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MGMT 276: Statistical Inference in Management Fall 2015 Welcome

Schedule of readings Before our next exam (November 10th) OpenStax Chapters 1 – 10 and Chapter 13 Plous (2, 3, & 4) Chapter 2: Cognitive Dissonance Chapter 3: Memory and Hindsight Bias Chapter 4: Context Dependence

Homework On class website: Please print and complete homework worksheet #13 Hypothesis Testing using two-sample t-tests Due: Tuesday November 3rd

By the end of lecture today 10/29/15 Logic of hypothesis testing Hypothesis testing with z-score and t-scores (one-sample) Hypothesis testing with t-scores (two independent samples) Constructing brief, complete summary statements

Homework Review 26.08 < µ < 33.92 mean + z σ = 30 ± (1.96)(2) 95% 26.08 < µ < 33.92 mean + z σ = 30 ± (1.96)(2) 99% 24.84 < µ < 35.16 mean + z σ = 30 ± (2.58)(2)

Melvin Melvin Mark Difference not due sample size because both samples same size Difference not due population variability because same population Yes! Difference is due to sloppiness and random error in Melvin’s sample Melvin

6 – 5 = 4.0 .25 Two tailed test 1.96 (α = .05) 1 1 = = .25 16 4 √ 4.0 z- score : because we know the population standard deviation Ho: µ = 5 Bags of potatoes from that plant are not different from other plants Ha: µ ≠ 5 Bags of potatoes from that plant are different from other plants Two tailed test 1.96 (α = .05) 1 1 = = .25 6 – 5 16 4 √ = 4.0 .25 4.0 -1.96 1.96

Because the observed z (4.0 ) is bigger than critical z (1.96) These three will always match Yes Yes Probability of Type I error is always equal to alpha Yes .05 1.64 No Because observed z (4.0) is still bigger than critical z (1.64) 2.58 No Because observed z (4.0) is still bigger than critical z(2.58) there is a difference there is not there is no difference there is 1.96 2.58

89 - 85 Two tailed test (α = .05) n – 1 =16 – 1 = 15 -2.13 2.13 t- score : because we don’t know the population standard deviation Two tailed test (α = .05) n – 1 =16 – 1 = 15 Critical t(15) = 2.131 89 - 85 2.667 6 √ 16

Because the observed z (2.67) is bigger than critical z (2.13) These three will always match Yes Yes Probability of Type I error is always equal to alpha Yes .05 1.753 No Because observed t (2.67) is still bigger than critical t (1.753) 2.947 Yes Because observed t (2.67) is not bigger than critical t(2.947) No These three will always match No No consultant did improve morale she did not consultant did not improve morale she did 2.131 2.947

Value of observed statistic Finish with statistical summary z = 4.0; p < 0.05 Or if it *were not* significant: z = 1.2 ; n.s. Start summary with two means (based on DV) for two levels of the IV Describe type of test (z-test versus t-test) with brief overview of results n.s. = “not significant” p<0.05 = “significant” The average weight of bags of potatoes from this particular plant is 6 pounds, while the average weight for population is 5 pounds. A z-test was completed and this difference was found to be statistically significant. We should fix the plant. (z = 4.0; p<0.05) Value of observed statistic

Value of observed statistic Finish with statistical summary t(15) = 2.67; p < 0.05 Or if it *were not* significant: t(15) = 1.07; n.s. Start summary with two means (based on DV) for two levels of the IV Describe type of test (z-test versus t-test) with brief overview of results n.s. = “not significant” p<0.05 = “significant” The average job-satisfaction score was 89 for the employees who went On the retreat, while the average score for population is 85. A t-test was completed and this difference was found to be statistically significant. We should hire the consultant. (t(15) = 2.67; p<0.05) Value of observed statistic df

Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses Step 2: Decision rule Alpha level? (α = .05 or .01)? One or two tailed test? Balance between Type I versus Type II error Critical statistic (e.g. z or t or F or r) value? Step 3: Calculations Step 4: Make decision whether or not to reject null hypothesis If observed z (or t) is bigger then critical z (or t) then reject null Step 5: Conclusion - tie findings back in to research problem

Hypothesis testing Is this a single sample or two sample test? Is it a z or a t test or an ANOVA? Hypothesis testing Is it a one-tail test or a two tail test? Step 1: Identify the research problem Did the sheriff keep her promise to change response times from the previous average of 30 minutes? Step 2: Describe the null and alternative hypotheses Ho: The response times are not changed H1: The response times did change As the new chief of police, I am going to change response times for traffic accidents. Before I started the average response time was 30 minutes 17

