BNAD 276 Statistical Inference in Management Winter 2015

BNAD 276: Statistical Inference in Management Winter, 2015 Green sheet Seating Chart http: //www. youtube. com/watch? v=Ahg 6 qcgoay 4&watch_response

Daily group portfolios Beginning of each lecture (first 5 minutes) • Meet in groups of 3 or 4 • Quiz one another on class material • Discuss the questions and determine the correct answer for each question • Five copies (one for each group member – and typed) 3 multiple choice questions based on lecture • Include 4 options (a, b, c, and d) • Include a name and describe a person in a certain situation Margaret was interested in taking a Statistics course. It is likely she was interested in studying which of the following? a. economic theories of communism b. theological perspectives of life after death c. musical compositions of the 12 th century d. statistical techniques and inference They can be funny or serious, and must be clear and have only one correct answer.

. Homework Assignment Go to D 2 L - Click on “Content” Click on “Interactive Online Homework Assignments”

Confidence Interval of 99% Has and alpha of 1% Area outside confidence interval is alpha Area in the tails is called alpha α =. 10 Critical z 1. 96 Critical z -1. 96 95% α =. 05 Confidence Interval of 90% Has and alpha of 10% Area associated with most extreme scores is called alpha 99% α =. 01 Confidence Interval of 95% Has and alpha of 5% Critical z 2. 58 Critical z -2. 58 Critical z 1. 64 Critical z -1. 64 90%

Moving from descriptive stats into inferential stats…. Measurements that occur within the middle part of the curve are ordinary (typical) and probably belong there Area outside confidence interval is alpha 99% 95% Measurements that occur outside this middle ranges are suspicious, may be an error or belong elsewhere 90%

How do we know if something is going on? How rare/weird is rare/weird enough? Every day examples about when is weird, weird enough to think something is going on? • Handing in blue versus white test forms • Psychic friend – guesses 999 out of 1000 coin tosses right • Cancer clusters – how many cases before investigation • Weight gain treatment – one group gained an average of 1 pound more than other group…what if 10?

Why do we care about the z scores that define the middle 95% of the curve? Inferential Statistics Hypothesis testing with z scores allows us to make inferences about whether the sample mean is consistent with the known population mean. • Is the mean of my observed sample consistent with the known population mean or did it come from some other distribution?

Why do we care about the z scores that define the middle 95% of the curve? If the z score falls outside the middle 95% of the curve, it must be from some other distribution Main assumption: We assume that weird, or unusual or rare things don’t happen If a score falls out into the 5% range we conclude that it “must be” actually a common score but from some other distribution That’s why we care about the z scores that define the middle 95% of the curve

. Main assumption: We assume that weird, or unusual or rare things don’t happen If a score falls out into the tails (low probability) we conclude that it “must be” a common score from some other distribution I’m not an outlier I just haven’t found my distribution yet

. Reject the null hypothesis . . 95% Relative to this distribution I am unusual maybe even an outlier X 95% Relative to this distribution I am utterly typical X Support for alternative hypothesis

Rejecting the null hypothesis null big z score x not. null If the observed z falls beyond the critical z in the distribution (curve): • then it is so rare, we conclude it must be from some other distribution • then we reject the null hypothesis Alternative • then we have support for our alternative hypothesis Hypothesis If the observed z falls within the critical z in the distribution (curve): • then we know it is a common score and is likely to be part of this distribution, we conclude it must be from this distribution • then we do not reject the null hypothesis • then we do not have support for our alternative null small z score x

Rejecting the null hypothesis If the observed z falls beyond the critical z in the distribution (curve): • then it is so rare, we conclude it must be from some other distribution • then we reject the null hypothesis • then we have support for our alternative hypothesis If the observed z falls within the critical z in the distribution (curve): • then we know it is a common score and is likely to be part of this distribution, we conclude it must be from this distribution • then we do not reject the null hypothesis • then we do not have support for our alternative hypothesis

α =. 01 α =. 10 α =. 05 99% 95% 90% Area in the tails is alpha How do we know how rare is rare enough? (α ) Level of significance is called alpha • The degree of rarity required for an observed outcome to be “weird enough” to reject the null hypothesis • Which alpha level would be associated with most “weird” or rare scores? Critical z: A z score that separates common from rare outcomes and hence dictates whether the null hypothesis should be retained (same logic will hold for “critical t”) If the observed z falls beyond the critical z in the distribution (curve) then it is so rare, we conclude it must be from some other distribution

