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

2. 381 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. 381 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 Population Parameter Test Statistic Sampling Distribution Standardized test statistic  Normal (n  30) Student t ztzt pNormalz Chi-square

4. 381 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. 381 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. 381 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. 381 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.

8. 381 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. 381 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. 381 Making Decisions Based on p-values-I 1. State the claim mathematically and verbally. Identify the null and alternative hypotheses. H 0 = ?; H a = ? 2. Specify the level of significance.  =? 3. Determine the standardized sampling distribution if the hypothesis is true (sketch it)

11. 381 Making Decisions Based on p-values-II 4. Calculate the test statistic and its standardized value (z in this case). (add it your sketch). 5. Find the p-value 6. Apply the decision rule: 7. Interpret the results Is the p-value less than or equal to  ? Yes Reject H 0 Fail to reject H 0 No

12. 381 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. 381 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.