1. For each of the following cases, determine the null hypothesis and the alternative hypothesis.

a) A murder suspect is on trial.

b) A patient is being tested for his or her mental illness.

c) A person with mental illness has been hospitalized for a certain period of time. Now the doctor wants to determine whether this patient may be discharged from the hospital.

d) You met a person at a blind date. You like her (or him). But you do not know whether she (or her) likes you or not.

e) You have been dating a person. You suspect that he (or she) no longer loves you.

2. Explain the following terms: type I error, type II error, power of a test.

3. You may want the probability of type I error to be zero. What happens to the probability of type II error? Answer to this question by considering a situation in which you are a judge and a murder suspect is under trial.

4. Suppose that, in the past years, the average height of all the UW freshmen used to be around 175.

Your friend claims that UW freshmen this year are taller than that in the past years. In order to test your friend's claim, you decided to test the following hypothesis: H₀: µ = 175;     H₁: µ > 175

Based on 9 observations, µ^{^} turned out to be 178 and σ^{^2} turned out to be 4 (In other words, the variance of µ^{^} turned out to be 4/9). Perform the test at the significance level of 5%. Explain your answer. (Draw a graph for the distribution of the t-value, and specify the critical value and the significance level on the graph.)

5. From Question #4, how does your answer change if the null and the alternative hypotheses are changed as follows?

H₀: µ = 175; H₁: µ ≠ 175

(Draw a graph for the distribution of the t-value, and specify the critical values and the significant level on the graph.)

6. Consider the following model:

Population Equation-

Y_i = β₁ + β₂X_i + e_i, i = 1, 2, ..., n  

Assumptions-

E[ei] = 0, ∀ i,

E[e_i²] = σ², ∀ i,

E[e_ie_j] = 0, i ≠ j

e_i ∼ normally distributed.

X_i is non - stochastic

Sample Regression Equation -

Y_t = β^{^}₁ + β^{^}₂X_i + e_i  (C)

a) Explain the last assumption given above.

b) We want to apply the OLS method to estimate β₁ and β₂. Set up an objective function, and provide the two first-order conditions.

c) From c), show that ∑e^{^}_i = 0, where e^{^}_i = y_i - β^{^}₁ - β^{^}₂X_i.

d) Show that the sample regression equation in equation (D) can be written in the following deviation-from-mean form:

(Y_i - Y^-) = β^{^}₂(X_i - X^-) + e^{^}_i

Or y_i = β^{^}₂x_i + e^{^}_i,              (D)

e) Apply the OLS method to equation (D) and derive β^{^}₂.