Nicole Dalzell. July 2, 2014

Transcription

1 UNIT 1: INTRODUCTION TO DATA LECTURE 3: EDA (CONT.) AND INTRODUCTION TO STATISTICAL INFERENCE VIA SIMULATION STATISTICS 101 Nicole Dalzell July 2, 2014

2 Teams and Announcements Team1 = Houdan Sai Cui Huanqi Team2 = Ludi Li Jackson Hannah Team3 = Christian Christine Sasha Office Hours: Today from 2-3 PM in Old Chem 211 A. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

3 Review and Recap 1 Review and Recap 2 Distribution of one numerical variable Standard Deviation 3 Distribution of one numerical variable Robust statistics 4 Relationship between a numerical and a categorical variable 5 Case study: Gender discrimination Study description and data Competing claims Testing via simulation Checking for independence 6 ] Statistics 101

4 Review and Recap Distribution Shape How would you describe the shape of the distribution of number of piercing college students have? frequency # of piercings Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

5 Distribution of one numerical variable 1 Review and Recap 2 Distribution of one numerical variable Standard Deviation 3 Distribution of one numerical variable Robust statistics 4 Relationship between a numerical and a categorical variable 5 Case study: Gender discrimination Study description and data Competing claims Testing via simulation Checking for independence 6 ] Statistics 101

6 Distribution of one numerical variable Standard Deviation Standard deviation Standard deviation, s Roughly the deviation around the mean, calculated as the square root of the variance, and has the same units as the data. s = s 2 = n i=1 (x i x) 2 n 1 Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

7 Distribution of one numerical variable Standard Deviation Standard deviation Standard deviation, s Roughly the deviation around the mean, calculated as the square root of the variance, and has the same units as the data. s = s 2 = n i=1 (x i x) 2 n 1 The standard deviation of energy use per capita can be calculated as: s = = Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

8 Distribution of one numerical variable Standard Deviation Standard Deviation The standard deviation gives a rough estimate of the typical distance of a data values from the mean. The larger the standard deviation, the more variability there is in the data and the more spread out the data are. Standard Deviation of 2 Standard Deviation of 4 Frequency Frequency rnorm(100, 0, 2) rnorm(100, 0, 4) Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

9 Distribution of one numerical variable Standard Deviation Variability in Student Sleep sleep, x = 4.6, s x = out of 86 students (80%) are within 1 SD of the mean. 80 out of 86 students (93%) are within 2 SDs of the mean. 86 out of 86 students (100%) are within 3 SDs of the mean. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

10 Distribution of one numerical variable Standard Deviation 95% Rule 95 % Rule If a distribution of data is approximately symmetric and bell-shaped, about 95% of the data should fall within two standard deviations of the mean. For a population, 95% of the data will be between µ 2σ and µ + 2σ rchsbowman.files.wordpress.com/ 2008/ 09/ empirical-rule-3.jpg Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

11 Distribution of one numerical variable Standard Deviation Notation Recap mean variance SD sample x s 2 s population µ σ 2 σ Do you see a trend in what types of letters are used for sample statistics vs. population parameters? Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

12 Distribution of one numerical variable Standard Deviation Notation Recap mean variance SD sample x s 2 s population µ σ 2 σ Do you see a trend in what types of letters are used for sample statistics vs. population parameters? Latin letters for sample statistics, Greek letters for population parameters. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

13 Distribution of one numerical variable Standard Deviation Z-Scores Z-Score The z-score for a data value, x i, is z = x i x s For a population, x is replaced with µ and s is replaced with σ. Values farther from 0 are more extreme. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

14 Distribution of one numerical variable Standard Deviation Z-Scores: Why? A z-score puts values on a common scale A z-score is the number of standard deviations a value falls from the mean 95% of all z-scores fall between -2 and 2. z-scores beyond -2 or 2 can be considered extreme Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

15 Distribution of one numerical variable Standard Deviation Z-Scores: Example Which is better, (A) an ACT score of 28 or (B) a combined SAT score of 2100? Assume ACT and SAT scores have approximately bell-shaped distributions. ACT: x = 21, s = 5 SAT: x = 1500, s = 325 Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

