MATH235 MATH235 Week 3 - Workshop problems MATH235 Week 3 - Moodle Quiz-assessed problems

MATH235 Week 3 - Assessed problems (coursework)

Submission is due on Tuesday in Week 4.

CW3.1

For the Olympics data in Table 0.12, a linear regression model in which long jump distance is used to predict high jump height is proposed,

Y_{i}=\beta_{1}+\beta_{2}x_{i}+\epsilon_{i},~{}~{}~{}i=1,\ldots,n,

where $Y_{i}$ is high jump height and $x_{i}$ is long jump distance.

Observation	Year	High Jump	Long Jump	Discus Throw
1	1896	71.25	249.750	1147.50
2	1900	74.80	282.875	1418.90
3	1904	71.00	289.000	1546.50
4	1952	80.32	298.000	2166.85
5	1960	85.00	319.750	2330.00
6	1964	85.75	317.750	2401.50
7	1972	87.75	324.500	2535.00
8	1976	88.50	328.500	2657.40
9	1980	92.75	336.250	2624.00
10	1984	92.50	336.250	2622.00

Table 0.12: Winning Olympic high jump heights and long jump and discus distances for a sample of 10 years. All measurements are in inches.

(a)

What assumptions are made about the residuals $\epsilon_{1},\ldots,\epsilon_{n}$ ?

[marks: 2]
(b)
- (i)
  
  Using the data in Table 0.12, write out the design matrix for this model.
  
  [marks: 1]
- (ii)
  
  Obtain the least squares estimates for $\beta=(\beta_{1},\beta_{2})^{\prime}$ .
  
  [marks: 2]

CW3.2

A survey on five colonies of ants was conducted to assess whether the mean ant length varied between colony. The results are shown in Table 0.13, and can also be found in the dataframe antLengths. Lengths are in $m m$ . Let $\mu_{i}$ denote the population mean length for colony $i$ .

Ant length ( $m m$ )	Group
8.1	1
7.7	1
8.1	1
7.8	1
7.6	1
8.2	1
8.0	1
7.6	1
9.9	2
10.0	2
9.9	2
9.7	2
10.0	2
9.5	2
9.1	2
11.4	2
9.9	3
10.0	3
10.3	3
9.6	3
10.6	3
10.6	3
14.4	3
13.6	3
14.3	4
13.5	4
13.0	4
13.7	4
13.9	4
14.8	4
16.7	4
16.5	4
15.3	5
16.3	5
15.3	5
15.9	5
16.0	5
19.2	5
18.1	5
17.1	5

Table 0.13: Ant lengths (

m m

) for ants from five colonies.

(a)

Give appropriate null and alternative hypotheses to test whether the mean ant length varies between colonies.

[marks: 1]
(b)

Calculate the sample group means for all five groups.

[marks: 1]
(c)

Given that the overall mean length of the samples is $12.03mm$ and that the total sum of squares $SS_{T}=453.864$ , carry out a one-way ANOVA to test your hypotheses in part $(i)$ . You should test at the 5% significance level, and state your conclusions clearly.

[marks: 3]

CW3.3

Challenge
Consider the following extension to the simple linear regression model,

Y_{i}=\beta_{1}+\beta_{2}x_{i}+\epsilon_{i},~{}~{}~{}i=1.\ldots,n

where $\epsilon_{i}$ are independent but not identically distributed Normal $(0,\sigma^{2}_{i})$ random variables, where $\sigma_{i}=\sigma\sqrt{x_{i}}$ .

(a)

Standardise $Y_{i}$ to create a random variable $Z_{i}$ which has Normal $(0,\sigma^{2})$ distribution.

[marks: 1]
(b)

By summing the squares of $Z_{1},\ldots,Z_{n}$ , create a sum of squares function for $(\beta_{1},\beta_{2})$ .

[marks: 2]
(c)

Using the first-order partial derivatives of the sum of squares function derived in part (b), show that the least squares estimator of $\beta_{2}$ is

$\hat{\beta}_{2}=\frac{\sum_{i=1}^{n}x_{i}^{-1}\sum_{i=1}^{n}Y_{i}-n\sum_{i=1}^% {n}x^{-1}_{i}Y_{i}}{\sum_{i=1}^{n}x_{i}\sum_{i=1}^{n}x_{i}^{-1}-n^{2}}.$

[marks: 2]