18 - Comparative Analytics

Question 1

How to estimate (µ₁ - µ₂)

Answer 1

(Xbar₁ - Xbar₂)

Question 2

t-statistic for the hypothesis test comparing 2 means

Answer 2

a standard error counter:

t = [(Xbar₁ - Xbar₂) - D₀] / se(Xbar₁ - Xbar₂)

Question 3

95% CI for (Xbar₁ - Xbar₂)

Answer 3

(Xbar₁ - Xbar₂) +/- 2se(Xbar₁ - Xbar₂)

Question 4

SE(Xbar₁ - Xbar₂)

Answer 4

sqr(σ₁²/n₁ + σ₂²/n₂)

Question 5

se(Xbar₁ - Xbar₂)

Answer 5

sqr(s₁²/n₁ + s₂²/n₂​)

Question 6

Assumptions for 2 sample t-test

Answer 6

1. Independence, within and between groups
   
   • variances of the 2 groups are allowed to be different
2. Constant variance within each group
3. Approximate normality of the raw data
   
   • ^as long as you have decent sample sizes, the test doesn’t require this 3^rd assumption due to the CLT

Question 7

test statistic for two-sample proportions

Answer 7

[Estimate - Null Hyp Value] / [std. error (Estimate)]

Question 8

95% CI for 2-sample proportion

Answer 8

Estimate +/- 2 std. error(Estimate)

Question 9

100(1-a)% confidence interval for 2 sample proportions

Answer 9

*+/- z_a/2

Question 10

Paired t-test

Answer 10

2 repeat observations on the same experimental unit

• *controls for unwanted variability between eusbjects**
• **a one-sample t-test on the pairwise differences***

Question 11

d_i

dbar

s_d

n

d₀

Answer 11

d_i = pairwise differences = (x_i - y_i)

dbar = mean of differences

s_d = SD of differences

n = number of pairs

d₀ = value of the diff in means under the Null

Question 12

test statistic for the paired t-test

Answer 12

t = (dbar - d₀)/(s_d/sqr(n)

Question 13

100(1 − α)% confidence interval for paired t-test

Answer 13

dbar +/- t_{a/2, n-1} (s_d/sqr(n))

t_{a/n, n-1} = 100(1-a/2) quantile of the t-distrib on (n-1) degrees of freedom

n = number of data pairs, not original data points

Question 14

pooled t-test

Answer 14

another 2-sample t-test that assumes the variances are the same