: p = 0.35, where p = the population proportion. 3.Alternative hypothesis: H1: p ≠ 0.35.4.A two-tail test is used because the null hypothesis is nondirectional6

Example 2◦Supplier’s assertion “No more than 60% of the customers dislike our services”◦Null hypothesis: H0: p ≤ 0.60, where p = the population proportion. ◦Alternative hypothesis: H1: p > 0.60. ◦A one-tail test is used because the null hypothesis is directional.7

Errors in Hypothesis Testing◦Whenever we reject a null hypothesis, there is a chance that we have made a mistake – i.e. that we have rejected a true statement. ◦Rejecting a true null hypothesis is referred to as a Type I error, and our probability of making such an error is represented by alpha (a). This probability, which is referred to as the significance level of the test, is of primary concern in hypothesis testing.