1. MARE 250 Dr. Jason Turner Hypothesis Testing

2. This is not a Test… Hypothesis testing – used for making decisions or judgments Hypothesis – a statement that something is true Hypothesis test typically involves two hypothesis: Null and Alternative Hypotheses

3. Testes…Testes…One…Two…Three? Null hypothesis – a hypothesis to be tested ( H 0 ) H 0 : μ = μ 0 (ALWAYS the NULL for MEANS TEST) Alternative hypothesis (research hypothesis) – a hypothesis to be considered as an alternative to the null hypothesis (H a ) Three possible choices: 1. Mean is Different From a specified value – two-tailed test H a : μ ≠ μ 0 2. Mean is Less Than a specified value – left-tailed test H a : μ < μ 0 3. Mean is Greater Than a specified value – right-tailed test H a : μ > μ 0 One-tailed tests

4. For Example… The FDA has issued fish consumption advisories for populations containing Hg levels greater than 1.0 ppm. Want to test whether Yellowfin tuna have levels of Hg below 1.0 ppm

5. For Example… 1. Determine the null hypothesis for the hypothesis test. 2. Determine the alternative for the hypothesis test. 3. Classify the hypothesis test as two-tailed, left-tailed, or right-tailed

6. For Example… One Sample t-test 1.The null hypothesis for this test is: “the mean Hg level for yellowfin tuna equals the FDA level of 1.0 ppm” H 0 : μ = 1.0 ppm 2. The alternative for the hypothesis test is: “the mean Hg level for yellowfin tuna is less than 1.0 ppm” H a : μ < 1.0 ppm 3. The hypothesis test is left-tailed because the less- than-sign (<) appears in the alternative hypothesis

7. Hypothesis Testing for the Rest of Us Hypothesis tests for one population mean when σ is unknown "A Festivus, For the Rest of Us!" - Frank Costanza One Sample t-test; t-test Used to testing a collected sample versus a number

8. Hypothesis Testing for Two Means “This is getting out of hand, now there are two of them!” – Lott Dod Two Sample t-test; t-test Used to testing a collected sample another collected sample One of the most common (and simple?) tests in statistics

9. For Example… 2 Sample t-test 1.The null hypothesis for this test is: “the mean Hg level for yellowfin tuna from Hilo equals the Hg level from Kona” H 0 : μ Hilo = μ Kona 2. The alternative for the hypothesis test is: “the mean Hg level for yellowfin tuna from Hilo does not equal the Hg level from Kona” H 0 : μ Hilo = μ Kona

10. Hold on, I have to p P-value approach – indicates how likely (or unlikely) the observation of the value obtained for the test statistic would be if the null hypothesis is true A small p-value (close to 0) the stronger the evidence against the null hypothesis It basically gives you odds that you sample test is a correct representation of your population

11. Didn’t you go before we left P-value – equals the smallest significance level at which the null hypothesis can be rejected - the smallest significance level for which the observed sample data results in rejection of H 0 If the p-value is less than or equal to the specified significance level, reject the null hypothesis, otherwise, do not (fail to) reject the null hypothesis

12. No, I didn’t have to go then

13. Critical Region-Defined We need to determine the critical value (s) for a hypothesis test at the 5% significance level (α=0.05) if the test is (a) two-tailed, (b) left tailed, (c) right tailed { 0.025 {{{ 0.05

14. First name Mr., last name t “I pity myself that I got to be with these fools!”– Mr. T A t-test is based upon at least 2 assumptions: 1. Data normally (or a least somewhat normally) distributed t-test is robust to moderate variations of the normality assumption 2. All outliers have been accounted for Should be controlled by normality assumption Mean and std. dev. not resistant to outliers – can skew

15. To ASSUME is to make an… Four assumptions for t-test hypothesis testing: 1. Random Samples 2. Independent Samples 3. Normal Populations (or large samples) 4. Variances (std. dev.) are equal

16. Let Us Review… We use a type of stats test called a means test It tests for differences in samples based upon their average (mean) and standard deviation The first one we learned is for 1 or 2 samples called a t-test

17. Let Us Review… There are two basic types (versions) of statistical tests: Parametric – has strict assumptions Non-parametric – no assumptions There are 4 types of t-tests: Pooled t-test Non-pooled t-test Paired t-test Non-parametric t-test

18. When do I do the what now? “Well, whenever I'm confused, I just check my underwear. It holds the answer to all the important questions.” – Grandpa Simpson If all 4 assumptions are met: Conduct a pooled t-test - you can “pool” the samples because the variances are assumed to be equal If the samples are not independent: Conduct a paired t-test If the variances (std. dev.) are not equal: Conduct a non-pooled t-test If the data is not normal or has small sample size: Conduct a non-parametric t-test (Mann-Whitney)

19. When to pool, when to not-pool “"We have a pool and a pond…The pond would be good for you.” – Ty Webb Both tests are run by Minitab as “2-sample t-test” For pooled test check box – “Assume Equal Variances” For non-pooled, do not check box

20. When to pair, when to not-pair “All I got's two fives!” - Jean LaRose Test is run by Minitab directly as “paired t-test” Used when there is a natural pairing of the members of two populations Each pair consists of a member from one population and that members corresponding member in the other population Use difference between the two sample means

21. When to pair, when to not-pair “All I got's two fives!” - Jean LaRose Paired t-test assumptions: 1. Random Sample 2. Paired difference normally distributed; large n 3. Outliers can confound results Tests whether the difference in the pairs is significantly different from zero

22. When to parametric… Non-parametric t-test (Mann-Whitney): 1. Random Sample – small sample size OK! 2. Do not require normally distributed data 3. Outliers do not confound results Tests whether the difference in the pairs is significantly different from zero Non-parametric test are used heavily in some disciplines – although not typically in the natural sciences – often the “last resort” when data is not collected correctly, low “power”