16 Distribution of one numerical variable Standard Deviation Z-Scores: Example Which is better, (A) an ACT score of 28 or (B) a combined SAT score of 2100? Assume ACT and SAT scores have approximately bell-shaped distributions. ACT: x = 21, s = 5 SAT: x = 1500, s = 325 Histogram of Z Scores ACT: SAT: z = z = = 7 5 = 1.4 = = 1.85 Frequency Z Score Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

17 Distribution of one numerical variable Standard Deviation Other Measures of Location The 25 th percentile is also called the first quartile, Q1. The 50 th percentile is also called the median. The 75 th percentile is also called the third quartile, Q3. summary ( energy$x2011 ) Min. 1 s t Qu. Median Mean 3 rd Qu. Max Between Q1 and Q3 is the middle 50% of the data. The range these data span is called the interquartile range, or the IQR. IQR = = Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

18 Distribution of one numerical variable Standard Deviation Participation question Which of the following is false about the distribution of average number of hours students study daily? Average number of hours students study daily Min. 1st Qu. Median Mean 3rd Qu. Max (a) There are no students who don t study at all. (b) 75% of the students study more than 5 hours daily, on average. (c) 25% of the students study less than 3 hours, on average. (d) IQR is 2 hours. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

19 Distribution of one numerical variable Standard Deviation Participation question Which of the following is false about the distribution of average number of hours students study daily? Average number of hours students study daily Min. 1st Qu. Median Mean 3rd Qu. Max (a) There are no students who don t study at all. (b) 75% of the students study more than 5 hours daily, on average. (c) 25% of the students study less than 3 hours, on average. (d) IQR is 2 hours. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

20 Distribution of one numerical variable Standard Deviation Box Plot The box in a box plot represents the middle 50% of the data, and the thick line in the box is the median # of study hours / week Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

21 Distribution of one numerical variable Standard Deviation Anatomy of a Box Plot # of study hours / week suspected outliers max whisker reach upper whisker Q 3 (third quartile) median Q 1 (first quartile) 0 lower whisker Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

22 Distribution of one numerical variable Standard Deviation Whiskers and Outliers Whiskers of a box plot can extend up to 1.5 * IQR away from the quartiles. max upper whisker reach : Q IQR = = 35 max lower whisker reach : Q1 1.5 IQR = = 5 An outlier is defined as an observation beyond the maximum reach of the whiskers. It is an observation that appears extreme relative to the rest of the data. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

23 Distribution of one numerical variable Standard Deviation Outliers (cont.) Why is it important to look for outliers? Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

24 Distribution of one numerical variable Standard Deviation Outliers (cont.) Why is it important to look for outliers? Identify extreme skew in the distribution. Identify data collection and entry errors. Provide insight into interesting features of the data. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

25 Distribution of one numerical variable Standard Deviation Example: Visualizing What does our Energy Data look like? Energy Use Data Boxplot Energy Usage Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

26 Distribution of one numerical variable Standard Deviation Who uses the most energy? Country.Name X Iceland Qatar Trinidad and Tobago Kuwait Brunei Darussalam Oman Luxembourg United Arab Emirates Bahrain Canada North America United States Saudi Arabia Singapore Finland Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

27 Distribution of one numerical variable 1 Review and Recap 2 Distribution of one numerical variable Standard Deviation 3 Distribution of one numerical variable Robust statistics 4 Relationship between a numerical and a categorical variable 5 Case study: Gender discrimination Study description and data Competing claims Testing via simulation Checking for independence 6 ] Statistics 101

28 Distribution of one numerical variable Robust statistics Range and IQR Range Range of the entire data. range = max min Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

29 Distribution of one numerical variable Robust statistics Range and IQR Range Range of the entire data. range = max min IQR Range of the middle 50% of the data. IQR = Q3 Q1 Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

30 Distribution of one numerical variable Robust statistics Range and IQR Range Range of the entire data. range = max min IQR Range of the middle 50% of the data. IQR = Q3 Q1 Is the range or the IQR more robust to outliers? Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

