Hypothesis Testing (Introduction-II)

1. Hypothesis Testing(Introduction-II) QSCI 381 – Lecture 26 (Larson and Farber, Sect 7.1)

2. Overview • To test a claim using data, we: • Develop two statistical hypotheses (the null and alternative hypotheses). • Select a level of significance (which determines the level of type I error – the probability of (unintentionally) rejecting the null hypothesis when it is true).

3. We now need to summarize the data in the form of a . Common examples of test statistics and their associated sampling distributions are: Statistics Tests -I test statistic

4. Statistics Tests -II • Therefore, given a claim related to , p or 2: • We select the appropriate test statistic from the previous table. • We choose the appropriate sampling distribution. • We standardize the test statistic.

5. Examples-I • We wish to test the claim that 30% of the diet of Pacific cod is walleye pollock. • Test statistic = • Sampling distribution = • Standardized test statistic =

6. Examples-II • Identify the null and alternative hypotheses, the test statistic, the sampling distribution, and comment on the appropriate levels of type I and type II error: • The probability of a building by a given contractor collapsing is less than 1%. • The density of a fish species based on a survey consisting of 15 trawls is 15 kg / ha. • The standard deviation of the survey is 5 kg / ha.

7. p-values • Assuming that the null hypothesis is true, the (or probability value) of a hypothesis test is the probability of obtaining a sample statistic with a value as extreme or more extreme than the one determined from the sample data. p-value

8. p-values and Tests-I • The nature of a hypothesis test depends on whether it is left-, right- or two-tailed. This in turn depends on the nature of the alternative hypothesis (one- or two-sided alternatives). • There are three cases: • The alternative hypothesis contains the symbol “<“ (i.e. the null hypothesis involves the symbol “”). • The alternative hypothesis contains the symbol “>” (i.e. the null hypothesis involves the symbol “”). • The alternative hypothesis contains the symbol “” (i.e. the null hypothesis involves the symbol “=”).

9. p-values and Tests-II Left tailed test: Null hypothesis involves a population parameter  something The p-value is the total shaded area and measures the probability of getting a test statistic as extreme or more extreme than the observed value.

10. Making Decisions Based on p-values-I • State the claim mathematically and verbally. Identify the null and alternative hypotheses. H0 = ?; Ha = ? • Specify the level of significance. =? • Determine the standardized sampling distribution if the hypothesis is true (sketch it)

11. Making Decisions Based on p-values-II • Calculate the test statistic and its standardized value (z in this case). (add it your sketch). • Find the p-value • Apply the decision rule: • Interpret the results Is the p-value less than or equal to ? Yes No Reject H0 Fail to reject H0

12. Making Decisions Based on p-values-III • Note that rejection of the null hypothesis is not proof that the null hypothesis is false, just that it is (very) unlikely. • Rejection of the null hypothesis is also not proof that the alternative hypothesis is true. • The following lectures cover various situations in which the algorithm outlined above is used to make decisions regarding hypotheses.

13. Caveat • The inability to reject the null hypothesis can arise because: • The null hypothesis is true (is a null hypothesis ever true?) • The sample size is too small to show that the null hypothesis is false. • The null hypothesis may be rejected even if it is not substantially false.