Rejecting the null hypothesis The result is “statistically significant” if: • the observed statistic is larger than the critical statistic (which can be a ‘z” or “t” or “r” or “F” or x 2) observed stat > critical stat If we want to reject the null, we want our t (or z or r or F or x 2) to be big • the p value is less than 0. 05 (which is our alpha) p < 0. 05 If we want to reject the null, we want our “p” to be small!! • we reject the null hypothesis • then we have support for our alternative hypothesis

Confidence Interval of 99% Has and alpha of 1% Area in the tails is called alpha Critical Z separates rare from common scores 99% α =. 01 Confidence Interval of 95% Has and alpha of 5% α =. 10 Critical z 1. 96 Critical z -1. 96 95% α =. 05 Confidence Interval of 90% Has and alpha of 10% Critical z 2. 58 Critical z -2. 58 Critical z 1. 64 Critical z -1. 64 90%

Deciding whether or not to reject the null hypothesis. 05 versus. 01 alpha levels What if our observed z = 2. 0? How would the critical z change? α = 0. 05 Remember, reject the null if the observed z is bigger than the critical z Significance level =. 05 α = 0. 01 Significance level =. 01 -1. 96 or +1. 96 p < 0. 05 Reject the null -2. 58 or Do not +2. 58 Reject the null Yes, Significant difference Not a Significant difference

Deciding whether or not to reject the null hypothesis. 05 versus. 01 alpha levels What if our observed z = 1. 5? How would the critical z change? α = 0. 05 Remember, reject the null if the observed z is bigger than the critical z Significance level =. 05 α = 0. 01 Significance level =. 01 -1. 96 or +1. 96 Do Not Reject the null -2. 58 or Do Not +2. 58 Reject the null Not a Significant difference

Deciding whether or not to reject the null hypothesis. 05 versus. 01 alpha levels What if our observed z = -3. 9? How would the critical z change? α = 0. 05 Remember, reject the null if the observed z is bigger than the critical z Significance level =. 05 α = 0. 01 Significance level =. 01 -1. 96 or +1. 96 p < 0. 05 Reject the null -2. 58 or Reject +2. 58 the null Yes, Significant difference p < 0. 01 Yes, Significant difference

Deciding whether or not to reject the null hypothesis. 05 versus. 01 alpha levels What if our observed z = -2. 52? How would the critical z change? α = 0. 05 Remember, reject the null if the observed z is bigger than the critical z Significance level =. 05 α = 0. 01 Significance level =. 01 -1. 96 or +1. 96 p < 0. 05 Reject the null -2. 58 or Do not +2. 58 Reject the null Yes, Significant difference Not a Significant difference

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)? • Critical z statistic? Step 3: Calculations of “observed z” score Step 4: Make decision whether or not to reject null hypothesis If observed z is bigger then critical z then we reject null Step 5: Conclusion - tie findings back in to research problem 3 parts: 1) IV and DV and means (descriptive stats) 2) Which type of test and whether it was a significant difference 3) Summary results phrase

Moving from descriptive stats Critical z into inferential stats…. -1. 64 Measurements that occur within the middle part of the curve are ordinary (typical) and probably belong there 5% Measurements that occur outside this middle ranges are suspicious, may be an error or belong elsewhere For scores that fall into the middle range, we do not Critical z reject the null 1. 64 90% 5% Critical For scores thatof fall What percent Values into regions of thethe distribution will rejection, fall in region weof reject the null rejection http: //today. msnbc. msn. com/id/33411196/ns/today-today_health/ http: //www. youtube. com/watch? v=0 r 7 NXEWpheg

Rejecting the null hypothesis The result is “statistically significant” if: • the observed statistic is larger than the critical statistic observed stat > critical stat If we want to reject the null, we want our t (or z or r or F or x 2) to be big!! • the p value is less than 0. 05 (which is our alpha) p < 0. 05 If we want to reject the null, we want our “p” to be small!! • we reject the null hypothesis • then we have support for our alternative hypothesis A note on decision making following procedure versus being right relative to the “TRUTH”

. Decision making: Procedures versus outcome Best guess versus “truth” What does it mean to be correct? Why do we say: • “innocent until proven guilty” • “not guilty” rather than “innocent” Is it possible we got a verdict wrong?

. We make decisions at Security Check Points .

