Chapter 8 Introduction to Hypothesis Testing Hypothesis Testing

Chapter 8: Introduction to Hypothesis Testing

Hypothesis Testing • A hypothesis test is a statistical method that uses sample data to evaluate a hypothesis about a population. • The general goal of a hypothesis test is to rule out chance (sampling error) as a plausible explanation for the results from a research study. • If M is a distance away from your expected μ, you need some tools to tell you whether your “guess” is “true H 0” or “false H 1”.

Hypothesis Test - Steps 1. State hypothesis (H 0, H 1) about the population. 2. Use hypothesis to predict the characteristics the sample should have. (formalize the decision process: choose α) 3. Obtain a sample from the population. (calculate M, s, and z) 4. Compare data with the hypothesis prediction. (make a decision: reject or failed to reject H 0)

Hypothesis Testing (cont'd. ) • If the individuals in the sample are noticeably different from the individuals in the original population, we have evidence that the treatment has an effect. • However, it is also possible that the difference between the sample and the population is simply sampling error

Example 8. 1 (p. 235) • neuropsychological tests: blueberry (high in antioxidants) v. s. aging (↓cognitive function) • age 65 and up: daily dos of a blueberry supplement for 6 months (n=25, μ=80, σ=20) • after 6 months, give another test M, z=(M-μ)/ σM, • noticeably different effective • if not effective

Hypothesis Testing (cont'd. ) • The purpose of the hypothesis test is to decide between two explanations: 1. The difference between the sample and the population can be explained by sampling error (there does not appear to be a treatment effect) 2. The difference between the sample and the population is too large to be explained by sampling error (there does appear to be a treatment effect).

The Hypothesis Test: Step 1 • State the hypothesis about the unknown population. – The null hypothesis, H 0, states that there is no change in the general population before and after an intervention. In the context of an experiment, H 0 predicts that the independent variable had no effect on the dependent variable. – The alternative hypothesis, H 1, states that there is a change in the general population following an intervention. In the context of an experiment, predicts that the independent variable did have an effect on the dependent variable. Mutually exclusive & collectively exhaustive

LO 10 -2 Step 1: State the Null and the Alternate Hypothesis NULL HYPOTHESIS A statement about the value of a population parameter developed for the purpose of testing numerical evidence. It is represented by H 0. ALTERNATE HYPOTHESIS A statement that is accepted if the sample data provide sufficient evidence that the null hypothesis is false. It is represented by H 1. 10 -10

Important Things to Remember about H 0 and H 1 • H 0 is the null hypothesis; H 1 is the alternate hypothesis. • H 0 and H 1 are mutually exclusive and collectively exhaustive. • H 0 is always presumed to be true. • H 1 has the burden of proof. • A random sample (n) is used to “reject H 0. ” • If we conclude “do not reject H 0, ” this does not necessarily mean that the null hypothesis is true, it only suggests that there is not sufficient evidence to reject H 0; rejecting the null hypothesis, suggests that the alternative hypothesis may be true given the probability of Type I error. • Equality is always part of H 0 (e. g. “=”, “≥”, “≤”). • Inequality is always part of H 1 (e. g. “≠”, “<”, “>”).

p. 236(example 8. 1) • H 0: μ = 80 • H 1: μ ≠ 80 (note: mutually exclusive and collectively exhaustive) • cannot both be true & one of them must be true • a two-tail test

The Hypothesis Test: Step 2 • The α level establishes a criterion, or "cut-off", for making a decision about the null hypothesis. The alpha level also determines the risk of a Type I error. α =. 01, α =. 05 (most used), α =. 001 • The critical region consists of outcomes that are very unlikely to occur if the null hypothesis is true. That is, the critical region is defined by sample means that are almost impossible to obtain if the treatment has no effect. • Once α is determined critical region is set for the hypothesis testing

p. 239 1. class size ↑ negative effect or not? H 0: ? 2. α↑ boundaries↑ (true/false? ) 3. α=0. 02 z*= ? (two-tail test) 1% 2. 575 2% 2. 33 5% 1. 96 10% 1. 645, 0. 1% 3. 3

The Hypothesis Test: Step 3 • Compare the sample means (data) with the null hypothesis. • Compute the test statistic. The test statistic (zscore) forms a ratio comparing the obtained difference between the sample mean and the hypothesized population mean versus the amount of difference we would expect without any treatment effect (the standard error), i. e. z.

