Introduction to Hypothesis Testing

MARE 250Dr. Jason Turner

We use inferential statistics to make decisions or judgments about data values

Hypothesis testing is the most commonly used method

Hypothesis testing is all about taking scientific questions and translating them into statistical hypotheses with “yes/no” answers

Hypothesis Testing

Hypothesis Testing

Start with a research question

Translate that question into a hypothesis - statement with a “yes/no” answer

Hypothesis crafted into two parts:

Null hypothesis and Alternative Hypothesis – mirror images of each other

Hypothesis Testing

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

Testing…Testing…One…TwoNull hypothesis – a hypothesis to be tested

Symbol (H0) represents Null hypothesis

Symbol (μ) represents Mean

H0: μ1 = μ2 (Null hypothesis = Mean 1 = Mean 2)

Testing…Testing…One…TwoResearch Question – Is there a difference in urchin densities across habitat types?

Null hypothesis – The mean number of urchins in the Deep region are equal to the mean number of urchins in the Shallow region H0: μurchins deep = μurchins shallow

In means tests – the null is always that means at equal

Three choices for Alternative hypotheses:

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

Testing…Testing…One…Two

Testing…Testing…One…Two

Testing…Testing…One…Two

Critical Region-DefinedWe 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

Testing…Testing…One…TwoAlternative hypothesis (research hypothesis) – a hypothesis to be considered as an alternative to the null hypothesis (Ha)

(Ha: μ1 ≠ μ2 )(Alt. hypothesis = Mean 1 ≠ Mean 2)

Testing…Testing…One…TwoResearch Question – Is there a difference in urchin densities across habitat types?

Null hypothesis – The mean number of urchins in the Deep region are equal to the mean number of urchins in the Shallow region

H0: μurchins deep = μurchins shallow

Alternative hypothesis - The mean number of urchins in the Deep region are not equal to the mean number of urchins in the Shallow region

Ha: μurchins deep ≠ μurchins shallow

Testing…Testing…One…TwoImportant terms:

Test statistic – answer unique to each statistical test; (t-test – t, ANOVA – F, correlation – r, regression – R2)

Alpha (α) – critical value; represents the line between “yes” and “no”; is 0.05

P-value – universal translator between test statistic and alpha

Hold on, I have to pP-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

Didn’t you go before we leftP-value – equals the smallest significance level at which the null hypothesis can be rejected

Didn’t you go before we leftP-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

If the p-value is less than or equal to the specified significance level (0.05), reject the null hypothesis, otherwise, do not (fail to) reject the null hypothesis

How to we use p?

Compare p-value from test to specified significance level (alpha, α=0.05)

If the p-value is less than or equal to α=0.05, reject the null hypothesis,

Otherwise, do not reject (fail to) the null hypothesis

No, I didn’t have to go then

p< 0.05 – Reject Null Hypothesis

p> 0.05 – Fail to Reject (Accept) Null

No, I didn’t have to go then

0.05 – value for Alpha (α)with fewest Type I and Type II Errors

Testing…Testing…One…TwoImportant terms:

Test statistic – answer unique to each statistical test; (t-test – t, ANOVA – F, correlation – r, regression – R2)

Alpha (α) – critical value; represents the line between “yes” and “no”; is 0.05

P-value – universal translator between test statistic and alpha

Testing…Testing…One…TwoThree steps:

1) You run a test (based upon your hypothesis) and calculate a Test statistic – T = 2.05

2) You then calculate a p value based upon your test statistic and sample size – p = 0.0001

3) Compare p value with alpha (α) (0.05)

Testing…Testing…One…TwoResearch Question – Is there a difference in urchin densities across habitat types?

Null hypothesis – The mean number of urchins in the Deep region are equal to the mean number of urchins in the Shallow region

H0: μurchins deep = μurchins shallow

Alternative hypothesis - The mean number of urchins in the Deep region are not equal to the mean number of urchins in the Shallow region

Ha: μurchins deep ≠ μurchins shallow

Testing…Testing…One…TwoMeans test is run

Output: T = 2.15 df = 59 p = 0.0001

Do we accept or reject the null hypothesis?

Testing…Testing…One…Twop< 0.05 – Reject Null Hypothesis

Output: T = 2.15 df = 59 p = 0.0001

Since P<0.05 – we reject the null thatH0: μurchins deep = μurchins shallow

and accept the alternative that Ha: μurchins deep ≠ μurchins shallow

Testing…Testing…One…TwoTherefore we reject the Null hypothesis and accept the Alternative hypothesis that:

The mean number of urchins in the Deep region are Significantly Different than the mean number of urchins in the Shallow region