MAT
137 Final Exam Example Problems:
1)
XYZ University requires all incoming students to take a math
placement test in order to determine the student’s mathematical skill level.
The score on the test can range from a 0 to a maximum score of 155 points. Historically
the average score on the math placement test is 90 points with a standard
deviation of 15 points. Typically XYZ University will place those students
finishing in the upper 2.5% on the test in an honors math section and those finishing
in the lower 2.5% may need to do additional course work prior to taking the
standard math classes. Anne, Bill and Steve each took the placement test and
scored 126 points, 104 points and 73 points, respectively.
a)
Based on the information given above do you believe Anne
will be placed in a standard math class, an honors math class or will be
required to do additional course work prior to the standard class? Show any
calculations and explain your answer.
b)
Based on the information given above do you believe Bill
will be placed in a standard math class, an honors math class or will be
required to do additional course work prior to the standard class? Show any
calculations and explain your answer.
c)
Based on the information given above do you believe Steve
will be placed in a standard math class, an honors math class or will be
required to do additional course work prior to the standard class? Show any
calculations and explain your answer.
2)
A food products manufacturer conducted a survey to observe
the amount of money spent by a family of four on groceries. One hundred
randomly selected families were picked from their population of interest. The
results were as follows:
Sample mean = $4,000
Sample standard deviation = $250
a) Calculate a 95% confidence interval for the above data and explain
the meaning of the interval as it applies to this particular data set. ($4000
+/- $49)
b) Calculate a 95% prediction interval (Empirical Rule) for the above
data and explain the meaning of the interval as it applies to this particular
data set. What assumption about the distribution of the data must be made for
this interval to be valid? ($4000 +/- $500)
c) What annual amount of money spent on groceries would you consider to
be very low for a family of four? Very high? Explain. Hint: use the results
from one of the above intervals.
3)
A sample of n = 16
Honda Civic automobiles with the same engines and transmissions yielded the
following sample statistics when fuel mileage tests were conducted on a test
track:
_
x
= 28.4 mpg
s
= 2.0 mpg
a) Calculate a 95% confidence interval for the average mileage for all
Honda Civics with this particular engine and transmission. (28.4 mpg +/- 1.0655
mpg)
b)
If we want to reduce the margin of error for the above
confidence interval to be about 0.5 mpg, approximately how many Honda Civics
would we need to test? Show your calculations. (approximately 64 cars)
4) A study of 300 recent new car purchasers yielded the following
results when asked the question, “Do you think the quality of Japanese cars is
better than American cars?”
Yes = 156
No
= 144
a)
Find a point estimate for the population percentage of all
new car purchasers that thinkthe quality of Japanese cars is better
than American cars. (52%)
b)
Calculate a 99% confidence interval for the population percentage
of all new car purchasers that thinkthe quality of Japanese cars is better
than American cars. (52% +/- 7.43%)
c)
Based on the interval in part b, can we clearly say that
most new car purchasers today still think the quality of Japanese cars is
better than American cars? Explain.
5)
The Bank of USA Credit Card Company has informed us that an
additional service charge will be deducted from our store’s card receipts if
our average credit card sale in a month is not over $20 per sale. To see if
this new rule will apply to our store, we conducted a random sample of size n =
49 charge sales from last month. The mean charge sale of the sample was equal
to $25 and the standard deviation was equal to $14.
a) Test the null hypothesis that µ = $20 vs. the alternative µ > $20
at a significance level ofa = 0.05. Note: this is an upper-tailed
test. (test statistic z = 2.5 compared to a critical value of 1.645, n >30
you can use Large/One Tail 0.05)
b) Based on the above test, should we be concerned about the new rule.
Explain.
6)
The administrators of school
district XYZ were interested in expanding the school facilities within their
school district. In order for them to obtain the funding for the expansion,
they were required to submit a request for a tax referendum to be voted on by
the residents of the school district. Acceptance of the referendum would result
in a tax increase for homeowners in the area. Prior to requesting the
referendum, the administrators decided to conduct a survey in order to
determine if the voters of the district would pass the school expansion
referendum. The survey asked the single following question:
“Would
you support the expansion and upgrade of the school facilities in your
district?”
The survey was conducted at a local
shopping mall in the district. A sample of 100 individuals was asked the above
question. The results were 55 yes and 45 no. They then calculated a test
statistic based on the null hypothesis that the population percentage of voters
in favor of the referendum, p, was equal to 50% vs. the alternative that p is
greater than 50%. That is:
p = .50
p >
.50
The test statistic was calculated as:
Test
Statistic = (.55 – .50 )/ sqrt( (.50) (.50) /100 ) = 1.00
Because the test statistic is much
less than 2.00, they were unable to reject the null hypothesis, and therefore,
concluded the voters would not pass the referendum. Based on this result, they
decided not to ask for the school tax referendum.
a) Comment on the question the administrators asked vs. what they were
trying to accomplish with the survey.
Could you suggest a better question?
b) Comment on their sampling. Is the sample representative of their
population of interest?
c) Was their conclusion correct? Should they drop the referendum based
on this survey? What should they do next? Explain.
