MARE 250 Dr. Jason Turner

1. Hypothesis Testing II MARE 250 Dr. Jason Turner

2. To ASSUME is to make an… However... 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

3. 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)

4. 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

5. Assessing Equal Variances… Equality of variance can checked by performing an F-test Often not recommended: Although pooled t-test is moderately robust to unequal variances, F test is extremely non-robust to such inequalities What? Pooled t-test will allow you to run an accurate test with some degree of unequal variance F-test is much more specific than pooled-t

6. Who did the What Now… What?

7. Assessing Equal Variances… F-test and Levene’s used to judge the equality of variances. In both tests, the null hypothesis (Ho) is that the population variances under consideration (or equivalently, the population standard deviations) are equal, and the alternative hypothesis (Ha) is that the two variances are not equal. The choice of test depends on distribution properties

8. What the F…? Use the F-test when the data come from a normal distribution - is not robust to departures from normality Use Levene's test when the data come from continuous, but not necessarily normal, distributions is less sensitive than the F-test, so use the F-test when your data are normal or nearly normal

9. When the F…? MINITAB calculates and displays a test statistic and p-value for both the F-test and Levene's test Ho: σ1 = σ2 2 population variances equal Ha: σ1 ≠σ2 2 variances are not equal High p-values (above α-level) Fail to Reject Null - indicate no statistically significant difference between the variances (equality or homogeneity of variances) Low p-values (below α-level) Reject Null - indicate a difference between the variances (inequality of variances)

10. How the F…? STAT – Basic Statistics – 2-Variances Enter columns of data as before Under “Options” can modify α-level of test (but why would you do that) Note that by default, MINITAB gives you the results of both the F-test and Levene’s Must decide a priori which test you plan to utilize

11. Significance Level The probability of making a TYPE I Error (rejection of a true null hypothesis) is called the significance level (α) of a hypothesis test TYPE II Error Probability (β) – nonrejection of a false null hypothesis For a fixed sample size, the smaller we specify the significance level (α) , the larger will be the probability (β) , of not rejecting a false hypothesis

12. I have the POWER!!! The power of a hypothesis test is the probability of not making a TYPE II error (rejecting a false null hypothesis) t evidence to support the alternative hypothesis POWER = 1 - β Produce a power curve

13. We need more POWER!!! For a fixed significance level, increasing the sample size increases the power Therefore, you can run a test to determine if your sample size HAS THE POWER!!! By using a sufficiently large sample size, we can obtain a hypothesis test with as much power as we want

14. Power - the probability of being able to detect an effect of a given size Sample size - the number of observations in each sample Difference (effect) - the difference between μ for one population and μ for the other

15. Increasing the power of the test • There are four factors that can increase the power of a two-sample t-test: • Larger effect size (difference) - The greater the real difference between m for the two populations, the more likely it is that the sample means will also be different. • Higher α-level (the level of significance) - If you choose a higher value for α, you increase the probability of rejecting the null hypothesis, and thus the power of the test. (However, you also increase your chance of type I error.) • 3. Less variability - When the standard deviation is smaller, smaller differences can be detected. • 4. Larger sample sizes - The more observations there are in your samples, the more confident you can be that the sample means represent m for the two populations. Thus, the test will be more sensitive to smaller differences.

16. Increasing the power of the test The most practical way to increase power is often to increase the sample size However, you can also try to decrease the standard deviation by making improvements in your process or measurement

17. Sample size Increasing the size of your samples increases the power of your test You want enough observations in your samples to achieve adequate power, but not so many that you waste time and money on unnecessary sampling If you provide the power that you want the test to have and the difference you want it to be able to detect, MINITAB will calculate how large your samples must be

18. 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

19. 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

20. Paired Test - Example For Example… If you are testing the effects of some experimental treatment upon a population e.g. – effect of new diet upon a single sample of fish However… Paired test must have equal sample sizes

21. When to parametric… Nonparametric procedures Statistical procedures that require very few assumptions about the underlying population. They are often used when the data are not from a normal population.

22. Non-Parametric Non-parametric t-test (Mann-Whitney): 1. Random Sample 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”

23. Drawbacks of Nonparametric Tests Nonparametric tests: Less powerful than parametric tests. Thus, you are less likely to reject the null hypothesis when it is false. Often require you to modify the hypotheses. For example, most nonparametric tests concerning the population center are tests about the median rather than the mean. The test does not answer the same question as the corresponding parametric procedure. When a choice exists and you are reasonably certain that the assumptions for the parametric procedure are satisfied, then use the parametric procedure.