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Unformatted text preview: UNIVERSITY OF TORONTO
QUANTITATIVE METHODS IN ECONOMICS II
ECMB12H3F (LEC02)
MIDTERM TEST – SOLUTIONS
DURATION: 90 MINUTES
OCTOBER 20, 2010
IMPORTANT:
(i) This test should be answered in BALLPOINT PEN and students who attempt the test in
pencil would surrender their rights to request for reassessment. (ii) Ensure HANDWRITING is LEGIBLE and show STEPBYSTEP CALCULATIONS.
Answer all questions only in the DESIGNATED PAGES. ROUGH WORK should be
done in the LAST TWO PAGES and contents in those pages will not be graded. (iii) AIDS ALLOWED – a HANDWRITTEN SINGLESIDED LETTER SIZE CRIB
SHEET (photocopies or printed copies not allowed) and a NONPROGRAMMABLE
CALCULATOR (other electronic devices with calculator functions not allowed).
(iv) DO NOT SEPARATE any page from this test. Student Name Student ID# Question 1 2 3 4 5 Total Marks 17 31 20 12 20 100 Score Page 1 of 8 1. As the vice president in operations of a major fast food chain, the current average time to
serve a customer using the drivethru service is 3 minutes and you are interested in cutting
down the average time by installing a new automated selfserve system, which costs
millions of dollars. The new system has been tested in a particular location where it served
a sample of 64 customers and the average time is 2.75 minutes with a standard deviation of
1 minute. (i) Follow the sixstep critical value approach to hypothesis testing at 5% level
of significance with the sample data to determine whether to invest in this new automated
selfserve system or not. (ii) Can you determine the range of the value of the test
statistic?
(i) 12 marks and (ii) 5 marks (i) Step 1 – the company will only invest in the new automated self serve system if it can
significantly reduce the average time to serve customers using the drivethru service, which
implies a one tail test.
and
Step 2 – the level of significance is 5% and the sample size . Step 3 – only the test is appropriate where there is no information about the population
variance.
Step 4 – for
and
for the one tail test is
Step 5 – with sample statistics with degrees of freedom , the critical value .
, , and . The test statistic Step 6 – since
and it is less than the critical value of
, the test statistic
falls in the rejection region and hence the null hypothesis is rejected. There is enough
evidence to conclude that the average time to serve a drivethru customer is significantly
less than 3 minutes and hence it is worthwhile to invest in the new automated selfserve
system.
(ii) With the degrees of freedom at 63, the probability of a value at
or smaller is
2.5% and the probability of a value at
or smaller is 1%. Since the test statistic of
is between
and
, which implies the range of the value of the test
statistic of
must be between 1% and 2.5%. Page 2 of 8 2. (i) Explain when a researcher may commit a Type I Error and how the probability of Type I
Error is being determined. (ii) Suppose the population variance is known and the
hypothesized population mean
is false when the true population mean is
in which
. Use two distribution graphs (with one above the other and each graph has two
value lines – one for units and one for sample unit of measurement for – as shown in
lecture discussion) to help you explain the relationship between Type I and Type II Errors
with
and
at
. (iii) Use two separate distribution graphs
(with the same set up as in part (ii)) to help you explain why the one tail test
is
more powerful than the two tail test
. Any intuitive explanation?
(i) 3 marks; (ii) graphs 4 marks and explanation 8 marks; and
(iii) graphs 4 marks and explanation 12 marks (i) A researcher commits a Type I Error if he rejects the null hypothesis when it is true. The
probability of Type I Error is , which is called the level of significance of the test, and is
set by the researcher in advance. (ii) Given the hypothesized population mean
is false in a lower tail test, any sample with a
test statistic
higher than
when
(defined as the Type I error) would
not reject the null hypothesis. When the true population mean
is lower than , the
probability of that occurring is defined as the Type II error (indicated in the diagram as ).
The lower the probability of Type I Error, the smaller the critical value, the more the red
line moves to the left and the higher the probability of Type II Error under the true
distribution in the lower graph. In other words, smaller Type I Error results in larger Type
II Error. Page 3 of 8 (iii) With
and
, any sample with a test statistic
higher than
would not reject the null hypothesis even though it is false when the true population mean
is lower than , and the probability of the Type II error for this lower tail test is the
area under the true distribution in the lower graph
(as pointed out in
part (ii)).
With and
, any sample with a test statistic
between
and
would not reject the null hypothesis even though it is false when the true population
mean
is lower than , and the probability of the Type II error for this two tail test is the
area under the true distribution in the lower graph
.
