WELCOME BACK! 2017 128 days From: Tuesday, January 3, 2017 To: Thursday, May 11, 2017 128 days Or 4 months, 8 days 11,059,200 seconds 184,320 minutes 3072 hours 18 weeks and 2 days 35.07% of 2017
Warm-Up: What makes a single die FAIR? H0: The Prop. Of 1’s =2’s = … = 6’s OR The Die is Fair! Ha: The Outcomes are not Uniformly distributed OR The Die is NOT Fair.
Two Proportions Z-Test What is the statistical inference test you would use to determine if there is a significant difference in the true proportions of males with brown eyes vs. females with brown eyes? Two Proportions Z-Test Is there a Statistical Inference Test you would use if you needed to determine if there was a Significant Difference between Three or More Proportions? YES!
(Degree of Freedom = df = n – 1). Multiple Proportions Chapter 26: Chi Square(d) or X2–Test The P-Values/Probabilities for the X 2 – Test come from a family of Chi Square Distributions, which only take Positive Values and are all Skewed Right. A specific distribution is specified by a parameter called the Degree of freedom (df). (Degree of Freedom = df = n – 1).
Calculating The X2 - Test Statistic: Calculating The X2 - P-Value: Or find the appropriate line on the X2 Table. Find the P-Value for a Chi-Square Statistic = 12.132 with df = 6. P-Value = X2cdf (12.132, E99, 6) = 0.0591
Ch. 26 - Multiple Proportions There are THREE types of Chi-Square Tests: 1. The Chi-Square Test for Goodness of Fit. 2. The Chi-Square Test for Independence. 3. The Chi-Square Test for Homogeneity.
GOODNESS OF FIT The Chi-Square Test for A test of whether the distribution of Counts in one categorical variable matches the distribution predicted (expected) by a model. (Degree of Freedom = df = n – 1) Where n = # of levels of the Category. CONDITIONS SRS All Expected Counts greater than 5.
EXAMPLE: Is there one month of the year that stands out as having more births occurring as compared to the others? If births were distributed uniformly across the year, we would expect 1/12 of them to occur each month. To test the claim, birth data was randomly collected and compiled. JAN. FEB. MAR APR. MAY JUN. JUL. AUG. SEP. OCT. NOV. DEC. OBS. DATA 75 87 91 88 76 98 74 81 70 83 EXP. DATA 82 H0: Births are uniformly distributed over the year. Ha: Births are NOT uniformly distributed over the year. or X2 Goodness of Fit Test H0: Proportion of births in Jan = Feb.= Mar.= • • • = Dec. Ha: Prop. of births are not all uniform. Not all pi’s equal. P-Value = X2cdf (9.5366, E99, 11) = 0.5725 X2 = 9.5366
82 CONDITIONS SRS √ All Expected Counts are 5 or greater. Since the P-Value is NOT less than α = 0.05 we Fail to REJECT H0 . No evidence that Births do NOT occur uniformly through out the year. CONDITIONS SRS √ All Expected Counts are 5 or greater. EXP. DATA 82
Homework Page 628: #3, 10
Homework: Page 628: #10
Homework Page 628: #3-5, 9
EXAMPLE: The NY Civil Liberties Union feel that the NYC Police Dept is not hiring an ‘ethnic composition’ representing the city. NYC is 29.2% White, 28.3% Black, 31.5% Latino, 9.1% Asian, and 2% other. If the NYC Police Dept. is composed of the following, does the Union have a case? White Black Latino Asian Other OBS. DATA 8560 7120 2762 1852 560 EXP. DATA 6089.4 5901.7 6569 1897.7 417.08 H0: The Police Dept. represents the Population of NYC. Ha: The Police Dept. Does NOT represents the Population of NYC. X2 Goodness of Fit Test P-Value = X2cdf (3510.3, E99, 4) = 0 X2 = 3510.3
Since the P-Value is less than α = 0. 05 the data IS significant Since the P-Value is less than α = 0.05 the data IS significant . There is STRONG evidence to REJECT H0 . The hiring practice of the NYC police dept. does NOT represent the ethnic composition of NYC. CONDITIONS SRS X All Expected Counts are 1 or greater. √ No more than 20% of the Expected Counts are less than 5. √ EXP. DATA 6089.4 5901.7 6569 1897.7 417.08