31 Distribution of one numerical variable Robust statistics Range and IQR Range Range of the entire data. range = max min IQR Range of the middle 50% of the data. IQR = Q3 Q1 Is the range or the IQR more robust to outliers? IQR Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

32 Distribution of one numerical variable Robust statistics Extreme observations How would sample statistics such as mean, median, SD, and IQR of household income be affected if the largest value was replaced with $10 million? What if the smallest value was replaced with $10 million? household income ($ thousands) Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

33 Distribution of one numerical variable Robust statistics Income Example household income ($ thousands) robust not robust scenario median IQR x s original data 165K 150K 211K 180K move largest to $10 million 165K 150K 398K 1,422K move smallest to $10 million 190K 163K 4,186K 1,424K Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

34 Distribution of one numerical variable Robust statistics Robust statistics Since the median and IQR are more robust to skewness and outliers than mean and SD: skewed median and IQR symmetric mean and SD Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

35 Distribution of one numerical variable Robust statistics Robust statistics Since the median and IQR are more robust to skewness and outliers than mean and SD: skewed median and IQR symmetric mean and SD If you were searching for a car, and you are price conscious, would you be more interested in the mean or median vehicle price when considering a car? Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

36 Distribution of one numerical variable Robust statistics Mean vs. median If the distribution is symmetric, center is the mean Symmetric: mean is roughly equal to the median If the distribution is skewed or has outliers center is the median Right-skewed: mean is likely greater than the median Left-skewed: mean is likely less than the median red solid - mean, black dashed - median ls rs sym Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

37 Relationship between a numerical and a categorical variable 1 Review and Recap 2 Distribution of one numerical variable Standard Deviation 3 Distribution of one numerical variable Robust statistics 4 Relationship between a numerical and a categorical variable 5 Case study: Gender discrimination Study description and data Competing claims Testing via simulation Checking for independence 6 ] Statistics 101

38 Relationship between a numerical and a categorical variable Side-by-side box plot How does the number of the average number of times students go out per week vary by involvement? Do the two variables appear to be associated or independent? Greek Independent SLG Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

39 Case study: Gender discrimination 1 Review and Recap 2 Distribution of one numerical variable Standard Deviation 3 Distribution of one numerical variable Robust statistics 4 Relationship between a numerical and a categorical variable 5 Case study: Gender discrimination Study description and data Competing claims Testing via simulation Checking for independence 6 ] Statistics 101

40 Case study: Gender discrimination Study description and data Gender discrimination In 1972, as a part of a study on gender discrimination, 48 male bank supervisors were each given the same personnel file and asked to judge whether the person should be promoted to a branch manager job that was described as routine. The files were identical except that half of the supervisors had files showing the person was male while the other half had files showing the person was female. It was randomly determined which supervisors got male applications and which got female applications. Of the 48 files reviewed, 35 were promoted. The study is testing whether females are unfairly discriminated against. Is this an observational study or an experiment? B.Rosen and T. Jerdee (1974), Influence of sex role stereotypes on personnel decisions, J.Applied Psychology, 59:9-14. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

41 Case study: Gender discrimination Study description and data Data At a first glance, does there appear to be a relatonship between promotion and gender? Gender Promotion Promoted Not Promoted Total Male Female Total Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

42 Case study: Gender discrimination Study description and data Data At a first glance, does there appear to be a relatonship between promotion and gender? Gender Promotion Promoted Not Promoted Total Male Female Total % of males promoted: 21/24 = % of females promoted: 14/24 = Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

43 Case study: Gender discrimination Study description and data Participation question We saw a difference of almost 30% (29.2% to be exact) between the proportion of male and female files that are promoted. Based on this information, which of the below is true? (a) If we were to repeat the experiment we will definitely see that more female files get promoted, this was a fluke. (b) Promotion is dependent on gender, males are more likely to be promoted, and hence there is gender discrimination against women in promotion decisions. (c) The difference in the proportions of promoted male and female files is due to chance, this is not evidence of gender discrimation against women in promotion decisions. (d) Women are less qualified than men, and this is why fewer females get promoted. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