Step 4: Formulate a Decision Rule: One-Tail vs. Two-Tail Tests CRITICAL VALUE Based on the selected level of significance, the critical value is the dividing point between the region where the null hypothesis is rejected and the region where it is not rejected. If the test statistic is greater than or less than the critical value (in the region of rejection), then reject the null hypothesis. 17 10 -17

One-Tailed Test versus Two-Tailed Test 10 -18

The Hypothesis Test: Step 4 • If the test statistic results are in the critical region, we conclude that the difference is significant or that the treatment has a significant effect. In this case we reject the null hypothesis. reject H 0 • If the mean difference is not in the critical region, we conclude that the evidence from the sample is not sufficient, and the decision is fail to reject the null hypothesis. cannot reject H 0

p. 241 (example 8. 1) • • • n=25, μ=80, σ=20, M=84 σM =20/5 = 4 H 0: μ = 80 H 1: μ ≠ 80 α =. 05 z = (84 -80)/4 = 1 not in the critical region failed to reject H 0

Analogy for Hypothesis Testing 1. begin with a null hypothesis • H 0: no treatment effect • H 0: innocent • H 0: original μ (before treatment) 2. gather evidence, data, . . . 3. choose acceptable “error” (type I) 4. decision: • enough evidence reject H 0 • not enough evidence failed to reject H 0

z score as. . . • a recipe 1. H 0: guess what’s in the recipe 2. cook and taste it 3. taste good: H 0: maybe true taste bad: H 0: maybe false • a ratio z = sample error / standard error = actual difference / standard difference

Errors in Hypothesis Tests • Just because the sample mean (following treatment) is different from the original population mean does not necessarily indicate that the treatment has caused a change. • You should recall that there usually is some discrepancy between a sample mean and the population mean simply as a result of sampling error.

Errors in Hypothesis Tests (cont'd. ) • Because the hypothesis test relies on sample data, and because sample data are not completely reliable, there is always the risk that misleading data will cause the hypothesis test to reach a wrong conclusion. • Two types of errors are possible.

Errors in Hypothesis Testing

Type I Errors • A Type I error occurs when the sample data appear to show a treatment effect when, in fact, there is none. – In this case the researcher will reject the null hypothesis and falsely conclude that the treatment has an effect. • Type I errors are caused by unusual, unrepresentative samples, falling in the critical region even though the treatment has no effect. • The hypothesis test is structured so that Type I errors are very unlikely; specifically, the probability of a Type I error is equal to the alpha level.

Type II Errors • A Type II error occurs when the sample does not appear to have been affected by the treatment when, in fact, the treatment does have an effect. – In this case, the researcher will fail to reject the null hypothesis and falsely conclude that the treatment does not have an effect. – Type II errors are commonly the result of a very small treatment effect. Although the treatment does have an effect, it is not large enough to show up in the research study.

Type I and Type II Errors Illustrated n = 100, σ = 400, α = 0. 05. H 0: μ = 10, 000, middle 95% z. C = (-1. 96, +1. 96) XC=10, 000+( 1. 96) σM XC = (9921. 6, 10078. 4) ≈ (9922, 10078) it is possible that a sample would have a sample mean greater than 9, 922. See Region B. So we could commit a Type II error: Fail to reject a false null hypothesis. Type II error is 0. 2912 when the population mean is 9, 900.

