MATH235 MATH235 Week 4 - Assessed problems (coursework) MATH235 Week 5 - Workshop problems

MATH235 Week 4 - Moodle Quiz-assessed problems

Using the lm function, fit the following model for log FEV ( $Y_{i}$ ), with age ( $x_{i}$ ) only as a covariate,

\mathbb{E}[\log(Y_{i})]=\beta_{1}+\beta_{2}x_{i}.

You should use the full data set of 654 records held in the file fev.

Q4.1

Least squares estimate
The least squares estimate $\hat{\beta}$ is

(a)

$(0.0505,0.0871)$
(b)

$(-2.27,0.0521)$
(c)

$(6.84,3.67)$
(d)

$(3.67,6.84)$
(e)

$(0.0871,0.0505)$

[marks: 2]

Q4.2

Standard error
Using your output from lm, what is the standard error of $\hat{\beta}_{2}$ ?

(a)

0.0291
(b)

0.0829
(c)

0.00281
(d)

1.74
(e)

31.0

[marks: 2]

Q4.3

Prediction
The predicted log FEV for a 10 year old child is

(a)

$10\hat{\beta}_{1}+\hat{\beta}_{2}$
(b)

$\hat{\beta}_{1}+\hat{\beta}_{2}+10$
(c)

$10(\hat{\beta}_{1}+\hat{\beta}_{2})$
(d)

$\hat{\beta}_{1}+10\hat{\beta}_{2}$
(e)

$10\hat{\beta}_{2}$

[marks: 2]

Q4.4

Prediction variance
The variance for the predicted log FEV for a 10-year old child is

(a)

$100\mbox{Var}(\hat{\beta}_{2})$
(b)

$\mbox{Var}(\hat{\beta}_{1})+100\mbox{Var}(\hat{\beta}_{2})$
(c)

$\mbox{Var}(\hat{\beta}_{1})+10\mbox{Var}(\hat{\beta}_{2})$
(d)

$\mbox{Var}(\hat{\beta}_{1})+10\mbox{Var}(\hat{\beta}_{2})+10\mbox{Cov}(\hat{% \beta}_{1},\hat{\beta}_{2})$
(e)

$\mbox{Var}(\hat{\beta}_{1})+100\mbox{Var}(\hat{\beta}_{2})+20\mbox{Cov}(\hat{% \beta}_{1},\hat{\beta}_{2})$

[marks: 2]

Q4.5

Prediction test statistic
If $\mbox{Var}(\hat{\beta}_{1})=0.000847$ , $\mbox{Var}(\hat{\beta}_{2})=0.00000789$ and $\mbox{Cov}(\hat{\beta}_{1},\hat{\beta}_{2})=-0.0000784$ , calculate the test statistic to test whether or not the log FEV of a 10 year old child is significantly different to 0.9.

(a)

$111.7$
(b)

$-2.60$
(c)

$-315.2$
(d)

$2.60$
(e)

$315.2$

[marks: 2]