Hypothesis testing Two-tailed test n = 10 df = 9 alpha = 0.05 Gather the data: We measured the time for police to respond to 10 accidents Two-tailed test One or two tailed test? What is our sample size What is size of our degrees of freedom? What is our alpha What is our critical t value? n = 10 (df = 9) Alpha = .05 Decision rule: critical t = 2.62 n = 10 df = 9 alpha = 0.05 18

Hypothesis testing Two-tailed test Alpha of 0.05 Gather the data: We measured the time for police to respond to 10 accidents One or two tailed test? What is our sample size What is size of our degrees of freedom? What is our alpha What is our critical t value? Decision rule: critical t = 2.262 Two-tailed test Alpha of 0.05 Critical t (9) = 2.262 19

Average time for response before 30 minutes Step 3: Calculations: Average time for response before 30 minutes Average time for response after 24 minutes Observed t = - 2.71 Step 4: Make decision whether or not to reject null hypothesis Observed t = - 2.71 Critical t = 2.262 -2.71 is farther out on the curve than 2.262 so, we do not reject the null hypothesis Step 5: Conclusion: There appears to be a significant difference between the sheriff’s times and 30 minutes 20

Hypothesis testing: Did the sheriff keep her promise to reduce response times to less than 30 minutes? Start summary with two means (based on DV) for two levels of the IV notice we are comparing a sample mean with a population mean: single sample t-test Finish with statistical summary t(9) = -5.71; p < 0.05 Describe type of test (t-test versus anova) with brief overview of results Or if it had been different results that *were not* significant: t(9) = -1.71; ns The mean response time for following the sheriff’s new plan was 24 minutes, while the mean response time prior to the new plan was 30 minutes. A t-test was completed and there appears to be a significant difference in the response time following the implementation of the new plan t(9) = -2.71; p < 0.05 Type of test with degrees of freedom n.s. = “not significant” p<0.05 = “significant” n.s. = “not significant” p<0.05 = “significant” Value of observed statistic 21

A note on z scores, and t score: . . A note on z scores, and t score: Numerator is always distance between means (how far away the distributions are) Denominator is always measure of variability (how wide or much overlap there is between distributions) Difference between means Difference between means Difference between means Variability of curve(s) Variability of curve(s) Variability of curve(s)

Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) How is a single sample t-test different than two sample t-test? Describe the null and alternative hypotheses How is a single sample t-test most similar to the two sample t-test? Step 2: Decision rule Alpha level? (α = .05 or .01)? Critical statistic (e.g. z or t) value? Step 3: Calculations Step 4: Make decision whether or not to reject null hypothesis If observed z (or t) is bigger then critical z (or t) then reject null Single sample standard deviation versus average standard deviation for two samples Single sample has one “n” while two samples will have an “n” for each sample Step 5: Conclusion - tie findings back in to research problem

Independent samples t-test Are the two means significantly different from each other, or is the difference just due to chance? Independent samples t-test Donald is a consultant and leads training sessions. As part of his training sessions, he provides the students with breakfast. He has noticed that when he provides a full breakfast people seem to learn better than when he provides just a small meal (donuts and muffins). So, he put his hunch to the test. He had two classes, both with three people enrolled. The one group was given a big meal and the other group was given only a small meal. He then compared their test performance at the end of the day. Please test with an alpha = .05 Big Meal 22 25 Small meal 19 23 21 Mean= 21 Mean= 24 Got to figure this part out: We want to average from 2 samples - Call it “pooled” t = 24 – 21 variability x1 – x2 t = variability 24

α = .05 Hypothesis testing Step 1: Identify the research problem Did the size of the meal affect the learning / test scores? Step 2: Describe the null and alternative hypotheses Ho: The size of the meal has no effect on test scores H1: The size of the meal does have an effect on test scores Step 3: Decision rule α = .05 One tail or two tail test? 25

