Q1: From experience it is known that a certain brand of tyre lasts, on average, 15,000 miles with standard deviation 1,250. A new compound is tried and a sample of 120 tyres yields an average life of 15,150 miles.

a) Write down the null and alternative hypotheses to test for an improvement in tyre technology, given the above information.

b) Use the 5% significance level to test the null in a). Interpret your finding.

Q2: A political opinion poll questions 1000 people. Some 464 declare they will vote Conservative. Find the 95% confidence interval estimate for the Conservative share of the vote.

Q3: Explain and compare point estimation and interval estimation.

Q4: What is the relationship between a statistical test, such as the t-test, and a confidence interval?

Q5: What are the Type I and Type II errors? Discuss and compare.

Q6: Discuss the following statement: "If H₀ cannot be rejected at the 5% significant level, a test at the 1% is required."

Q7: Discuss the following statement:

"The linear regression model is excessively restrictive since it only allows for a linear relation between the dependent variable ??t and the explanatory variables x_1,t, x_2,t, ... x_k,t"

Q8: What are dummy variables? How can they be used to test for structural breaks/regime changes in the data?

Q9: The data given below gives annual income (X) of 10 families and their annual expenditures on entertainment and recreation (Y), all expressed in thousands of dollars. It is hypothesized that the expenditure on entertainment and recreation is linearly related to income, i.e.

Y_i = α + βX_i + u_i.

The following question should be answered using Excel, or a calculator, by applying relevant formulas given in lecture notes. Make sure that you clearly write out which formula you apply and present your calculations.

a) Estimate α^{^} and β^{^} (show all computations) write out the estimated regression line and graph it. Interpret the coefficients.

b) Calculate σ^{^2}u = s², var(α^), var(β^{^}).

c) For each value of X calculate

w_i = ∑(X_i - X¯)/∑(X_i - X¯)²

and show that the OLS estimate of beta is a linear function of the data y, i.e. β^{^} = ∑ⁿ_i=1w_iY_i.

d) Calculate R², and demonstrate that R² = ρ^², where ρ^ is the sample correlation coefficient.

e) Use the 5% significance level to test the following hypotheses:

H₀: β = 0
H₁: β ≠ 0

Q10: Use the data in the file "Test2Data.xlsx" in the "Test 2" folder on iLearn and Eviews to estimate the following model using 1000 observations on wage, education, experience and gender.

W_agei = β₁ + β_2EDUci + β_3Experi + β_4Femalei + u_i

where

W_agei = earnings per hour for person i ($)
EDUc_i = years of education
Exper_i = years of experience
Female_i = 1 if female , 0 otherwise.
u_i = random error

a) Present the estimated equation, comment on the statistical significance of the estimated coefficients (at the 5% level) and interpret the estimated coefficients.

b) Compute a 95% confidence interval for the coefficient β4 and interpret the interval.

c) Is there evidence of wage discrimination between the sexes? Test your claim at the 1% significance level and comment on it. (Show all steps in hypothesis testing, including the hypotheses, the test statistic, the distribution of the test statistic under the null with exact degrees of freedom, critical value, decision rule and your conclusion.)

d) Compute the elasticity of Wage with respect to ??x??????i???????? at the mean values and interpret your answer.

e) Predict the expected wage for a male and a female worker who both have 12 years of education and 5 years experience. By how much the expected wages differ between the two groups for the given education and experience levels?

Attachment:- Test2Data.xlsx