. Type I or Type II error? . Does this airline passenger have a snow globe? Null Hypothesis means she does not have a snow globe (that nothing unusual is happening) – Should we reject it? ? ? !! As detectives, do we accuse her of brandishing a snow globe?

. Does this airline passenger have a snow globe? Are we correct or have we made a Type I or Type II error? Decision made by experimenter Status of Null Hypothesis (actually, via magic truth-line) True Ho No snow globe False Ho Yes snow globe You are right! “no snow globe move on” Correct decision You are wrong! Type II error (miss) You are wrong! Type I error (false alarm) You are right! Correct decision Do not reject Ho Reject Ho “yes snow globe, stop!” Note: Null Hypothesis means she does not have a snow globe (that nothing unusual is happening) – Should we reject it? ? ? !!

. Type I or type II error? Does this airline passenger Decision made have a snow globe? by experimenter True Ho Do not Reject Ho False Ho You are right! Correct decision You are wrong! Type II error (miss) You are wrong! You are right! Type I error Correct Reject Ho (false alarm) decision Two ways to be correct: • Say she does have snow globe when she does have snow globe • Say she doesn’t have any when she doesn’t have any Two ways to be incorrect: • Say she does when she doesn’t (false alarm) • Say she does not have any when she does (miss) What would null hypothesis be? This passenger does not have any snow globe Which is worse? Type I error: Rejecting a true null hypothesis Saying the she does have snow globe when in fact she does not (false alarm) Type II error: Not rejecting a false null hypothesis Saying she does not have snow globe when in fact she does (miss)

True Ho . Type I or type II error Does advertising affect sales? Do not Reject Ho Decision made by experimenter You are right! Correct decision False Ho You are wrong! Type II error (miss) You are wrong! You are right! Type I error Correct Reject Ho (false alarm) decision Two ways to be correct: • Say it helps when it does • Say it does not help when it doesn’t help Two ways to be incorrect: • Say it helps when it doesn’t • Say it does not help when it does Which is worse? What would null hypothesis be? This new advertising has no effect on sales Type I error: Rejecting a true null hypothesis Saying the advertising would help sales, when it really wouldn’t help people (false alarm) Type II error: Not rejecting a false null hypothesis Saying the advertising would not help when in fact it would (miss)

. What is worse a type I or type II error? What if we were looking at a new HIV drug that had no unpleasant side affects Do not Reject Ho Decision made by experimenter True Ho False Ho You are right! Correct decision You are wrong! Type II error (miss) You are wrong! You are right! Type I error Correct Reject Ho (false alarm) decision Two ways to be correct: • Say it helps when it does • Say it does not help when it doesn’t help Two ways to be incorrect: • Say it helps when it doesn’t • Say it does not help when it does Which is worse? What would null hypothesis be? This new drug has no effect on HIV Type I error: Rejecting a true null hypothesis Saying the drug would help people, when it really wouldn’t help people (false alarm) Type II error: Not rejecting a false null hypothesis Saying the drug would not help when in fact it would (miss)

. Type I or type II error Which is What if we were looking to see if there is a fire worse? burning in an apartment building full of cute puppies Two ways to be correct: • Say “fire” when it’s really there • Say “no fire” when there isn’t one Two ways to be incorrect: • Say “fire” when there’s no fire (false alarm) • Say “no fire” when there is one (miss) What would null hypothesis be? No fire is occurring Type I error: Rejecting a true null hypothesis (false alarm) Type II error: Not rejecting a false null hypothesis (miss)

. Type I or type II error What if we were looking to see if an individual were guilty of a crime? Which is worse? Two ways to be correct: • Say they are guilty when they are guilty • Say they are not guilty when they are innocent Two ways to be incorrect: • Say they are guilty when they are not • Say they are not guilty when they are What would null hypothesis be? This person is innocent - there is no crime here Type I error: Rejecting a true null hypothesis Saying the person is guilty when they are not (false alarm) Sending an innocent person to jail (& guilty person to stays free) Type II error: Not rejecting a false null hypothesis Saying the person in innocent when they are guilty (miss) Allowing a guilty person to stay free

. The null hypothesis is typically that something is not present, that there is no effect, that there is no difference between population and sample or between treatment and control. Null Hypothesis. A measure of sickness people taking drug people not taking drug people taking drug AA measure of sickness Drug does have effect people not taking drug Null is FALSE Something going on There is a difference between the groups (There are two distributions here, they are just on top of each other) (overlapping) Null is TRUE No effect of drug Nothing going on There is no difference between the groups

