Question 1: Hypothesis Testing
Records show that the mean farm size in southern NSW has increased over the last 70 years. An agribusiness researcher believes that the mean size of farms has increased since 2007 from a mean of 471 hectares. To test this, a random sample of 36 farms was selected. The sample had a mean farm size of 498.78 hectares and a standard deviation of 46.94. Use a significance level of 0.05 to test this hypothesis.
a. State the null and alternative hypotheses.
b. State your decision rule.
c. Test your hypotheses and state your conclusion.
d. What is your test decision and conclusion if the significance level is 0.01?
e. Calculate a 90% confidence interval for the population mean, μ.
Question 2: Simple Linear Regression
A car magazine is interested in the relationship between the new list price of a car and the amount of power (kW) the car has. The following data has been collected for a range of new cars.
|
Car price ($)
|
Power (kW)
|
| 99990 |
141 |
| 292800 |
298 |
| 49790 |
141 |
| 269900 |
285 |
| 115500 |
200 |
| 62990 |
230 |
| 41990 |
116 |
| 227600 |
261 |
| 184874 |
280 |
| 82900 |
191 |
|
Mean car price
$142,833.40
|
Mean kW 214.30
|
Linear regression analysis has been undertaken with Excel and is shown in the following tables.
|
SUMMARY OUTPUT
|
|
|
|
|
|
|
|
Regression Statistics
|
|
Multiple R
|
0.866920453
|
|
|
|
|
|
|
R Square
|
0.751551071
|
|
|
|
|
|
|
Adjusted R Square
|
0.720494955
|
|
|
|
|
|
|
Standard Error
|
49521.10928
|
|
|
|
|
|
|
Observations
|
10
|
|
|
|
|
|
|
ANOVA
|
|
|
|
|
|
|
|
|
df
|
SS
|
MS
|
F
|
Significance F
|
|
|
Regression
|
1
|
59346086604
|
59346086604
|
24.1997766
|
0.001165002
|
|
|
Residual
|
8
|
19618722117
|
2452340265
|
|
|
|
|
Total
|
9
|
78964808720
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Coefficients
|
Standard Error
|
t Stat
|
P-value
|
Lower 95%
|
Upper 95%
|
|
Intercept
|
-118508.7466
|
55385.57987
|
-2.139704
|
0.06480627
|
-246228.1228
|
9210.629612
|
|
Power (kW)
|
1219.515383
|
247.9028983
|
4.919326846
|
0.001165
|
647.8502744
|
1791.180492
|
a. State the linear regression equation for this data.
b. Interpret the meaning of the slope, b1 in this problem.
c. Predict the mean price of a car with power of (i) 130 kW, and (ii) 220 kW. Explain which of these two predictions (i.e., (i) or (ii)) is likely to be more reliable and why this may be the case.
d. Comment on the goodness of fit of the estimated regression model (e.g., R2 and SYX).
e. At the 0.01 level of significance, is there evidence of a linear relationship between the power of the car (kW) and its price?
Question 3: Multiple Linear Regression
The salary of workers is expected to be dependent, among other things, the number of years they have spent undertaking education and their work experience. The following table provides information from a randomly selected sample of employees' annual salaries (in thousands of dollars), the number of years each of them has spent in education, and the total number of years of work experience.
|
Salary
($'000)
|
Education X1
(Years)
|
Experience X2
(Years)
|
|
52
|
16
|
6
|
|
44
|
12
|
10
|
|
48
|
13
|
15
|
|
77
|
20
|
8
|
|
68
|
18
|
11
|
|
48
|
16
|
2
|
|
59
|
14
|
12
|
|
83
|
18
|
4
|
|
28
|
12
|
6
|
|
61
|
16
|
9
|
|
27
|
12
|
2
|
|
69
|
16
|
18
|
|
Mean salary ('000)
$55.33
|
Mean education 15.25
|
Mean experience 8.58
|
Linear regression analysis has been undertaken with Excel and is shown in the following tables.
|
SUMMARY OUTPUT
|
|
Regression Statistics
|
|
Multiple R
|
0.921915904
|
|
|
|
|
|
R Square
|
0.849928935
|
|
|
|
|
|
Adjusted R Square
|
0.816579809
|
|
|
|
|
|
Standard Error
|
7.556773078
|
|
|
|
|
|
Observations
|
12
|
|
|
|
|
|
ANOVA
|
|
|
|
|
|
|
|
df
|
SS
|
MS
|
F
|
Significance F
|
|
Regression
|
2
|
2910.723292
|
1455.36165
|
25.48579372
|
0.000196488
|
|
Residual
|
9
|
513.9433742
|
57.1048194
|
|
|
|
Total
|
11
|
3424.666667
|
|
|
|
|
|
Coefficients
|
Standard Error
|
t Stat
|
P-value
|
Lower 95%
|
Upper 95%
|
|
Intercept
|
-42.1095481
|
13.82993282
|
-3.04481219
|
0.013909163
|
-73.3950297
|
-10.824067
|
|
Education X1
|
5.775214858
|
0.854329848
|
6.75993572
|
8.27131E-05
|
3.842586472
|
7.70784324
|
|
Experience X2
|
1.091750081
|
0.459145034
|
2.37778915
|
0.04137736
|
0.053091854
|
2.13040831
|
a. State the multiple regression equation for this data.
b. Interpret the meaning of the slopes in this problem.
c. Predict the salary for an employee who has 14 years education and 17 years work experience.
d. Comment on the goodness of fit of the estimated regression model.
e. Is there a significant relationship between salary and the two independent variables (i.e., education and work experience) at the 0.01 level of significance?
f. Explain how you think the above model could be improved?