Introduction to Hypothesis Testing for μ Research Problem: Infant Touch Intervention Designed to increase child growth/weight Weight at age 2: Known population: μ = 26 σ = 4 Sample data: n = 16 = 30 Did intervention increase weight?

Hypothesis Testing: Using sample data to evaluate an hypothesis about a population parameter. Usually in the context of a research study evaluate effect of a “treatment” Compare to known μ Can’t take difference at face value Differences between and μ expected simply on the basis of chance sampling variability How do we know if it’s just chance? Sampling distributions!

Research Problem: Infant Touch Intervention Known population: μ = 26 σ = 4 Assume intervention does NOT affect weight Sample means ( ) should be close to population μ

Compare Sample Data to know population: z-test = How much does deviate from μ? What is the probability of this occurrence? How do we determine this probability?

Distribution of Sample Means (DSM)! in the tails are low probability How do we judge “low” probability of occurrence? Widely accepted convention..... < 5 in a 100 p <.05

Logic of Hypothesis Testing Rules for deciding how to decide! Easier to prove something is false Assume opposite of what you believe… try to discredit this assumption…. Two competing hypotheses: (1)Null Hypothesis (H 0 ) The one you assume is true The one you hope to discredit (2)Alternative Hypothesis (H 1 ) The one you think is true

Inferential statistics: Procedures revolve around H 0 Rules for deciding when to reject or retain H 0 Test statistics or significance tests: Many types: z-test t-test F-test Depends on type of data and research design Based on sampling distributions, assumes H 0 is true If observed statistic is improbable given H 0, then H 0 is rejected

Hypothesis Testing Steps: (1)State the Research Problem Derived from theory example: Does touch increase child growth/weight? (2)State statistical hypotheses Two contradictory hypotheses: (a)Null Hypothesis: H 0 There is no effect (b)Scientific Hypothesis: H 1 There is an effect Also called alternative hypothesis

Form of Ho and H1 for one-sample mean: H 0 : μ = 26 H 1 : μ <> 26 Always about a population parameter, not a statistic H 0 : μ = population value H 1 : μ <> population value non-directional (two-tailed) hypothesis mutually exclusive :cannot both be true

Example: Infant Touch Intervention Known population:μ = 26 σ = 4 Did intervention affect child weight? Statistical Hypotheses: H 0 :μ = 26 H 1 :μ <> 26

Hypothesis Testing Steps: (3)Create decision rule Decision rule revolves around H 0, not H 1 When will you reject Ho? …when values of are unlikely given H 0 Look in tails of sampling distribution Divide distribution into two parts: Values that are likely if H 0 is true Values close to H 0 Values that are very unlikely if H 0 is true Values far from H 0 Values in the tails How do we decide what is likely and unlikely?

Level of significance = alpha level = α Probability chosen as criteria for “unlikely” Common convention: α =.05 (5%) Critical value = boundary between likely/unlikely outcomes Critical region = area beyond the critical value

Decision rule: Reject H 0 when observed test- statistic (z) equals or exceeds the Critical Value (when z falls within the Critical Region) Otherwise, Retain H 0

Hypothesis Testing Steps: (4) Collect data and Calculate “observed” test statistic z-test for one sample mean: A closer look at z: z = sample mean – hypothesized population μ standard error z = observed difference difference due to chance

Hypothesis Testing Steps: (5)Make a decision Two possible decisions: Reject H 0 Retain (Fail to Reject) H 0 Does observed z equal or exceed CV? (Does it fall in the critical region?) If YES, Reject H 0 = “statistically significant” finding If NO, Fail to Reject H 0 = “non- significant” finding

Hypothesis Testing Steps: (6)Interpret results Return to research question statistical significance = not likely to be due to chance Never “prove” or H 0 or H 1

Example (1)Does touch increase weight? Population:μ = 26 σ = 4 (2)Statistical Hypotheses: H 0 : μ = H 1 : μ <> (3)Decision Rule: α =.05 Critical value: (4)Collect sample data: n = 16 = 30 Compute z-statistic: (5)Make a decision: (6)Interpret results: Intervention appears to increase weight. Difference not likely to be due to chance.

