University of British Columbia
Final Examination
STAT 305 Introduction to Statistical Inference 2016{17 Term 2
Instructor: William J. Welch
Student FAMILY Name:
(Please PRINT)
Student Given Names:
(Please PRINT)
Student ID Number:
Signature:
Date of Exam: April 13, 2017
Time Period: 12:00{2:30 pm
Number of Exam Pages: 12, including this cover sheet
(please check for completeness)
Additional Materials Allowed: Calculator; formula sheet (81
2
11, 2-sided)
Question Marks Score
1 12
2 13
3 12
4 13
Total 50
Student Conduct During Examinations
1. Each examination candidate must be prepared to produce, upon request of the invigilator or
examiner, his or her UBCcard for identication.
2. Examination candidates are not permitted to ask questions of the examiners or invigilators,
except in cases of supposed errors or ambiguities in examination questions, illegible material,
or the like.
3. No examination candidate shall be permitted to enter the examination room after the expira-
tion of one-half hour from the scheduled starting time, or to leave during the rst half hour of
the examination. Should the examination run forty-ve (45) minutes or less, no examination
candidate shall be permitted to enter the examination room once the examination has begun.
STAT 305, April 13, 2017 Page 2
4. Examination candidates must conduct themselves honestly and in accordance with established
rules for a given examination, which will be articulated by the examiner or invigilator prior
to the examination commencing. Should dishonest behaviour be observed by the examiner(s)
or invigilator(s), pleas of accident or forgetfulness shall not be received.
5. Examination candidates suspected of any of the following, or any other similar practices, may
be immediately dismissed from the examination by the examiner/invigilator, and may be
subject to disciplinary action:
(a) speaking or communicating with other candidates, unless otherwise authorized;
(b) purposely exposing written papers to the view of other examination candidates or imag-
ing devices;
(c) purposely viewing the written papers of other examination candidates;
(d) using or having visible at the place of writing any books, papers or other memory aid
devices other than those authorized by the examiner(s); and,
(e) using or operating electronic devices including but not limited to telephones, calculators,
computers, or similar devices other than those authorized by the examiner(s)|(electronic
devices other than those authorized by the examiner(s) must be completely powered down
if present at the place of writing).
6. Examination candidates must not destroy or damage any examination material, must hand
in all examination papers, and must not take any examination material from the examination
room without permission of the examiner or invigilator.
7. Notwithstanding the above, for any mode of examination that does not fall into the traditional,
paper-based method, examination candidates shall adhere to any special rules for conduct as
established and articulated by the examiner.
8. Examination candidates must follow any additional examination rules or directions commu-
nicated by the examiner(s) or invigilator(
s).
STAT 305, April 13, 2017 Page 3
Tables 1 and 2 at the end of the booklet contain some common discrete and continuous distributions,
along with their properties. They are the same as Tables 1.2 and 1.3 in the Course Notes, except
that the Laplace distribution is added to Table 2.
For full marks in questions asking you to show or derive a property, you must be clear about any
RESULT you are using, including any CONDITIONS for it to hold, and HOW the result is applied.
1. The United States PR/HACCP Act prescribes an inspection plan to monitor a facility’s
control of E. coli in the production of beef carcasses. It involves taking a sample of n = 13
carcasses periodically from production and testing them.
Let be the probability a randomly chosen carcass from a large batch of carcasses has an
unacceptable test result, and assume the 13 test results in a sample are statistically indepen-
dent. Then Y , the number of carcasses in the random sample with unacceptable levels of E.
coli, has a Bin (n = 13; ) distribution.
Suppose we want to test = 0:1822 (safe”) against = 2 0:1822 = 0:3644 (unsafe”).
(a) [1 mark] Write down the null hypothesis, H0, and the alternative hypothesis, Ha, of
interest here.
(b) [2 marks] Write down the likelihood ratio for testing H0 versus Ha and simplify it.
(c) [2 marks] Let y be the observed value of Y . Do small or large values of y give more
evidence against H0 in favour of Ha What theorem or lemma justies your answer
STAT 305, April 13, 2017 Page 4
(d) Suppose the hypothesis test rejects H0 if y is 3 or more.
i. [2 marks] Write down an expression for the probability of a Type I error. Be specic
but do not evaluate the expression numerically.
ii. [2 marks] Write down an expression for the power of the test. Be specic but do not
evaluate the expression numerically.
iii. [1 mark] Give an R expression to evaluate the power of the test numerically.
(e) Now consider a test of = 0:1822 against > 0:1822. We still have n = 13, and H0 is
still rejected if y is 3 or more.
i. [1 mark] Is the probability of a Type I error the same as in question 1(d)i Why or
why not
ii. [1 mark] Is the power the same as in question 1(d)ii Explain.
STAT 305, April 13, 2017 Page 5
2. A contingency table of frequency data has a row variable” with levels indexed by i = 1; : : : ;
I and a column variable” with levels j = 1; : : : ; J . Thus, there are frequencies yij for IJ
categories generated by all combinations of the levels of the row and column variables, as set
out in the following table.
Level Frequency
of row Level of column variable
variable 1 2 : : : J Total
1 y11 y12 : : : y1J y1:
2 y21 y22 : : : y2J y2:
…
…
…
. . .
…
…
I yI1 yI2 : : : yIJ yI:
Total y:1 y:2 : : : y:J n
The table also denes the following quantities: yi:, the total frequency for row i; y:j, the total
frequency for column j; and n, the total sample size across all categories.
We assume that y1j; : : : ; yIJ are frequencies for a random sample of size n from a multino-
mial distribution with category probabilities 11; : : : ; IJ . This multinomial distribution has
probability mass function
fY11;:::;YIJ (y11; : : : ; yIJ j n; 11; : : : ; IJ) =