Problem solvingFor each question, you must show the derivation of your answer step-by-step to obtain full mark as specified.
L1: In a sample of 25 classes, the following numbers of students were observed:
40 50 42 20 29
39 49 46 52 45
51 64 43 37 35
44 10 40 36 20
20 29 58 51 54
(10 Points) Calculate the median , mode, range, mean, variance, and standard deviation, for this sample.
L2. A random sample of twelve automobiles showed the following figures for miles achieved on a gallon of gas. Assume the population distribution is normal. From the data:
n=12; sum_{i=1}^{12}X_i=232.9 ; sum_{i=1}^{12}{X_i^2=4514.1 ; and sum_{i=1}^{12}{{(X_i-bar{X})}^2=13.2867} }
a) (1 Point) Find the expected miles per gallon of gas.
b) (1 Point) Find the sample variance.
c) (3 Point) Find an approximate 95% confidence interval for the population mean.
L3. The following simple bivariate linear regression model was estimated explaining a firm’s sales revenue (SR) to the income of its customer’s (INC) using annual data over a nine-year period:
SR = 81.38 + .23(INC),
(0.018)
where the standard error of the slope estimate is reported in parentheses below the coefficient estimate.
a) (2 Points) Interpret the estimated slope term in the fitted regression equation.
b) (3 Points) Is the coefficient on the income variable significantly different from zero at a 95-percent confidence level using a one-tailed test?
c) (2 Points) What level of sales would you forecast if income were $4,200?
d) (3 Points) Find an approximate 95 percent prediction interval for sales at income level $4,200, given that the standard error of the regression was 63.67.
L4. The following relationship was estimated between grade-point average Y and family income X:
Y = 1.38 + .12X
(.03)
The sum of squared residuals was .6528, and the sample size was 8. The sum of squared deviations of X about its sample mean was 162. The standard error of the estimated coefficient is reported in parentheses.
a) (2 Points) Find the estimated variance of the regression line.
b) (3 Points) Test the null that the slope coefficient is zero at the .05 level of significance.
L5. Develop a multiple-regression model for auto sales (AS) as a function of household income (INC) and population (POP) from the following data for 10 metropolitan areas:
Area Auto Sales Household Income Population Your Task
1 185792 23409 133.17
2 85643 19215 110.86 Mr. Ermon
3 97101 20374 68.04
4 100249 16107 99.59 Mr. Livingston
5 527817 23423 289.52
6 403916 19426 339.98
7 78283 18742 89.53 Miss Rassif
8 188756 18553 155.78
9 329531 21953 248.95
10 91944 16358 102.13 Mr. Anthony
To assure that each of you run your own regression problem, please replace the values on the row that I assigned your name on it but leave other rows unchanged.
Area Auto Sales Household Income Population
2 85650 19415 115.86 Mr. Ermon
4 100259 16117 100.00 Mr. Livingston
7 78263 18730 80.53 Miss Rassif
10 92944 16558 122.13 Mr. Anthony
The Empirical model is,
{AS}_i=beta_0+beta_1{POP}_i+beta_2{INC}_i+epsilon_i
Where I = 1, 2, 3,……, 10 and epsilon_i is the error term of observation ith
a) (10 Points) Using Excel to run the regression to obtain the estimated coefficients: {hat{beta}}_0, {hat{beta}}_1, and {hat{beta}}_2 (you need to submit your excel output as we did in class)
b) (1 Point) Base on your Excel result, what percentage of the variation in AS is explained by this model?
c) (4 Points) Use your Excel result, what is the fitted line? And base on the fitted line, What level of auto sales would you forecast if household income is$23,175 and population is 128.07?