ECMB12H3 Lec30 and Lec60
Quantitative Methods in Economics II
Division of Management
University of Toronto at Scarborough
January – May 2010
Date: Saturday February 6, 2010, 9:00 – 11:00 am
Time allowed: Two (2) hours
Aids allowed: Calculator and one asheet (two 8.5”x11” pages) prepared by
• This test consists of 22 questions in 10 pages including the cover page.
• It is the stu dent’s responsibility to hand in all pages of this test. Any m issing page
will get zero mark.
• Statistical tables are provided separately. Do not hand in the tables. Only hand in
your test papers.
• Show your work in each question in Part II.
• This test is worth 30% of your course grade.
Print Last Name Solution
Circle your Section L30 Wednesday at 7 pm L60 Online
Do not write on the space below, for markers only
Page Question Max Marks
2-5 1-17 51
6 18 9
7 19 10
8 20 10
9 21 10
10 22 10
1 Part I. Multiple Choice. 3 marks in each question. No part mark.
Circle only one answer. If there are more than one correct answer, circle the best one.
1. Which one of the following statements is true?
(a) The margin of error for a 95% confidence interval for the mean µ
increases as the sample size increases.
(b) The margin of error for a 95% confidence interval for a proportion p
increases as the proportion gets close to 0 or 1.
√(c) The sample size required to obtain a confidence interval of specified
margin of error increases as the confidence level increases from 95% to
(d) Only (a) and (b) are true.
(e) All (a), (b) and (c) are true.
2. A polling organization is planning to in crease the size of its random sample of
voters from about 1200 people to ab out 3000 people. What will be the effect of
(a) Reduce the bias of the estimate.
(b) Increase the confidence interval width for the parameter.
(c) Reduce the risk of ecological fallacy.
√ (d) Reduce the standard error of the estimate.
(e) Have no effect because the population size is the same.
3. In a hypothesis test, the p-value is:
(a) the significance level.
(b) the complement of β , the probability of a type II error.
(c) compared to β so as to decide whether to reject or accept the null
(d) the probability that an outcome of the data will happen purely by chance
when the alternative hypothesis is true.
√(e) the probability that an outcome of the data will happen purely by chance
when the null hypothesis is true.
4. You are conducting a one-sided test of the null hypothesis that the population
mean is 532 versus the alternative that the population mean is less than 532. If the
sample mean is 529 and the p-value is 0.01, which of the following statements is
(a) There is a 0.01 probability that the population mean is smaller than 529.
√(b) The probability of observing a sample mean smaller than 529 when the
population mean is 532 is 0.01.
(c) There is a 0.01 probability that the population mean is smaller than 532.
(d) If the significance level is 0.05, you will accept the null hypothesis.
(e) All of the above are true.
2 5. A scientist computes a 90% confidence interval for the population m ean µ to be
(4.38, 6.02). Using the sam e data, she also computes a 95% confidence interval
for µ to be (4.22, 6.18), and a 99% confidence interval for µ to be (3.91, 6.49).
Now she wants to test for H 0:µ = 4 versus H1: µ ≠ 4 . Regarding the p-value,
which one of the following statements is true?
(a) p-value < 0.01
√(b) 0.01< p-value < 0.05
(c) 0.05< p-value < 0.10
(d) 0.10< p-value < 0.50
(e) 0.50< p-value < 1.00
6. A random sample of 100 observations is selected from a population with mean µ
and standard deviation σ = 80 . Let X be the sam ple mean. Suppose you are
interested in testingH 0:µ = 100 against H 1 µ >100 at a significance level 0.05.
Which one of the following statements is true?
√ (a) Reject H 0f X >113.16
(b) Reject H 0f X <113.16
(c) Accept H if X > 86.64
(d) Accept H if X < 86.64
(e) None of the above is true
7. Suppose H : µ = µ is tested aga inst H : µ ≠ µ at a significance level α .
0 0 1 0
Which one of the following statements is true?
(a) If H0is rejected at α =0.05, then H 0ust also be rejected at α =0.01.
√ (b) If H is rejected at α =0.05, then H must also be rejected at α =0.10.
(c) If H0is accepted at α =0.05, then H m0st also be accepted at α =0.10
(d) If H0 is accepted atα =0.05, then H 0 must be rejected atα =0.01.
(e) None of the above is true.