Type I and Type II Errors Illustrated

p < α significant • • z. M = (M-μ)/σM as critical value p = Prob(|z|>z. M) for 2 -tailed test p = Prob(z>z. M) for 1 -tailed test (right-hand tail) p = Prob(z<z. M) for 1 -tailed test (left-hand tail) • if p < α should reject H 0 statistically significant

p < α significant ↓rejection region p < α significant

z test is be influenced by 1. σ↑ σM ↑ z ↓ less likely to reject H 0 test statistics ↑ ↑ ↓ 2. n ↑ σM↓ z ↑ more likely to reject H 0 ↑ ↓ ↑

Basic assumption for Hypothesis Testing • Random sampling • Independent observations Box 8. 1 • σ unchanged by the treatment • Normal distribution

p. 255 1. μ = 10. 5, σ = 4. 8, n = 16, M = 15. 9, normal a. α = 0. 01, significant or not? z = (15. 9 -10. 5)/(4. 8/4) = 4. 5 b. write a report. Texting had a significant effect on driving and p < 0. 01. 5. σ = 2, σ = 10, which is more likely to reject H 0? σ↑ z ↓ more difficult to reject H 0 σ↓ z ↑ more likely to reject H 0

Directional Tests (one-tailed test) • When a research study predicts a specific direction for the treatment effect (increase or decrease), it is possible to incorporate the directional prediction into the hypothesis test. • The result is called a directional test or a onetailed test. A directional test includes the directional prediction in the statement of the hypotheses and in the location of the critical region.

Directional Tests (cont'd. ) • For example, if the original population has a mean of μ = 80 and the treatment is predicted to increase the scores, then the null hypothesis would state that after treatment: H 0: μ ≤ 80 (there is no increase) • In this case, the entire critical region would be located in the right-hand tail of the distribution because large values for M would demonstrate that there is an increase and would tend to reject the null hypothesis. H 1: μ > 80

example 8. 4 & p. 257 -258 • • μ = 80, σ = 20, n = 25 σM = 20/5 = 4 if α = 0. 01 critical value: z* = 2. 33 if α = 0. 025 critical value: z* = 1. 96 if α = 0. 05 critical value: z* = 1. 645 • Now α=0. 05, M=87, H 1: μ > 80, z* = 1. 645 test statistics: z=(87 -80)/4=1. 75 reject H 0 if H 1: μ ≠ 80, α=0. 05, M=87, z* = 1. 96 test statistics: z=1. 75 failed to reject H 0

~ One-Tailed Test versus Two-Tailed Test for p. 258 10 -39

two-tailed vs. one-tailed • 2 -tailed test: - more rigorous, more convincing when H 0 is rejected - need more evidence (i. e. ∆=(M-μ) ) to reject H 0, ∆=(M-μ) : treatment effect • 1 -tailed test: - more sensitive (small ∆ can be significant) - more precise (test a specific directional effect)

Box 8. 2 (p. 260) • type I error (α) is “true” only if H 0 is true. • If H 0 is false, then α tells you nothing about the population distribution and your hypothesis. • Suppose: 80% H 0 is true, and 20% H 0 is false. for 125 tests, 100 H 0 is true, 25 H 0 is false • if α = 0. 05 5 out of 100’s H 0 is wrongly rejected • Suppose: when H 0 is false, 60% is correctly rejected 15 out of 25 H 0 is correctly rejected 20 out of 125’s H 0 is reject (20 significant results) • True probability of type I error (H 0 true but rejected) = 5/20 = 0. 25 • So, ¼ of significant research results has type I error!!

Limitations of Hypothesis Testing 1. the test depend on data rather than the hypothesis reject H 0 ≈ M is very unlikely to be so far away from μ ≈ H 0 is very likely to be false ≠ H 0 is truly false 2. significant ≠ big effect (treatment effect maybe small) (M- μ)↑ z ↑ more likely to be significant n↑ σM↓ z ↑ more likely to be significant σ↓ z ↑ more likely to be significant

example 8. 5 (p. 261) • μ =5, σ = 10, M = 51, n = 25, • treatment effect = 51 -50 = 1 (quite small) • 2 -tailed test: n = 25, z = (51 -50)/(10/5)=0. 5 < 1. 96 failed to reject H 0 but if n = 400 z = (51 -50)/(10/20) = 2 reject H 0

