1. Exam2 A learning experience….

2. Scores Raw Scores went from 68 to 147 As percentage of total….40% to 86% Scaled scores went from 60.5 to 100 Some still left to be graded… 90s8 80s23 70s23 60s5 TOTAL59

3. Question by Question Question123456789rawscaled count59 min42244101206860.5 max20 34182015102147100 avg13.716.412.923.210.516.08.94.30.8106.779.8 s4.54.14.47.54.11.73.02.70.617.38.7 avail20 4020 15105170100

4. Data for Q1 to Q3 Car ClassDisplacementFuel TypeHwy MPG 1 Midsize3.5R28 2 Midsize3R26 3 Large3P26 4 Large3.5P25.......... 58 Compact6P20 59 Midsize2.5R30 60 Midsize2R32 Categorical Numerical n=60

5. Q1 expect that the size of the car engine (measured by displacement) would change based on car class (compact, midsize, large) H0: MU(compact)=MU(mid)=MU(large) Ha: not all equal ANOVA single factor (3 samples) Unstack the data, excel data analysis

6. Q2 expect to see a relationship between car class and recommended fuel type Relationship between two categorical variables (car class and fuel type) Chi-sq independence test – 3x2 contingency table of counts…summing to 60 PR Compact16319 Large11516 Midsize91625 362460

7. Q3. Fuel type and mpg expect that because premium gasoline is higher quality, cars for which it is recommended will get higher gas mileage (on average) than cars for which regular fuel is recommended Ho: MU(prem) = MU(reg) Ha: MU(prem) > MU(reg) Unstack, T-test two sample NOTE: We guessed the wrong tail. Do not reject HO in favor of THIS Ha. t-Test: Two-Sample Assuming Equal Variances PR Mean24.3333327.70833 Variance12.49.519928 Observations3624 Pooled Variance11.2579 Hypothesized Mean Difference0 df58 t Stat-3.81704 P(T<=t) one-tail0.0001650.999835 t Critical one-tail1.671553 P(T<=t) two-tail0.000331 t Critical two-tail2.001717 R got higher sample mean The wrong p value The correct p value

8. Q4a Aspirin and Heart Attack Relationship between two 0/1 variable. 2x2 contingency table from the facts in the question (like lights and myopia). Chi-sq independence test for 2T alternative. Half the pvalue if you want a 1T alternative (Paspirin < Pplacebo) Heart AttackNo Heart Attack Aspirin1041089611000 Placebo1891081111000 2932170722000

9. Q4b. How many heart attacks using new design (given Ps) It is easy to calculate the mean (most likely) of 250.5. Tell me that the actual number is a random variable Provide a probability distribution for that random variable Number of Heart Attacks GroupNumberProbabilitymeanvariancestd dev aspirin165000.009454545156154.5312.4 placebo55000.01718181894.592.889.6 TOTAL22000 250.5247.4015.7 Normal approx to binomial

10. Q4c. Will new design affect p-value? Yes. We will be more certain about Aspirin’s effect and LESS certain about Placebo’s effect. The test is focused on the difference. The gain in accuracy for aspirin is not as great as the loss in accuracy for placebo (diminishing returns) Our test will be less powerful. P-value will go up. 50/50 v 75/25 v 100/0 Best designWorst design

11. Q5. Is Di significantly better than El? Not about whether P=0.5 About whether P(di)=P(el) 2x2 chi-squared independence test InOut Di10 20 El51722 152742 Expected7.112.9 7.914.1 Distances1.1428570.634921 1.0389610.577201 calculated chi- suared3.393939 Pvalue0.065436 Pvlaue/20.032718 2 tailed p- value 1 tailed p- value

12. Q6. Rportfolio Rportfolio = (R1+R2+R3)/3 StockMean ReturnVarianceStandard Deviation 10.10.010.1 20.050.00160.04 30.20.160.4 TOTAL 0.350.17160.414 Rportfolio0.1170.0190666670.138 Sum of variances (independent).414/3 R1, R2, R3 Will not be Independent.

13. Q7. Total (Avg) weight of n=20 Mean = 20*μ Variance = 20*σ 2 Normal (sum of normals) MeanVarianceStd Dev One guest150160040 Total of 20 guests300032000178.8854 Pr(total<3500) = NORMDIST(3500,3000,178.9,true) = 0.9974 Family hotel means….. Weights in elevator not independent. More likely to be under 3500.

14. Q8. Al and Bo

15. Q9 If students don’t cheat, then their IQs are independent identically distributed N(100,15) The null hypothesis (mean men = mean women) IS TRUE!!! When H0 is true, and we do any test correctly, we reject with probability 0.05. We will reject H0 with probability 0.05 and fail to reject with probability 0.95 What will happen under H0 is “easy” What will happen under Ha is very difficult…