8. A random sam ple of size 9 is taken fr om a norm ally distri buted population.
Suppose that 9 (x − x 2=140. The following hypothesis test is conducted:
∑ i= i
H 0: µ = 20 ,H 1 µ ≠ 20 . Suppose that the null hypothesis is accepted for values
of xsuch that 17.406 < x < 22.594 . Then th e significance level of this test is
(a) 1% (b) 2.5% (c) 5% √ (d) 10% (e) 12.5%
3 9. The following hypothesis test is conduc ted at a 5% significance level.
H 0:µ = 60 , H 1 µ > 60 . A random sa mple of size 16 is taken and the sam ple
mean is calculated to be x = 62. Assume that the population variance is 64. If
the power of this test is 0.9 when the true value ofµµi1 then µ1 is closest to
(a) 54 (b) 58 (c) 62 √ (d) 66 (e) 70
10. The hypotheses H 0:µ = 800 , H1: µ ≠ 800 are tested using a large random
sample. If the value of the test statistic is equal to 1.75, then the p-value is closest
(a) 0.0401 √ (b) 0.0802 (c) 0.10 (d) 0.4599 (e) 0.9198
11. In testing the null hypothesisH0 :µ = 10 versus the alternativeH1: µ ≠ 10 with
significance levelα =0.05. Which one of the following statements is false?
(a) If H is rejected, the p-value is smaller than 0.05.
√ (b) The probability of a type II error is β =0.95.
(c) If H0is accepted, a 95% confidence interval must contain 10.
(d) If H is accepted, then H will also be accepted at any α < 0.05.
(e) If the p-value is 0.07, then0H is accepted.
12. To test H 0 : p =0.5 versus H 1 p < 0.5 whe re p is the proportion of the
personal com puters sold this y ear com ing from relatively unknown
manufacturers. The fi rm selects a new random sa mple of 2,500 personal
computers sold this year. The p-value calculated from this sample is 0.016. The
number of personal computers sold this year in this sample is closest to
(a) 850 (b) 950 (c) 1050 √ (d) 1197 (e) 1250
Solution: Since the p-value is 0.016, from the Z-table,2.145 = p−0.5 ,
0.5 0.5 /2500
∧ 0.5 0.5
p = 0.5− 2.145 2500 =0.47855, hence (2500)(0.47855)=1196.375
4 13. A random sample is drawn from a normal population with mean µ and standard
deviation σ =5. A 95 percent confid ence interval for µ is (29.9, 30.3). If the
same sample data are u sed to test H :µ = 30 versus H : µ > 30 , the smallest
value of α that H 0ill be rejected is closest to
(a) 0.01 (b) 0.05 (c) 0.102 (d) 0.125 √ (e) 0.1635
14. A random sample X1,X 2...,X 6 is selected from a normal population with mean
µ and standard deviation σ . Let X be the sam ple m ean and
∑ (X i X =120. For what values of X will the hypothesis H : 0 =10 be
accepted at the 5% level of significance?
(a) between 6.08 and 13.92
(b) between 5.12 and 14.43
√ (c) between 4.858 and 15.142
(d) between 5.863 and 16.235
(e) none of these
Questions 15-17, let X ,X , X , X be a random sample from a normal population with
1 2 3 4
mean µ and variance 1. We wish to test H :0µ = 2 versus H : µa< 2 using the
significance level 0.05.
15. Suppose the decision rule rejectsH if X ≤ c, the value of c is closest to
0 ∑i=1 i
(a) -3.29 (b) -1.645 (c) 2.355 √ (d) 4.71 (e) 9.42
16. The power of the test when µ =1 is closest to
(a) 0.1406 (b) 0.3594 (c) 0.4255 √ (d) 0.6406 (e) 0.8594
17. If thep-value is 0.1075, the value of∑ Xiis closest to
(a) -1.645 (b) 1.645 (c) 1.96 (d) 2.76 √ (e) 5.52
5 Part 2 Show your work in each question.
18. (9 marks)
A production line operates with a mean filling weight of 16 ounces per container.
Overfilling or underfilling presents a seri ous problem and when detected requires
the operator to shut down th e production line to readjust the filling mechanism .
From past data, a population standard deviation σ = 0.8 ounces is assum ed. A
quality control inspector selects a sample of 36 items every hour and at that time
makes the decision of whether to shut dow n the line for readjustm ent. The level
of significance isα = 0.05 .
Suppose a sample result shows an average filling weight of 16.32 ounces.
State the null and alternative hypotheses, write down the decision rule, calcu late
the test statistic and draw a conclusion.