SECTION A

Instructions to students: Section A is completed on Blackboard. Go to the ’FINAL EXAM’ tab on Blackboard and click on ’Final Exam Section A’ to see the questions   and submit your answers.

SECTION B

Instructions to students: Section B is submitted on Blackboard under ’Final Exam Section B’ . Write out your answers, scan to a single pdf file, and upload.

1.  In a study of 2246 randomly selected drivers, 964 said that they use their mobile phone regularly while driving. It was believed that more than 40% of drivers       regularly use their mobile phone while driving. Carry out a hypothesis test to test this claim.

(a) State the null and alternative hypotheses of interest.

(b)  Compute the test statistic.

[4 marks]

[4 marks]

(c)  Find the p-value, use it to evaluate the test statistic, and give the               conclusion. Use a = 0.05.                                                              [5 marks]

(d)  Briefly (in 2-3 sentences) explain the role of the Central Limit Theorem in  this hypothesis test.                                                                       [7 marks]

(20 marks)

2. A pot experiment was conducted to examine the relationship between the           potassium found in plants (y) and two soil characteristics, soil potassium (z1 )     and soil acidity (z2 ; coded 1 for high and 0 for low). Values were recorded for 14 pots where soil potassium and soil acidity were manipulated at establishment.

The y and zi  values are for high acidity:

i         1         2         3        4         5         6         7

yi

x1i

and for low acidity:

i

yi

x1i

Page 2 of 4

Oc Trinity College Dublin, The University of Dublin 2022

A multiple regression model was fitted with soil potassium and soil acidity as predictors, with the following output from R:

Call:

lm(formula  =  y  ~  x1  +  x2)

Residuals:

Min            1Q   Median           3Q         Max

-4 .5179  -3 .0321  -0 .1821    2 .4688    5 .0357

Coefficients:

Estimate  Std .  Error  t  value  Pr(>|t |)

(Intercept)    27 .0571         4 .0274      6 .718  3 .29e-05  *** x1                       1 .0982          0 .2373     4 .628  0 .000731  *** x2                       7 .2286          1 .8985      3 .807  0 .002906  **

---

Signif .  codes:    0  ***  0 .001  **  0 .01  *  0 .05  .  0 .1      1

Residual  standard  error:  3 .552  on  11  degrees  of  freedom Multiple  R-squared:    0 .7655,Adjusted  R-squared:    0 .7229 F-statistic:  17 .96  on  2  and  11  DF,   p-value:  0 .0003433

(a)  Express the multiple regression model in matrix notation, using numeric

values in the y and x matrices.                                                    [6 marks]

(b)  Interpret the estimate for the coefficient of z1  and the associated

hypothesis test.                                                                              [6 marks]

(c) Show how the standard error for the parameter estimate associated with z2

is calculated.

Hint:

_ 1.28571429 (xT x)− 1 =  '(') -0.07142857

'-0.14285714

-0.071428571 0.004464286

0.000000000

[4 marks]

-0.1428571_

0.0000000  '(')

0.2857143

(d)  Give a brief (2-3 sentences) explanation of the F statistic in the final line of the R output, written for a non-statistician to understand.             [4 marks]