. (There are two distributions here, they are just on top of each other) (overlapping) A measure of sickness Remember: “procedure” vs “TRUTH” people taking drug AA measure of sickness Drug does have effect people taking drug people not taking drug Null is FALSE critical stat Score should fall in one of these regions people not taking drug Null is TRUE critical stat Score should fall in one of these regions Null is FALSE Something going on Score should fall in this critical stat region Null is TRUE No effect of drug Nothing going on

Deciding whether or not to reject the null hypothesis. 05 versus. 01 alpha levels What if our observed z = -3. 9? How would the critical z change? α = 0. 05 Remember, reject the null if the observed z is bigger than the critical z Significance level =. 05 α = 0. 01 Significance level =. 01 -1. 96 or +1. 96 p < 0. 05 Reject the null -2. 58 or Reject +2. 58 the null Yes, Significant difference p < 0. 01 Yes, Significant difference ew i v Re

Deciding whether or not to reject the null hypothesis. 05 versus. 01 alpha levels What if our observed z = -2. 52? How would the critical z change? α = 0. 05 Remember, reject the null if the observed z is bigger than the critical z Significance level =. 05 α = 0. 01 Significance level =. 01 -1. 96 or +1. 96 p < 0. 05 Reject the null -2. 58 or Do not +2. 58 Reject Yes, Significant difference Not a Significant difference the null ew i v Re

One versus two tail test of significance: Comparing different critical scores (but same alpha level – e. g. alpha = 5%) One versus two tailed test of significance z score = 1. 64 95% 5% 2. 5% One-tailed test: If your hypothesis is “directional” claiming that one group will have a bigger mean the other group Two-tailed test: If your hypothesis is “non-directional” claiming only that the two groups have different means How would the critical z change? Pros and cons…

One versus two tail test of significance 5% versus 1% alpha levels How would the critical z change? One-tailed 5% 1% α = 0. 05 Significance level =. 05 α = 0. 01 Significance level =. 01 Two-tailed 2. 5%. 5% -1. 64 or +1. 64 -1. 96 or +1. 96 -2. 33 or +2. 33 -2. 58 or +2. 58

One versus two tail test of significance 5% versus 1% alpha levels What if our observed z = 2. 0? How would the critical z change? One-tailed α = 0. 05 Remember, reject the null if the observed z is bigger than the critical z Significance level =. 05 -1. 64 or Reject +1. 64 the null α = 0. 01 -2. 33 or Significance Do not +2. 33 level =. 01 Reject the null Two-tailed -1. 96 or +1. 96 Reject the null -2. 58 or Do not +2. 58 Reject the null

One versus two tail test of significance 5% versus 1% alpha levels What if our observed z = 1. 75? How would the critical z change? One-tailed α = 0. 05 Remember, reject the null if the observed z is bigger than the critical z Significance level =. 05 -1. 64 or Reject +1. 64 the null α = 0. 01 -2. 33 or Significance Do not +2. 33 level =. 01 Reject the null Two-tailed -1. 96 or +1. 96 Do not Reject the null -2. 58 or Do not +2. 58 Reject the null

One versus two tail test of significance 5% versus 1% alpha levels What if our observed z = 2. 45? How would the critical z change? One-tailed α = 0. 05 Remember, reject the null if the observed z is bigger than the critical z Significance level =. 05 α = 0. 01 -1. 64 or Reject +1. 64 the null -2. 33 or Significance Reject +2. 33 level =. 01 the null Two-tailed -1. 96 or +1. 96 Reject the null -2. 58 or Do not +2. 58 Reject the null

Standard deviation and Variance For Sample and Population These would be helpful to know by heart – please memorize these formula w ie v Re

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

Summary Template There are three parts to the summary (below) Start summary with two means (based on DV) for two levels of the IV Finish with statistical summary t(4) = 1. 96; ns Describe type of test (t-test versus anova) with brief overview of results Or if it *were* significant: t(9) = 3. 93; p < 0. 05 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 no significant difference in the response time following the implementation of the new plan t(9) = -1. 71; n. s. Type of test with degrees of freedom Value of observed statistic n. s. = “not significant” p<0. 05 = “significant”

Degrees of Freedom We lose one degree of freedom for every parameter we estimate Degrees of Freedom (d. f. ) is a parameter based on the sample size that is used to determine the value of the t statistic. Degrees of freedom tell how many observations are used to calculate s, less the number of intermediate estimates used in the calculation.