44 Case study: Gender discrimination Study description and data Participation question We saw a difference of almost 30% (29.2% to be exact) between the proportion of male and female files that are promoted. Based on this information, which of the below is true? (a) If we were to repeat the experiment we will definitely see that more female files get promoted, this was a fluke. (b) Promotion is dependent on gender, males are more likely to be promoted, and hence there is gender discrimination against women in promotion decisions. Maybe (c) The difference in the proportions of promoted male and female files is due to chance, this is not evidence of gender discrimation against women in promotion decisions. Maybe (d) Women are less qualified than men, and this is why fewer females get promoted. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

45 Case study: Gender discrimination Competing claims Two competing claims 1 There is nothing going on. Promotion and gender are independent, no gender discrimination, observed difference in proportions is simply due to chance. Null hypothesis Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

46 Case study: Gender discrimination Competing claims Two competing claims 1 There is nothing going on. Promotion and gender are independent, no gender discrimination, observed difference in proportions is simply due to chance. Null hypothesis 2 There is something going on. Promotion and gender are dependent, there is gender discrimination, observed difference in proportions is not due to chance. Alternative hypothesis Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

47 Case study: Gender discrimination Competing claims A trial as a hypothesis test Hypothesis testing is very much like a court trial. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

48 Case study: Gender discrimination Competing claims A trial as a hypothesis test Hypothesis testing is very much like a court trial. H 0 : Defendant is innocent Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

49 Case study: Gender discrimination Competing claims A trial as a hypothesis test Hypothesis testing is very much like a court trial. H 0 : Defendant is innocent H A : Defendant is guilty Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

50 Case study: Gender discrimination Competing claims A trial as a hypothesis test Hypothesis testing is very much like a court trial. H 0 : Defendant is innocent H A : Defendant is guilty Present the evidence: collect data. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

51 Case study: Gender discrimination Competing claims A trial as a hypothesis test Hypothesis testing is very much like a court trial. H 0 : Defendant is innocent H A : Defendant is guilty Present the evidence: collect data. Judge the evidence: Could these data plausibly have happened by chance if the null hypothesis were true? Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

52 Case study: Gender discrimination Competing claims A trial as a hypothesis test Hypothesis testing is very much like a court trial. H 0 : Defendant is innocent H A : Defendant is guilty Present the evidence: collect data. Judge the evidence: Could these data plausibly have happened by chance if the null hypothesis were true? Make a decision: How unlikely is unlikely? Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

53 Case study: Gender discrimination Competing claims A trial as a hypothesis test Hypothesis testing is very much like a court trial. H 0 : Defendant is innocent H A : Defendant is guilty Present the evidence: collect data. Judge the evidence: Could these data plausibly have happened by chance if the null hypothesis were true? Make a decision: How unlikely is unlikely? Evidence not strong enough to reject the assumption of innocence verdict: not guilty The jury does not say that the defendant is innocent, just that there is not enough evidence to convict. The defendant may, in fact, be innocent, but the jury has no way of being sure. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

54 Case study: Gender discrimination Competing claims Recap: hypothesis testing framework We start with a null hypothesis (H 0 ) that represents the status quo. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

55 Case study: Gender discrimination Competing claims Recap: hypothesis testing framework We start with a null hypothesis (H 0 ) that represents the status quo. We also have an alternative hypothesis (H A ) that represents our research question, i.e. what we re testing for. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

56 Case study: Gender discrimination Competing claims Recap: hypothesis testing framework We start with a null hypothesis (H 0 ) that represents the status quo. We also have an alternative hypothesis (H A ) that represents our research question, i.e. what we re testing for. We conduct a hypothesis test under the assumption that the null hypothesis is true, either via simulation (today) or theoretical methods (later in the course). Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

