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1) 2) 95% of confidence intervals I could do with 40 trials would capture the true average 95% of the time this method is used, it will correctly capture the population average 3) Reject if xbar is above 2.33+2.33*20/sqrt(36) so reject if xbar is above 127.8 The z-score for that is z=(127.8-125)/(20/sqrt(20))=0.83 Power is 0.2033 4) H0: µ ≥ 3.93 Ha: µ < 3.93 α=0.05 t145=(3.85-3.93)/.85/sqrt(146) 0.10 < p-value < 0.15 Fail to Reject No, this class is not significantly less work than other classes in the department 5) H0: Social skills is not related to intelligence Ha: Social skills are related to higher intelligence α=0.05 z-test: (56/125-73/135)/sqrt{ 129/260*(1-129/260)/125 + 129/260*(1-129/260)/135 } = -1.49 Chisq: Obs: 56, 69, 73, 62 Exp: 62.02, 62.98, 66.98, 68.02 Chi: 0.584, 0.575, 0.541, 0.533 p-value = 0.1362 0.10 < p-value < 0.15 Fail to reject We cannot show poor social skill is related to intelligence 6) (263.14-277.24)+-2.626*7.2/sqrt(600) = (-14.87, 13.33) 7) There are multiple answers, but here are the main two possibilities. The main thing to notice is how interconnected everything is. You can’t change the MSE without needing to change either SSE or DFE (and the standard deviation too) Regression Statistics Multiple R 0.698729 R Square 0.488222 Standard Error 77.28074 Observations 20 ANOVA Df Regression Residual Total SS MS 1 102553.6 102553.6 18 107501.6 5972.313 19 210055.2 F 17.1715 P-value 0.00061 Coefficients Std Error t Stat P-value Intercept -2.48371 32.63094 -0.07612 0.940167 length 5.615491 1.355138 4.143851 0.00061 Regression Statistics Multiple R 0.698729 R Square 0.488222 Standard Error 77.28074 Observations 20 ANOVA Df Regression Residual Total SS MS 1 102553.6 102553.6 18 107501.6 5972.313 19 210055.2 F 17.1715 Coefficients Std Error t Stat P-value Intercept -2.48371 32.63094 -0.07612 0.940167 length 5.615491 1.355138 4.143851 0.00061 8) Group DF 1 SS 82.559 MS 82.559 F 5.934 P-value 0.00061 Error Total 26 27 361.7354 444.29 13.9129 9) 256/400+-1.75*sqrt{ 256/400*(1-256/400)/400} = (0.598, 0.682) 10) A z-test if I assume normality and assume the standard deviation is the true sigma A t-test if I assume normality 11) OBS Mechanical Civil Electrical Chemical City 26 44 63 45 Suburb 36 33 45 59 EXP Mechanical Civil Electrical Chemical City 31.44 39.05 54.8 52.8 Suburb 30.6 38.0 53.2 51.2 X2 Mechanical Civil Electrical Chemical City 0.94 0.63 1.24 1.14 Suburb 0.97 0.65 1.27 1.17 12) None of the above 13) .03/2=1.645*sqrt(.01*(1-0.01)/n) Then n=119.06 Need 120 engineers 14) 0.08<p-value<0.10 15) You can’t get negative values, but the normal sticks out past the 0, which means there is always a risk of getting negative (unrealistic) answers If the confidence level is too high then the results may not be realistic, and 99% must be too high The rule of n > 30 doesn’t mean it’s exactly normal, and this shows a time when that rule fails 16) (4.3-3.4)+-2.021*sqrt(1.7^2/50+0.9^2/70) = (0.367, 1.432) 17) Students should be able to explain where they are seeing clumping – Not random There is a fan shape at the right end – non constant variance The clump on the top left could be part of a fan shape with clumping – or students may say that is a clump around an outlier calling normality into question. In that case they may describe a swerve pattern from bottom left to mid right which violates linearity. 18) The biggest winners were telling the matched pairs from the pooled tests, ANOVA, and the definition of confidence.