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

. A note on variability versus effect size Difference between means Variability of curve(s) Variability (withinof group variability) curve(s)

. A note on variability versus effect size Difference between means Variability of curve(s) Variability (withinof group variability) curve(s) .

. Effect size is considered relative to variability of distributions 1. Larger variance harder to find significant difference Treatment Effect x 2. Smaller variance easier to find significant difference Treatment Effect x

. Effect size is considered relative to variability of distributions Treatment Effect x Difference between means Treatment Effect x Variability of curve(s) (within group variability)

Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) How is a t score Describe different the null and alternative hypotheses than a z score? Step 2: Decision rule: find “critical z” score • Alpha level? (α =. 05 or. 01)? • One versus two-tailed test Step 3: Calculations Step 4: Make decision whether or not to reject null hypothesis versus sample If observed z (or t) is bigger. Population then critical z (or t) then standard deviation reject null Step 5: Conclusion - tie findings back in to research problem

Comparing z score distributions with t-score distributions z-scores Similarities include: Using bell-shaped distributions to make confidence interval estimations and decisions in hypothesis testing Use table to find areas under the curve (different table, though – areas often differ from z scores) t-scores Summary of 2 main differences: • We are now estimating standard deviation from the sample ü (We don’t know population standard deviation) • We have to deal with degrees of freedom

Comparing z score distributions with t-score distributions Differences include: 1) We use t-distribution when we don’t know standard deviation of population, and have to estimate it from our sample 2) The shape of the sampling distribution is very sensitive to small sample sizes (it actually changes shape depending on n) Please notice: as sample sizes get smaller, the tails get thicker. As sample sizes get bigger tails get thinner and look more like the z-distribution

Comparing z score distributions with t-score distributions Differences include: 1) We use t-distribution when we don’t know standard deviation of population, and have to estimate it from our sample Critical t (just like critical z) separates common from rare scores Critical t used to define both common scores “confidence interval” and rare scores “region of rejection”

Comparing z score distributions with t-score distributions Differences include: 1) We use t-distribution when we don’t know standard deviation of population, and have to estimate it from our sample 2) The shape of the sampling distribution is very sensitive to small sample sizes (it actually changes shape depending on n) Please notice: as sample sizes get smaller, the tails get thicker. As sample sizes get bigger tails get thinner and look more like the z-distribution

Comparing z score distributions with t-score distributions Differences include: Please note: Once sample sizes get big enough the t distribution (curve) starts to look exactly like the z distribution (curve) scores 1) We use t-distribution when we don’t know standard deviation of population, and have to estimate it from our sample 2) The shape of the sampling distribution is very sensitive to small sample sizes (it actually changes shape depending on n) 3) Because the shape changes, the relationship between the scores and proportions under the curve change (So, we would have a different table for all the different possible n’s but just the important ones are summarized in our t-table)

A quick re-visit with the law of large numbers Relationship between • increased sample size • decreased variability • smaller “critical values” As n goes up variability goes down

Law of large numbers: As the number of measurements increases the data becomes more stable and a better approximation of the true signal (e. g. mean) As the number of observations (n) increases or the number of times the experiment is performed, the signal will become more clear (static cancels out) With only a few people any little error is noticed (becomes exaggerated when we look at whole group) With many people any little error is corrected (becomes minimized when we look at whole group) http: //www. youtube. com/watch? v=ne 6 t. B 2 Ki. Zuk

Interpreting t-table Technically, we have a different t-distribution for each sample size We use degrees of freedom (df) to approximate sample size This t-table summarizes the most useful values for several distributions This t-table presents useful values for distributions (organized by degrees of freedom) Each curve is based on its own degrees of freedom (df) - based on sample size, and its own table tying together t -scores with area under the curve . 1. 64 1. 96 2. 58 Remember these useful values for zscores? n = 17 n=5

Areabetween two twoscores Area beyond two scores (out in tails) Area in each tail (out in tails) df

Areabetween two twoscores Area beyond two scores (out in tails) Area in each tail (out in tails) Notice with large sample size it is same values as z-score df Remember these useful values for zscores? . 1. 64 1. 96 2. 58

Degrees of Freedom (d. f. ) is a parameter based on the sample size that is used to determine the value of the t statistic. Degrees of freedom tell how many observations are used to calculate s, less the number of intermediate estimates used in the calculation.

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

In-class writing assignment