P&S Exam 2

Question 1

Sample Mean

Answer 1

Xbar = (X1+…+Xn)/n

Question 2

Sample Variance

Answer 2

S^2 = 1/n-1* ∑(Xk-Xbar)^2

Question 3

Bias

Answer 3

B(𝜣^) = E(𝜣^) - 𝜽

Question 4

Mean Squared Error(MSE)

Answer 4

MSE(𝜣^) = E[(𝜣^ - 𝜽)^2]

Question 5

Bias-Variance Trade-Off

Answer 5

MSE(𝜣^) = Var(𝜣^) + B(𝜣^)^2

Question 6

Consistent Estimator

Answer 6

limn→∞ P(|𝜣^n - 𝜽|< ∈) = 1 for ∈ > 0

Question 7

Likelihood Function

Answer 7

• For Discrete: ∏PXi(xi)
• For Continuous: ∏fXi(xi)

Question 8

Maximum Likelihood Estimate (MLE)

Answer 8

• 𝜽^ML
• The value for 𝜽^ that maximizes the likelihood function (L’(𝜽 *) = 0 and L’‘(𝜽 *) < 0)

Question 9

(1 - 𝛼)100% Confidence Interval

Answer 9

• [𝜣^l, 𝜣^h]
• P(𝜣^l ≤ 𝜽 ≤ 𝜣^h) ≥ 1- 𝛼

Question 10

Pivot Quantity Q

Answer 10

• Q is a function of the data given X1,…,Xn and the unknown parameter, but doesn’t depend on the unknown parameter..
• The probability distribution of Q also does not depend on the unknown parameter or any other unknown parameter.

Question 11

Methodology for Confidence Interval

Answer 11

1. Find Pivotal Quantity Q
2. Find interval for Q such that P(𝜣^l ≤ Q ≤𝜣^h) = 1 - 𝛼

Question 12

Methodology for Hypothesis Testing

Answer 12

5 Cs:
1. Create Hypotheses
2. Check Conditions
3. Calculate
4. Compare
5. Concluse

Question 13

HT Comparisons

Answer 13

Reject H0 if |W|>z𝛼/2