57 Case study: Gender discrimination Competing claims Recap: hypothesis testing framework We start with a null hypothesis (H 0 ) that represents the status quo. We also have an alternative hypothesis (H A ) that represents our research question, i.e. what we re testing for. We conduct a hypothesis test under the assumption that the null hypothesis is true, either via simulation (today) or theoretical methods (later in the course). If the test results suggest that the data do not provide convincing evidence for the alternative hypothesis, we stick with the null hypothesis. If they do, then we reject the null hypothesis in favor of the alternative. We never declare the null hypothesis to be true, because we simply do not know whether it s true or not. Therefore we never accept the null hypothesis. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

58 Case study: Gender discrimination Testing via simulation Checking for Independence using Simulation Instead of repeating the experiment (difficult in this case), we instead simulate our results under the assumption of independence. If results from the simulations based on the chance model look like our data, then we can say that the differences between men and women were simply due to chance. If results from the simulations do not look like the data, then we can say that the difference between men and women was not due to chance. Gender Promotion Promoted Not Promoted Total Male Female Total Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

59 Case study: Gender discrimination Testing via simulation Simulating the experiment... Simulate the experiment in a way that satisfies the null hypothesis (in this case, in a way that there is no discrimination against females) Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

60 Case study: Gender discrimination Testing via simulation Simulating the experiment... Simulate the experiment in a way that satisfies the null hypothesis (in this case, in a way that there is no discrimination against females) Determine if the observed outcome from the original experiment (roughly 30% more males being promoted) is a likely outcome when things are left up to chance. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

61 Case study: Gender discrimination Testing via simulation Simulating the experiment... Simulate the experiment in a way that satisfies the null hypothesis (in this case, in a way that there is no discrimination against females) Determine if the observed outcome from the original experiment (roughly 30% more males being promoted) is a likely outcome when things are left up to chance. If the results from the simulations based on the chance model do not look like the data, determine that the observed difference between males and females was due to an actual effect of gender (promotion and gender are dependent). Gender Promotion Promoted Not Promoted Total Male Female Total Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

62 Case study: Gender discrimination Testing via simulation Simulation setup 1 We ll let a face card represent not promoted and a non-face card represent a promoted. Consider aces as face cards. Set aside the jokers. Take out 3 aces there are exactly 13 face cards left in the deck (face cards: A, K, Q, J): NOT PROMOTED Take out a number card there are exactly 35 number (non-face) cards left in the deck (number cards: 2-10): PROMOTED Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

63 Case study: Gender discrimination Testing via simulation Simulation setup 1 We ll let a face card represent not promoted and a non-face card represent a promoted. Consider aces as face cards. Set aside the jokers. Take out 3 aces there are exactly 13 face cards left in the deck (face cards: A, K, Q, J): NOT PROMOTED Take out a number card there are exactly 35 number (non-face) cards left in the deck (number cards: 2-10): PROMOTED 2 Shuffle the cards and deal them into two groups of size 24, representing males and females. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

64 Case study: Gender discrimination Testing via simulation Simulation setup 1 We ll let a face card represent not promoted and a non-face card represent a promoted. Consider aces as face cards. Set aside the jokers. Take out 3 aces there are exactly 13 face cards left in the deck (face cards: A, K, Q, J): NOT PROMOTED Take out a number card there are exactly 35 number (non-face) cards left in the deck (number cards: 2-10): PROMOTED 2 Shuffle the cards and deal them into two groups of size 24, representing males and females. 3 Count and record how many files in each group are promoted (number cards). Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

65 Case study: Gender discrimination Testing via simulation Simulation setup 1 We ll let a face card represent not promoted and a non-face card represent a promoted. Consider aces as face cards. Set aside the jokers. Take out 3 aces there are exactly 13 face cards left in the deck (face cards: A, K, Q, J): NOT PROMOTED Take out a number card there are exactly 35 number (non-face) cards left in the deck (number cards: 2-10): PROMOTED 2 Shuffle the cards and deal them into two groups of size 24, representing males and females. 3 Count and record how many files in each group are promoted (number cards). 4 Calculate the proportion of promoted files in each group and take the difference (male - female). Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