Since
,
. The power of the test for the one tail test is
and
for
the two tail test. Therefore,
for
and the one tail test is more
powerful than the two tail test.
A two tail test only aims to test if there is any difference from the hypothesized population
parameter in either direction (larger or smaller) when the researcher does not have enough
information to determine the direction of the test. A one tail test has specific direction of
the difference because the researcher has more information in which more information
often leads to better decision. Hence, the one tail test is more powerful than the two test
tail. Page 4 of 8 3. Suppose you are working on a PhD dissertation regarding customer satisfaction of new car
buyers between domestic brands and imported brands. You randomly selected a sample of
61 new car buyers of domestic brands (with sample average of 2.8 minor repairs and
sample standard deviation 1.7 during the first three months) and 41 new car buyers of
imported brands (with sample average of 2.4 minor repairs and sample standard deviation
1.2 during the first three months). Explain your decision on the selection of appropriate
tests (all at
level of significance) in each step to examine if the average number of
minor repairs of new domestic cars is different from that of new imported cars.
20 marks Since there is no information on whether the population variances of the average minor repairs of
these two groups of new cars are equal or not, an test on the difference in population variances
of these groups
and
must be performed before a decision can be
made on whether to use the pooled or separatevariance test with unknown population
variances.
Given the sample variance of the minor repairs on new domestic cars
is greater
than that of new imported cars
, the sample of new domestic cars is set as sample 1
and the sample of new imported cars is set as sample 2
.
The critical value of this two tail test at
is 1.80. The test statistic level of significance with and Since
is greater than the critical value, reject . There is enough evidence to conclude
that the population variance of minor repairs on new domestic cars is different from that of new
imported cars. Hence, the separatevariance test for the difference between two means
and
is the appropriate test.
The degrees of freedom must be calculated before we can determine the critical value: The standard error of the separatevariance: Page 5 of 8 The critical value for this two tail test at 5% level of significance with degrees of freedom
is
. The test statistic Since
is not in the rejection region, do not reject . There is not enough evidence that the
average number of minor repairs of new domestic cars is different from that of new imported
cars. Page 6 of 8 4. A survey done in 2006 indicated that 50% the females surveyed were impulsive shoppers
when the economy was booming. The same marketing firm conducted the same survey
earlier this year on a different random sample with 12 female respondents and found that 4
of them were impulsive shoppers. Use the value approach to determine if the current
recession has reduced the proportion of females as impulsive shoppers at 5% level of
significance. (Hint: If you cannot find the exact test statistic in the table, then use the
approximation method discussed in class)
12 marks According to the 2006 survey, 50% of the females were impulsive shoppers
. The first
step is to check if the test is appropriate. That is, if
and
. Using
and
,
and
. Passing this check, the
test is appropriate. It may look like there are two populations, one from 2006 and one from
this year, but there is only one population so that there is no need to know the sample size of the
survey done in 2006 because there is no need to check.
To determine if recession has reduced the proportion of females as impulsive shoppers implies
this will be a one tail test in the lower tail such that
and
From the sample taken earlier this year, and , . Since the
of
is not printed in the table, we must find the closet values from
the table to approximate the value of the
. Given
, if
, then
is definitely larger
than
and hence do not reject the null hypothesis. There is not enough evidence that the
recent recession has reduced the proportion of females as impulsive shoppers. Page 7 of 8 5. A randomized block experiment has groups
and
blocks
. (i) Complete the
table below. (ii) What are the hypotheses in testing whether the randomized block design
is advantageous to use and your conclusion at 5% level of significance? (iii) Compute the
estimated relative efficiency and explain what it means.
(i) 10 marks; (ii) 5 marks; and (iii) 5 marks
Source Degrees of
Freedom Sum of
Squares Mean Square
(Variance) Among
Groups Among
Blocks Error Total (ii) To test whether the randomized block design is advantageous to use,
and Not all are equal (where With
and
, the critical value
. Since
, reject
. There is enough evidence to conclude the randomized block design is advantageous to
use.
(iii) The estimated relative efficiency is defined as This value for relative efficiency means that it would take approximately 1.4 times as many
observations in a oneway ANOVA design as compared to the randomized block design in
order to have the same precision in comparing the groups. Page 8 of 8 ...
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This note was uploaded on 05/31/2011 for the course MGT C06 taught by Professor A.stawinoga during the Fall '10 term at University of Toronto Toronto.
 Fall '10
 A.Stawinoga

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