10/4/2013. Hypothesis Testing & z-test. Hypothesis Testing. Hypothesis Testing

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1 & z-test Lecture Set 11 We have a coin and are trying to determine if it is biased or unbiased What should we assume? Why? Flip coin n = 100 times E(Heads) = 50 Why? Assume we count 53 Heads... What could this be due to? Is this likely or unlikely? What if we counted 21 Heads? 77 Heads? 89 Heads? At some point, the discrepancy is statistically significant 1

2 In hypothesis testing: 1. State a hypothesis that assumed to be true 2. Collect data and calculate an observed result 3. Determine probability of observed result, which is, P(Observed Result Hypothesis = True) 4. If the probability is extremely low, the result is extreme rare given the hypothesis is true How low? If P(Observed Result Hypothesis = True).05 Alpha-level (α) Competing hypotheses are established Null hypothesis (H 0 ) Predicts no difference Assumed to be true Alternate hypothesis (H 1 ) Predicts a difference Accepted if H 0 is rejected Non-directional alternate hypothesis Directional alternate hypothesis H 0 : µ A =µ B H 1 : µ A µ B H 1 : µ A <µ B H 1 : µ A >µ B or Z-Test Do Yale students score different on the Quantitative Reasoning section of the GREs compared to all graduate students taking the GREs? Score Range Null H 0 : μ Yale = μ QR or μ Yale = Alternate H 1 :μ Yale μ QR or μ Yale Choose α =.05 2

3 Z-Test Gathering Data Randomly select n = 100 Yale students and obtain GREs M = What could this be due to? We need to know P(M = μ Yale = μ QR ) Use z-test Standard error of mean σ M = Test statistic / obtained value z Obt = = = =2.833 = =0.879 z-test Use test statistic to determine P(M = μ Yale = μ QR ) Go to z-tables: Column 1 Column 2 Column 3 Column 4 Column p-value P(Observed Result H 0 = True) P(M = μ Yale = μ QR ) = z-test P(M = μ Yale = μ QR ) = is less than the chosen alpha-level of α =.05 Statistically significant? Decisions on hypotheses? Reject H 0 Accept H 1 What does statistical significance' mean? 3

4 Z-Test Example 2 Do Harvard students score different on the Verbal Reasoning section of the GREs? Null H 0 : μ Harvard = μ VR or μ Harvard = Alternate H 1 :μ Harvard μ VR or μ Harvard Randomly select n = 81 students M = p =.0601 = =0.933 = = =1.876 Z-Test Example 3 Do Birthday clowns score different on the Verbal Reasoning section of the GREs? Null H 0 : μ Clowns = μ VR or μ Clowns = Alternate H 1 :μ Clowns μ VR or μ Clowns Randomly select n = 25 students M = 147 p =.0257 = =1.68 = = = Directional vs. non-directional Hypotheses Non-directional alternate hypotheses H 1 : µ A µ B Direction of predicted difference is not specified Two-tailed tests Retain H 0 Decision Reject H 0 Retain H 0 Decision Reject H 0 Probability α/2 α/2 4

5 Directional vs. non-directional Hypotheses Directional alternate hypotheses H 1 : µ A <µ B or H 1 :µ A >µ B Direction of predicted difference is specified One-tailed tests Retain H 0 Decision Reject H 0 Probability α Z-Test Example 4 Do UofS students score greater on the Quantitative Reasoning section of the GREs? Null H 0 : μ US = μ QR or μ US = Alternate H 1 :μ US > μ QR or μ US > Randomly select n = 49 students M = 154 p =.0485 = =1.256 = = =1.664 Z-Test Example 4 Do East Buzzkill students score lower on the Quantitative Reasoning section of the GREs? Null H 0 : μ EB = μ QR or μ EB = Alternate H 1 :μ EB < μ QR or μ EB < Randomly select n = 81 students M = 149 p =.0014 = =0.977 = = =

6 Reporting Results in APA Format Descriptive Statistic Symbols for APA Formatting Mean (M) Standard Deviation (SD) Variance (VAR) Inferential Statistics Results Should Include: Test statistic (z, t, F, r) Degrees of freedom (df)* Measure of error (SE, SEM, MSE) p-value or your chosen α-level with number of tails in test* Reporting Results in APA Format Z-Test Generic Format z =, SE =, p <=> ( tails) From Examples Example 1: z = 2.833, SE = 0.879, p =.0047 (two tails) Example 2: z = 1.876, SE = 0.933, p =.0601 (two tails) Example 3: z = , SE = 1.68, p =.0257 (two tails) Example 4: z = 1.664, SE = 1.256, p =.0485 (one tail) Example 5: z = , SE = 0.977, p =.0014 (one tail) Alternate Method for Determining Significance Back to Example 1: H 0 : μ Yale = μ QR H 1 :μ Yale μ QR α =.05 n = 100 M = Locate critical value in z-tables based on alpha-level (α) Column 1 Column 2 Column 3 Column 4 Column z α = ±1.96 6

7 Alternate Method for Determining Significance Perform z-test σ M = z Obt = Because z Obt > z α difference is statistically significant Reject null hypothesis Accept alternate hypothesis 7