66 Case study: Gender discrimination Testing via simulation Simulation setup 1 We ll let a face card represent not promoted and a non-face card represent a promoted. Consider aces as face cards. Set aside the jokers. Take out 3 aces there are exactly 13 face cards left in the deck (face cards: A, K, Q, J): NOT PROMOTED Take out a number card there are exactly 35 number (non-face) cards left in the deck (number cards: 2-10): PROMOTED 2 Shuffle the cards and deal them into two groups of size 24, representing males and females. 3 Count and record how many files in each group are promoted (number cards). 4 Calculate the proportion of promoted files in each group and take the difference (male - female). 5 Report the difference on the board (up to 2 decimal places, only 1 submission for each simulation). Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

67 Case study: Gender discrimination Testing via simulation Step 1 Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

68 Case study: Gender discrimination Testing via simulation Step 2-4 Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

69 Case study: Gender discrimination Testing via simulation Simulating with StatKey lock5stat.com/ statkey/ randomization 2 cat/ randomization 2 cat.html Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

70 Case study: Gender discrimination Checking for independence Participation question Do the data provide convincing evidence of gender discrimination against women, i.e. dependence between gender and promotion decisions? (a) No, the data do not provide convincing evidence for the alternative hypothesis, therefore we can t reject the null hypothesis of independence between gender and promotion decisions. The observed difference between the two proportions was due to chance. (b) Yes, the data provide convincing evidence for the alternative hypothesis of gender discrimination against women in promotion decisions. The observed difference between the two proportions was due to a real effect of gender. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

71 Case study: Gender discrimination Checking for independence Participation question Do the data provide convincing evidence of gender discrimination against women, i.e. dependence between gender and promotion decisions? (a) No, the data do not provide convincing evidence for the alternative hypothesis, therefore we can t reject the null hypothesis of independence between gender and promotion decisions. The observed difference between the two proportions was due to chance. (b) Yes, the data provide convincing evidence for the alternative hypothesis of gender discrimination against women in promotion decisions. The observed difference between the two proportions was due to a real effect of gender. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

72 Case study: Gender discrimination Checking for independence Making a decision The probability of observing a difference at least as favorable to the alternative hypothesis as the one observed in the original data (a difference of 29.2%) if H 0 is true is called the p-value. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

73 Case study: Gender discrimination Checking for independence Making a decision The probability of observing a difference at least as favorable to the alternative hypothesis as the one observed in the original data (a difference of 29.2%) if H 0 is true is called the p-value. The significance level is the threshold against which we compare the p-value to determine if it s small enough to reject the null hypothesis (this is usually 5%). Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

74 Case study: Gender discrimination Checking for independence Making a decision The probability of observing a difference at least as favorable to the alternative hypothesis as the one observed in the original data (a difference of 29.2%) if H 0 is true is called the p-value. The significance level is the threshold against which we compare the p-value to determine if it s small enough to reject the null hypothesis (this is usually 5%). Difference in promotion rates Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

75 Case study: Gender discrimination Checking for independence Making a decision The probability of observing a difference at least as favorable to the alternative hypothesis as the one observed in the original data (a difference of 29.2%) if H 0 is true is called the p-value. The significance level is the threshold against which we compare the p-value to determine if it s small enough to reject the null hypothesis (this is usually 5%). Difference in promotion rates p value= 1/100 = 0.01 Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

76 Case study: Gender discrimination Checking for independence To Do: Complete Lab 1, submit on Sakai by today at 5 PM Work on Problem Set (PS) 1, due at 11 AM in class tomorrow. Please staple your work! Reminder: Office Hours today from 2-3 PM in Old Chem 211 A Reading: Begin Reading Chapter 2 in preparation for tomorrow s lecture. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

77 Case study: Tapping on caffeine [Time permitting 1 Review and Recap 2 Distribution of one numerical variable Standard Deviation 3 Distribution of one numerical variable Robust statistics 4 Relationship between a numerical and a categorical variable 5 Case study: Gender discrimination Study description and data Competing claims Testing via simulation Checking for independence 6 ] Statistics 101

78 Case study: Tapping on caffeine [Time permitting Case study: Tapping on caffeine [Time permitting] Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

