Hypothesis Testing MARE 250 Dr. Jason Turner

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 hypothesis – a hypothesis to be tested (H0) H0: μ = μ0 Alternative hypothesis (research hypothesis) – a hypothesis to be considered as an alternative to the null hypothesis (Ha) Three possible choices: 1. Mean is Different From a specified value – two-tailed test Ha: μ ≠ μ0 2. Mean is Less Than a specified value – left-tailed test Ha: μ < μ0 3. Mean is Greater Than a specified value – right-tailed test Ha: μ > μ0 One-tailed tests

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 1 2 3 4 Blue marlin Mako shark Little tunny Warsaw grouper Greater amberjack Blackfin tuna Yellowfin tuna Dolphin 5 10 15 8.3 Key Mean FDA (1.0) Wahoo King mackerel Cobia Gag grouper Hg (ppm) SD

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 1 2 3 4 Blue marlin Mako shark Little tunny Warsaw grouper Greater amberjack Blackfin tuna Yellowfin tuna Dolphin 5 10 15 8.3 Key Mean FDA (1.0) Wahoo King mackerel Cobia Gag grouper Hg (ppm) SD

For Example… One Sample t-test The null hypothesis for this test is: “the mean Hg level for yellowfin tuna equals the FDA level of 1.0 ppm” H0: 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” Ha: μ < 1.0 ppm 3. The hypothesis test is left-tailed because the less-than-sign (<) appears in the alternative hypothesis

Hypothesis Testing for the Rest of Us Hypothesis tests for one population mean when σ is unknown

Hypothesis Testing for Two Means

For Example… 2 Sample t-test The null hypothesis for this test is: “the mean Hg level for yellowfin tuna from Hilo equals the Hg level from Kona” H0: μHilo = μKona 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”

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 It basically gives you odds that you sample test is a correct representation of your population

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 H0

No, I didn’t have to go then

{ { { { 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 0.025 0.05 0.05 { { { {

First name Mr., last name 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

To ASSUME is to make an… However... Four assumptions for t-test hypothesis testing:

Let Us Review…

Let Us Review…

When do I do the what now? 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)

When to pool, when to not-pool 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

When to pair, when to not-pair 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

When to pair, when to not-pair 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

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”