Download Math 116 - Final Exam - Spring 2007

MA 116 - Final Exam – Summer 2011
Your exam will be graded as: #correct / 174.
Name: _____________________________________
Time: 8:00 ~ 9:45.
Use your time wisely.
Q1. Each year, thousands of college seniors take the Graduate Record Examination (GRE). The scores are
transformed so they have a mean of 500 and a standard deviation of 100. Furthermore, the scores are known to
be normally distributed.
[28 points]
a) What is the probability that a student scores higher than 525? Graph, shade, and label.
Show procedure
Now check with calculator feature. Write calculator input and answer.
b) Complete the following:
The percentage of students that scores higher than 525 is _________________________
c) A GRE report includes the score and the percentile. The percentile is the percent of students who scored at or
below the score of 525. What is the percentile rank of the student who scored 525 points in the GRE? (Round to
the nearest percentile)
The percentile corresponding to a GRE score of 525 is ______________________
d) Let’s continue with the GRE story which is written here again: Each year, thousands of college seniors take
the Graduate Record Examination (GRE). The scores are transformed so they have a mean of 500 and a
standard deviation of 100. Furthermore, the scores are known to be normally distributed. A certain university
accepts students with GRE scores on the top 10%. Find the acceptance cut-off GRE score for this
university.
Show procedure
Now check with calculator feature. Write calculator input and answer.
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Q2. A study shows that social exclusion causes “real pain”. That is, activity in an area of the brain that
responds to physical pain goes up as distress from social exclusion goes up. A scatterplot and the linear
regression results are displayed in the table
[16 points]
The Linear regression feature of the
calculator produced the following output:
y = a + bx
a = - 0.126
b = 0.0608
r^2 = 0.771
Use the table displays to answer the following:
(a) Identify the explanatory variable and the response variable
(b) Use the regression equation to predict brain activity for social distress score 2.0. Show work.
(c) What percent of the variation in brain activity among these subjects is explained by the straight-line
relationship with social distress score?
(d) Use the calculator output shown on the table to find the correlation r between social distress score and
brain activity. How do you know whether the sign of r is + or −?
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Q3. Assume the lengths of life of people who have a certain kind of skin cancer are normally distributed with a
mean mu = 18 months and a standard deviation sigma = 5 months. If we select 45 patients with this type of skin
cancer at random, what is the probability of observing a mean length of life of more than 20.5 months?
[15 points]
a) Before answering the question, describe the shape (explain why), and give the mean, and standard deviation
of the distribution of sample means for samples of size 45
b) Now answer: what is the probability of observing a mean length of life of more than 20.5 months in a sample
of size 45? Graph, shade and label
Show procedure:
Now check with calculator feature. Write calculator input and answer.
Q4. The mean length of life of people who have a certain kind of skin cancer is 18 months. A new medication
has been developed and the developer claims that for people using the new drug, the average length of life is
more than 18 months. In order to test this claim, from the group of patients who received the medication 45
patients were selected at random and it was observed that they lived an average of 20.5 months. Assume the
standard deviation of the population is 5 months. Test the claim of the developer at the 1% significance level.
Do all of the following:
[15 points]
a) Write hypotheses, graph, shade, label and possible locations of the point estimate.
b) Use a calculator feature to test the claim and write the results
c) Interpret results within context.
d) Compare the test statistic and the p-value obtained in part (b) with the values obtained in problem Q3.
Hopefully, there is a match!!!
MATCH or NO MATCH? There is nothing else to do here!!!!
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Q5. State University requires an admission test of all its students. A simple random sample of 121 students from
the University was selected to estimate the average test score of all University students. The average (mean) for
the sample was found to be 23.4 with a sample standard deviation of 3.65.
[20 points]
(a)
Construct a 95% confidence interval for the average (mean) test score of all University students.
(b)
What does the 95% confidence interval that you calculated in (a) tell you about the average (mean) test
score of all University students?
(c)
Which of the following would result in a narrower confidence interval for the average (mean) test score of
all University students? (select ALL that apply)
(i)
Maintain the confidence level at 95%, but increase the sample size to 151.
(ii)
Maintain the confidence level at 95%, but decrease the sample size to 91.
(iii) Keep the sample size at 121, but increase the confidence level to 99%.
(iv) Keep the sample size at 121, but decrease the confidence level to 90%.
(d)
When we say “95% confidence” in a 95% confidence interval, we believe that (select only one)
(i)
95% of the intervals constructed using this process based on samples (of the same size) from this
population will include the population mean.
(ii)
There is a 95% probability that a particular confidence interval will include the sample mean.
(iii) 95% of the possible population means will be included in the interval.
(iv) 95% of the possible sample means will be included in the interval.
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Q6. A study was conducted to compare the abilities of men and women to perform the strenuous tasks of a fire
fighter. Female and male fire fighters who were of equal weight were matched together, thus producing data for
a matched pairs experiment. Given in the table below are the pulling forces (in newtons) that each fire fighter
was able to exert in pulling the starter cord of a P-250 fire pump.
Do the data provide sufficient evidence to indicate that male firefighters have a greater mean pulling force
than female firefighters of equal weight? Conduct the appropriate hypothesis test at the 5% significance level.
[20 points]
Pair
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2
3
4
Male
84.51
80.06
102.3
88.96
Female
40.03
75.62
53.38
62.27
Be sure to show all your work.