Notice: Two different ways to think about it Hypothesis testing Step 3: Decision rule α = .05 n1 = 3; n2 = 3 Degrees of freedom total (df total) = (n1 - 1) + (n2 – 1) = (3 - 1) + (3 – 1) = 4 Degrees of freedom total (df total) = (n total - 2) two tailed test Notice: Two different ways to think about it Critical t(4) = 2.776 26

two tail test α= .05 (df) = 4 Critical t(4) = 2.776 27

Notice: Simple Average = 3.5 Mean= 21 Mean= 24 Big Meal Deviation From mean -2 1 Small Meal Deviation From mean -2 2 Squared deviation 4 1 Squared Deviation 4 Big Meal 22 25 Small meal 19 23 21 Σ = 6 Σ = 8 = 1.732 6 Notice: s2 = 3.0 1 2 1 Notice: Simple Average = 3.5 = 2.0 8 Notice: s2 = 4.0 2 2 2 S2pooled = (n1 – 1) s12 + (n2 – 1) s22 n1 + n2 - 2 S2pooled = (3 – 1) (1.732) 2 + (3 – 1) (2)2 31 + 32 - 2 = 3.5 28

Mean= 21 Mean= 24 Big Meal Deviation From mean -2 1 Small Meal Deviation From mean -2 2 Squared deviation 4 1 Squared Deviation 4 Participant 1 2 3 Big Meal 22 25 Small meal 19 23 21 Σ = 6 Σ = 8 S2p = 3.5 = 24 – 21 1.5275 24 - 21 = 1.964 3.5 3.5 3 3 Observed t 29

Hypothesis testing Step 5: Make decision whether or not to reject null hypothesis Observed t = 1.964 Critical t = 2.776 1.964 is not farther out on the curve than 2.776 so, we do not reject the null hypothesis t(4) = 1.964; n.s. Step 6: Conclusion: There appears to be no difference in test scores between the two groups 30

How to report the findings for a t-test Mean of small meal was 21 How to report the findings for a t-test Mean of big meal was 24 One paragraph summary of this study. Describe the IV & DV. Present the two means, which type of test was conducted, and the statistical results. Finish with statistical summary t(4) = 1.96; ns Start summary with two means (based on DV) for two levels of the IV Observed t = 1.964 df = 4 Or if it *were* significant: t(9) = 3.93; p < 0.05 Describe type of test (t-test versus anova) with brief overview of results We compared test scores for large and small meals. The mean test scores for the big meal was 24, and was 21 for the small meal. A t-test was calculated and there appears to be no significant difference in test scores between the two types of meals t(4) = 1.964; n.s. Type of test with degrees of freedom n.s. = “not significant” p<0.05 = “significant” n.s. = “not significant” p<0.05 = “significant” Value of observed statistic 31

Figuring out the two means using Excel Participant 1 2 3 Big Meal 22 25 Small meal 19 23 21 32

Figuring out the two means Participant 1 2 3 Big Meal 22 25 Small meal 19 23 21 33

Figuring out the two means Participant 1 2 3 Big Meal 22 25 Small meal 19 23 21 Mean= 21 Mean= 24 34

May want to remove means so don’t include in the t-test Complete a t-test Mean= 24 Participant 1 2 3 Big Meal 22 25 Small meal 19 23 21 May want to remove means so don’t include in the t-test 35

Complete a t-test Mean= 21 Mean= 24 Participant 1 2 3 Big Meal 22 25 Small meal 19 23 21 If checked you’ll want to include the labels in your variable range If checked, you’ll want to include the labels in your variable range If checked you’ll want to include the labels in your variable range 36

Finding Means Finding Means 37

Finding degrees of freedom 38

Finding Observed t 39

Finding Critical t 40

Finding p value (Is it less than .05?) 41

42

How to report the findings for a t-test Mean of small meal was 21 How to report the findings for a t-test Mean of big meal was 24 One paragraph summary of this study. Describe the IV & DV. Present the two means, which type of test was conducted, and the statistical results. Finish with statistical summary t(4) = 1.96; ns Start summary with two means (based on DV) for two levels of the IV Observed t = 1.964 df = 4 Or if it *were* significant: t(9) = 3.93; p < 0.05 Describe type of test (t-test versus anova) with brief overview of results We compared test scores for large and small meals. The mean test scores for the big meal was 24, and was 21 for the small meal. A t-test was calculated and there appears to be no significant difference in test scores between the two types of meals t(4) = 1.964; n.s. Type of test with degrees of freedom n.s. = “not significant” p<0.05 = “significant” n.s. = “not significant” p<0.05 = “significant” Value of observed statistic 43

Thank you! See you next time!!