Statistical Analysis: Two-Sample t-Test and Simple Linear Regression - Prof. Radu Lazar, Exams of Business Statistics

Download Statistical Analysis: Two-Sample t-Test and Simple Linear Regression - Prof. Radu Lazar and more Exams Business Statistics in PDF only on Docsity! ∆ Comparing 2 Means ∆ Two Sample t-test (Different Variances) Pooled t-test (Equal Variances) t= x́1− x́2 √ s1 2 n1 + s2 2 n2 = x́1− x́2 SE t= x́1− x́2 √ (n1−1 ) s1 2 +(n2−1)s2 2 n1+n2−2 √ 1 n1 + 1 n2 = x́1− x́2 s p√ 1n1 + 1 n2 df ≈minimum of n1−1∧n2−1 df =n1+n2−2 CI=( x́1− x́2)± tmin ¿ √ s1 2 n1 + s2 2 n2 CI=( x́1− x́2 )± t n1+n2−2 ¿ √ ( n1−1 ) s1 2+(n2−1 ) s2 2 n1+n2−2 √ 1 n1 + 1 n2 =( x́1− x́2)± t n1+n2−2 ¿ (s¿¿ p)√ 1n1 + 1 n2 ¿ Assumptions: Normal, Independent Assumptions: Normal, Independent, EQUAL VARIANCE Paired t-Test d= x1−x2 di = x1i - x2i d= ∑ i=1 n d i n and sd=√∑i=1 n ( di−d ) 2 n−1 t n-1= d−μd sd √n and µd=0 almost always CI= d ± tn-1 sd √n Assumptions: Both populations are normal, or the population of the differences is normal H0 can be =, ≤, or ≥ in this case only If it’s not given, sx, the estimate of σ, is ¿√∑ (x i− x́) 2 n−1 T-table Assistance (If Radu doesn’t have it on the exam) Two-tail .20 .10 .05 .02 .01 One-tail .10 .05 .025 .01 .005 Confidence Int. 80% 90% 95% 98% 99% □ Simple Linear Regression □ Regression line  The line that minimizes the sum of the squared residual  Points on regression line are predicted variable  R is unit-less  R is between one and negative one μy=β0+β1 x≈ ŷ=b0+b1 x̂ b1=slope=r Standarddeviation of y Standard deviation of x =r SD( y ) SD( x) b0=(meanof y )– b1 (meanof x )= ý –b1 x́ r2=percent of variability∈ y (response ) that is explained by the change∈x (explanatory) CI for β0 CI for β1 CI=b0± tn−2 ¿ se√ 1 n + x́2 (n−1 ) sx 2 CI=b1±t n−2 ¿ se sx√n−1 se ≈σ Testing for Linearity  H0:β1=0 (no linear association)  HA:β1≠0 (linear association) t n−2= b1 se sx√n−1 Confidence interval with x* (a specific desired value of x) CI= ŷ± t n−2 ¿ se √ 1 n + x¿− x́ (n−1 ) sx 2 Prediction interval with x* (a specific desired value of x) PI= ŷ ± t n−2 ¿ √ sesx √n−1 2 +(x¿− x́ )+ se 2 n +se 2 Confidence intervals are intervals constructed about the predicted value of y, at a given level of x, which are used to measure the accuracy of the mean response of all the individuals in the population.