79 Case study: Tapping on caffeine [Time permitting Tapping on caffeine In a double-blind experiment a sample of male college students were asked to tap their fingers at a rapid rate. The sample was then divided at random into two groups of 10 students each. Each student drank the equivalent of about two cups of coffee, which included about 200 mg of caffeine for the students in one group but was decaffeinated coffee for the second group. After a two hour period, each student was tested to measure finger tapping rate (taps per minute). Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

80 Case study: Tapping on caffeine [Time permitting Data Taps Group Caffeine Caffeine Caffeine Caffeine Caffeine Caffeine NoCaffeine NoCaffeine NoCaffeine NoCaffeine NoCaffeine Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

81 Case study: Tapping on caffeine [Time permitting Participation question What type of plot would be useful to visualize the distributions of tapping rate in the caffeine and no caffeine groups. (a) Bar plot (b) Mosaic plot (c) Pie chart (d) Side-by-side box plots (e) Single box plot Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

82 Case study: Tapping on caffeine [Time permitting Participation question What type of plot would be useful to visualize the distributions of tapping rate in the caffeine and no caffeine groups. (a) Bar plot (b) Mosaic plot (c) Pie chart (d) Side-by-side box plots (e) Single box plot Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

83 Case study: Tapping on caffeine [Time permitting Exploratory data analysis Compare the distributions of tapping rates in the caffeine and no caffeine groups. Caffeine No Caffeine Difference mean SD median IQR Caffeine NoCaffeine Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

84 Case study: Tapping on caffeine [Time permitting Participation question We are interested in finding out if caffeine increases tapping rate. Which of the following are the correct set of hypotheses? (a) H 0 : µ caff = µ no caff H A : µ caff < µ no caff (b) H 0 : µ caff = µ no caff H A : µ caff > µ no caff (c) H 0 : x caff = x no caff H A : x caff > x no caff (d) H 0 : µ caff > µ no caff H A : µ caff = µ no caff (e) H 0 : µ caff = µ no caff H A : µ caff µ no caff Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

85 Case study: Tapping on caffeine [Time permitting Participation question We are interested in finding out if caffeine increases tapping rate. Which of the following are the correct set of hypotheses? (a) H 0 : µ caff = µ no caff H A : µ caff < µ no caff (b) H 0 : µ caff = µ no caff H A : µ caff > µ no caff (c) H 0 : x caff = x no caff H A : x caff > x no caff (d) H 0 : µ caff > µ no caff H A : µ caff = µ no caff (e) H 0 : µ caff = µ no caff H A : µ caff µ no caff Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

86 Case study: Tapping on caffeine [Time permitting Simulation scheme On 20 index cards write the tapping rate of each subject in the study. Shuffle the cards and divide them into two stacks of 10 cards each, label one stack caffeine and the other stack no caffeine. Calculate the average tapping rates in the two simulated groups, and record the difference on a dot plot. Repeat steps (2) and (3) many times to build a randomization distribution. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

87 Case study: Tapping on caffeine [Time permitting Making a decision Calculate the p-value based on the randomization distribution below and determine the conclusion of the hypothesis test. (100 simulations) Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

88 Case study: Tapping on caffeine [Time permitting Making a decision Calculate the p-value based on the randomization distribution below and determine the conclusion of the hypothesis test. (100 simulations) /100 = 0.01 Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

89 Case study: Tapping on caffeine [Time permitting Testing for the median Describe how could we use the same approach to test whether the median tapping rate is higher for the caffeine group? Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

90 Case study: Tapping on caffeine [Time permitting Testing for the median Describe how could we use the same approach to test whether the median tapping rate is higher for the caffeine group? Use the same simulation scheme but record the difference between the medians instead of the means, and calculate the p-value as the proportion of simulations where the simulated difference in medians is at least 3. Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48

91 Case study: Tapping on caffeine [Time permitting Testing for the median (cont.) Using the randomization distribution below of simulated differences in means, determine whether the data provide convincing evidence that caffeine increases median tapping rate. Caffeine No Caffeine Difference median affeine Randomization distribution Statistics 101 ( Nicole Dalzell) U1 - L3: EDA + Inference July 2, / 48