More about alpha ( α ) levels: most common : α =.05 more stringent : α =.01 α =.001 Critical values for two-tailed z- test: α =.05α =.01α =.001 ± 1.96± 2.58± 3.30

More About Hypothesis Testing I. Two-tailed vs. One-tailed hypotheses A.Two-tailed (non-directional): H 0 :  = 26 H 1 :   26 Region of rejection in both tails: Divide α in half: probability in each tail = α / 2 p=.025   =.05

B.One-tailed (directional): H 0 :   26 H 1 :  > 26 Upper tail critical: H 0 :   26 H 1 :  < 26 Lower tail critical: p=.05 z  1.65 p=.05 z

Examples: Research hypotheses regarding IQ, where  hyp = 100 (1)Living next to a power station will lower IQ? H 0 : H 1 : (2)Living next to a power station will increase IQ? H 0 : H 1 : (3)Living next to a power station will affect IQ? H 0 : H 1 : When in doubt, choose two-tailed!

II. Selecting a critical value Will be based on two pieces of information: (a) Desired level of significance (α)? α =alpha level (b)Is H 0 one-tailed or two-tailed? If one-tailed: find CV for α CV will be either + or - If two-tailed: find CV for α /2 CV will be both +/ - Most Common choices: α =.05 two-tailed test

Commonly used Critical Values for the z-statistic Hypothesis α =.05 α =.01 ______________________________________________ Two-tailed  1.96  2.58 H0:  = x H1:   x One-tailed upper H0:   x H1:  > x One-tailed lower  1.65  2.33 H0:   x H1:  < x ______________________________________________ Where x = any hypothesized value of  under H0 Note: critical values are larger when: a more stringent (.01 vs..05) test is two-tailed vs. one-tailed

III.Outcomes of Hypothesis Testing Four possible outcomes: True status of H0 No EffectEffect H 0 true H 0 false Reject H 0 Decision Retain H 0 Type I Error:Rejecting H0 when it’s actually true Type II Error:Retaining H0 when it’s actually false We never know the “truth” Try to minimize probability of making a mistake

A.Assume Ho is true Only one mistake is relevant  Type I error α =level of significance p (Type I error) 1- α = level of confidence p(correct decision), when H0 true if α =.05, confidence =.95 if α =.01, confidence =.99 So, mistakes will be rare when H 0 is true! How do we minimize Type I error? WE control error by choosing level of significance (α) Choose α =.01 or.001 if error would be very serious Otherwise, α =.05 is small but reasonable risk

B.Assume Ho is false Only one mistake is relevant  Type II error  = probability of Type II error 1-  = ”Power” p(correct decision), when H0 false How big is the “treatment effect”? When “effect size” is big: Effect is easy to detect  is small (power is high) When “effect size” is small: Effect is easy to “miss”  is large (power is low)

How do you determine  and power (1-  ) No single value for any hypothesis test Requires us to guess how big the “effect” is Power = probability of making a correct decision when H 0 is FALSE C.How do we increase POWER? Power will be greater (and Type II error smaller): Larger sample size (n) Single best way to increase power! Larger treatment effect Less stringent a level e.g., choose.05 vs..01 One-tailed vs. two-tailed tests

Four Possible Outcomes of an Hypothesis Test True status of H0 H 0 true H 0 false Reject H 0 Decision Retain H 0 α = level of significance probability of Type I Error risk of rejecting a true H 0 1- α =level of confidence p (making correct decision), if H 0 true  = probability of Type II Error risk of retaining a false H 0 1-  = power p(making correct decision), if H 0 false ability to detect true effect  1-  Type I ErrorPower 1-   Confidence Type II Error

IV.Additional Comments A.Statistical significance vs. practical significance “Statistically Significant” = H 0 rejected B.Assumptions of the z-test (see book for review): DSM is normal Known  (and  unaffected by treatment) Random sampling Independent observations Rare to actually know  ! Preview  use t statistic when  unknown

V.Reporting Results of an Hypothesis Test If you reject H 0 : “There was a statistically significant difference in weight between children in the intervention sample (M = 30 lbs) and the general population (M = 30 lbs), z = 4.0, p <.05, two-tailed.” If you fail to reject H 0 : “There was no significant difference in weight between children in the intervention sample (M = 30 lbs) and the general population (M = 30 lbs), z = 1.0, p >.05, two-tailed.”

A closer look… z = 4.0, p <.05 test statistic observed value level of significance

VI.Effect Size Statistical significance vs. practical importance How large is the effect, in practical terms? Effect size = descriptive statistics that indicate the magnitude of an effect Cohen’s d Difference between means in standard deviation units Guidelines for interpreting Cohen’s d Effect Sized Small .20 Medium.20 < d .80 Larged >.80