(a) Write the relevant statistics using calculator.
(b) Write the hypotheses, sketch graph, shade, label.
(c) Use a calculator feature to test the claim and write the results
(d) Interpret results within context.
(e) Will the conclusion be different at the 1% significance level? Explain.
.
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Q7. Up-County Associates is hired to estimate the proportion of county households that own three or more
automobiles. They conducted their survey using a simple random sample of 250 randomly selected households
and found that 23% of those households owned three or more automobiles. Answer the following questions,
giving reasons for your answers.
[16 points]
(a)
Is the proportion, 23%, a parameter or a statistic? Give reason for your answer
(b)
If Up-County Associates were to continually choose simple random samples of 250 randomly selected
households, each time recording the proportion of household in the sample owning three or more
automobiles, what would be the “shape” of the resulting distribution of all these sample proportions?
Give reason for your answer
Approximately normal
(c)
Right skewed
Left skewed
Suppose now that survey company Lower-County Polling also conducted a survey to determine the
proportion of county households that own three or more automobiles, but they surveyed only 100
randomly selected households. If Lower-County Polling were also to sample over and over again (using
simple random samples of size 100) would the standard deviation of its distribution of sample proportions
(i)
(ii)
(iii)
(iv)
in general, be lower than that of Up-County Associates?
in general, be higher than that of Up-County Associates?
in general, be about the same as that of Up-County Associates?
in general, not have any predictable relationship to that of Up-County Polling?
Give reason for your answer
(d)
Which company’s estimate is more likely to be closer to the actual proportion of households owning
three or more automobiles? Give reason for your answer
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Q8. In 1965, about 44% of the U.S. adult population had never smoked cigarettes. A national health survey of
1205 U.S. adults (presumably randomly selected) during 2006 revealed that 615 had never smoked cigarettes.
Suppose you wished to test whether there has been a change since 1965 in the proportion of U.S. adults
that have never smoked cigarettes.
[20 points]
(a)
Which of the following are the appropriate hypotheses?
H o : p  0.51
H o : p  0.51
H o : p  0.44
H o : p  0.44
H o : p  0.51
H a : p  0.51
H a : p  0.44
H a : p  0.44
H a : p  0.44
H a : p  0.51
(b) Sketch the graph, shade and label the sample statistic in the graph.
(c) Run the test in the calculator and write the results.
(d) Write the conclusion to the problem within context.
(e) Which of the following represents the p-value? Circle one.
p = P(p-hat < 0.44)
p = P(p-hat > 0.44)
p =2* P(p-hat > 0.44)
p = P(p-hat < 0.51)
p = P(p-hat > 0.51)
p =2* P(p-hat > 0.51)
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Q9. To study if there is any relation between the gender of a student and the major that they choose in the
science division, a random sample of 141 students had the following response.
[24 points]
Mathematics
Biology
Computer Science
Chemistry
Total
Female
12
27
15
9
64
Male
29
16
19
14
78
Total
41
43
34
23
141
a) Write the two hypotheses
Ho:
Ha:
b) The degrees of freedom for the Chi-square test are _____________. Show how you find it
c) Under the null hypotheses listed in part (a) the expected number of female students majoring in Computer
Sciences is ___________________. Show how you find it.
d) Calculate the contribution to the chi-square statistic of the cell implied in part (c). Show how you find it.
e) Run the chi-square test in the calculator and give the results..
f) Write the conclusion within context at the 5% significance level.
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Q10. WRITING QUESTION:
A long held medical standard is that normal body temperature is 98.6 degrees Fahrenheit. However,
recent medical research has challenged this commonly held belief. Researchers now suspect that the
normal body temperature is less than 98.6 degrees. The displays below show the results of a sample of
50 healthy individuals along with the results of various statistical inference procedures that were
performed on this data.
[25 points]
Sample Statistics
sample size = 50
mean = 98.23
standard deviation = 0.7484
Test of Significance
Dot Plot
Sample of Normal Body Temperature
H 0 : population mean = 98.6
H a : population mean < 98.6
test statistic: t = -3.496
p-value = 0.00051
Confidence Interval
confidence level = 95%
estimate: 98.0173 to 98.4427
96.0
97.0
98.0
99.0
100.0
101.0
BodyTemperature
Your task is to write a paragraph or two which answers the following question and provides an
explanation for your answer:
Do the results above provide statistical evidence that the normal body temperature is less than 98.6. Justify
your answer. (Note: If the evidence suggests a normal body temperature less than 98.6, you must provide a
possible reasonable alternative value or set of values and provide a justification for it.)
Direct your writing to a general audience—one that is unfamiliar with statistical terminology—so use clear,
everyday language. Begin your paragraph with a statement of the problem the researchers were investigating,
including statements of their hypotheses in written form. In addition, here are a few things you also ought to
include in your discussion.
 Does the confidence interval contain the value 98.6? What does that suggest?
 What is the meaning of the confidence level, 95%?
 What is the meaning of the p-value, 0.00051?
 How do these things help you answer the above question?
One last thing: remember your task—to write about this study in clear, simple, but correct paragraphs.
WRITE ON THE PAPER PROVIDED,
WRITE A FIRST DRAFT, MAKE CORRECTIONS IF NECESSARY. REWRITE AGAIN IF YOU THINK IT NECESSARY
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