Measuring Effect Size • A hypothesis test evaluates the statistical significance of the results from a research study. • That is, the test determines whether or not it is likely that the obtained sample mean occurred without any contribution from a treatment effect. • The hypothesis test is influenced not only by the size of the treatment effect (M-μ) but also by the size of the sample (σM ). • Thus, even a very small effect can be significant if it is observed in a very large sample. • n ↑ σM↓ z ↑ more likely to reject H 0

Measuring Effect Size • Because a significant effect does not necessarily mean a large effect, it is recommended that the hypothesis test be accompanied by a measure of the effect size. • We use Cohen’s d as a standardized measure of effect size. • Much like a z-score, Cohen’s d measures the size of the mean difference in terms of the standard deviation.

Measuring Effect Size • • Effect size = absolute size of treatment effect Effect size should be independent of n simplest, most direct effect size measure = d Cohen’s d :

example 8. 5 (p. 261) • μ =5, σ = 10, M = 51, n = 25, • treatment effect = 51 -50 = 1 • 2 -tailed test : z = (51 -50)/(10/5)=0. 5 < 1. 96 failed to reject H 0 if n = 400 z = (51 -50)/(10/20) = 2 reject H 0 • effect size: Cohen’s d = (M - μ) / σ M: estimated population mean with/after treatment μ: population mean without/before treatment Cohen’s d = (51 -50)/10 = 0. 1 (for both n)

p. 262 -263 Case 1 (Fig. 8. 11 (a)) • no treatment: μ =500, σ = 100, • after treatment: μ =515, σ = 100, d = 15/100 = 0. 15 (the size of treatment effect is 0. 15 standard deviation) Case 2 (Fig. 8. 11 (b)) • no treatment: μ =100, σ = 15, • after treatment: μ =115, σ = 15, d = 15/15 = 1 (the size of treatment effect is 1 standard deviation)

effect size: Cohen’s d • mean difference ↑ Cohen’s d ↑ • σ ↓ Cohen’s d ↑

effect size: Cohen’s d • d = 0. 2 small effect • d = 0. 5 medium effect • d = 0. 8 large effect

p. 265 1. n↑ σM ↓ z ↑ more likely to reject H 0 • n↑ Cohen’s d ? 2. μ = 45, σ = 8, M = 47 • d = (47 -45)/8 = 0. 25

Power of a Hypothesis Test • The power of a hypothesis test is defined is the probability that the test will reject the null hypothesis when the treatment does have an effect. P(reject H 0 | H 0 is false) = 1 -β • The power of a test depends on a variety of factors, including the size of the treatment effect and the size of the sample. • β = P(failed to reject H 0 | H 0 is false)

Example 8. 6 (p. 266 -267) • normal: μ = 80, σ = 10, H 1: μ ≠ 80 (2 -tailed test) Case 1: n = 25, α = 0. 05 Zc = 1. 96 Xc = 80 1. 96*(10/5) Xc = (76. 08, 83. 92) if true μ = 88 recalculate zc upper Zc = (83. 92 - 88)/2= -2. 04 lower Zc = (76. 08 - 88)/2 = -5. 96 1 -β = P(z < -5. 96)+P(z > -2. 04) ≈ P(z > -2. 04) = 0. 4793+0. 5= 0. 9793

-5. 96 -2. 04

Example 8. 6 (p. 268 -269) Case 2: n = 4, α = 0. 05, μ = 80, σ = 10 Xc = 80 1. 96*(10/2) Xc = (70. 2, 89. 8) upper Zc = (89. 8 - 88)/5= 0. 36, lower Zc = (70. 2 - 88)/5 = -3. 56 1 -β = P(z<-3. 56)+P(z>0. 36) ≈ P(z > 0. 36) = 0. 5 – 0. 1406 = 0. 3594

p. 270 1. power of test = 1 -β = 0. 5, M-μ = 5 for M-μ = 10, 1 -β ↑↓? (see Fig 8. 13 and Fig 8. 12) 2. 1 -β ↑ type II error ↑↓? 3. n ↑ 1 -β ↑↓? Other things being equal, the greater the sample size, the greater the power of the test. 4. Fig 8. 13, find 1 -β = ?