Download Practice worksheet 1 In a given year, the average annual salary of a

Practice worksheet
1
In a given year, the average annual salary of a famous football player was $2,000,000
with a standard deviation of $245,000. If a simple random sample of 50 players was
taken, what is the probability that the sample mean will exceed $2,100,000?
Practice worksheet
2
2. Bill Clinton, the former President of the United States, believes that the proportion
of voters who will vote for Hillary Clinton, the former Secretary of State, in the year
2016 presidential elections is 0.51. A sample of 500 voters is selected at random. What
is the probability that the number of voters in the sample who will vote for Hillary
Clinton in the year 2016 is between 260 and 280?
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3. 25% of all children affected by an infectious disease recover from it without any
intervention. A big pharmaceutical company develops a new drug for that infectious
disease and ten children who have the disease were randomly selected and received the
medication. Nine of these children recovered from the disease shortly thereafter.
Assume that the medication was ineffective and absolutely worthless. What is the
probability that at least nine of ten children receiving the new drug will recover? Based
on this probability, explain what your finding would mean.
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4
4. Assume that you just uploaded a video you made on Youtube and further assume that
you have observed that the number of hits you get for your video occur at a rate of 2 in
one hour.
a. What is the probability that your video will get no hits in a one hour period after you
uploaded it?
b. What is the probability that it will be viewed once in a one hour period after you
uploaded your video?
c. What is the probability that it will be viewed at least once in a one hour period after
you uploaded your video?
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5. The length of time patients must wait to see a doctor at an emergency room in a
Dubai hospital has a uniform distribution between 30 minutes and 2 hours.
a. What is the probability that a patient would have to wait no more than one hour?
b. What is the probability that a patient would have to wait exactly one hour?
c. What is the probability that a patient would have to wait between one and three hours?
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6. Researchers studying the effects of a new diet found that the weight loss over a oneyear period by those on the diet was normally distributed with a mean of 10 kg and a
standard deviation of 5 kg.
a. What proportion of the dieters lost more than 13 kg?
b. What proportion of the dieters gained weight?
c. What is the probability that the mean weight loss of a random sample of 36 dieters is
greater than 11 kg?
d. What is the minimum amount of weight lost by those highest 3% of weight losers on
the diet?
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7. Times spent studying by students in the week before final exams follow a normal
distribution with standard deviation 9 hours. A random sample of 5 students was taken
in order to estimate the mean study time for the population of all students. What is the
probability that the sample mean exceeds the population mean by more than 2.1 hours?
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8. A medical researcher wants to investigate the amount of time it takes for patients'
headache pain to be relieved after taking a new prescription painkiller. She plans to use
statistical methods to estimate the mean of the population of relief times. She believes
that the population is normally distributed with a standard deviation of 25 minutes. How
large a sample should she take to estimate the mean time to within 2 minute with 99%
confidence?
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9. The assembly line that produces an electronic component of a missile system has
historically resulted in a 3% defective rate. A random sample of 500 components is
drawn. What is the probability that the defective rate is greater than 5%? Suppose that
in the random sample the defective rate is 5%. Explain what the result you found
suggests about the defective rate on the assembly line?
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10. A business student claims that, on average, business students have to take more
classes than students in other colleges. She claims that business students take more than
4 classes per semester. To examine this claim, a statistics professor asks a random
sample of 10 business students to report the number of classes that they take. The results
are exhibited below. Can the professor conclude, at the 5% level of significance, that
the claim is true? Assume that the number of classes is normally distributed with a
standard deviation of 1.5.
3 6 4 5 5 6 4 3 4 5
a. State the null and alternative hypotheses.
b. Test the student’s claim by calculating the p-value. Show all intermediate steps.
c. How do you know whether the student’s claim is supported or not?
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11. A federal agency responsible for enforcing laws governing weights and measures
routinely inspects packages to determine whether the weight of the contents is at least
as great as that advertised on the package. A random sample of 18 containers whose
packaging states that the contents weigh 8 ounces was drawn. The contents were
weighed and the results follow.
a. Can we conclude at the 5% significance level that on average the containers are
mislabeled?
b. How do you know? Explain.
7.8
7.91
7.93
7.99
7.94
7.75
7.97
7.95
7.79
8.06
7.82
7.89
7.92
7.87
7.92
7.98
8.05
7.91
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12. Several students and faculty at AUS make international calls during the academic
year. A survey of 12 students and faculty revealed the following information about the
number of times an international call is made during the spring semester. Assume that
the number of calls made is normally distributed with a standard deviation of 12 calls.
3 41 17 1 33 37 18 15 17 12 29 51
a. Estimate with 90% confidence the average number of calls during the spring semester.
b. How large a sample should we take so that the mean number of calls is within 2 calls
with 95% confidence?
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13. A fast-food company is considering building a restaurant at a certain location. Based
on financial analyses, a site is acceptable only if the number of pedestrians passing the
location averages more than 100 per hour. The number of pedestrians observed for each
of 40 different one-hour periods. The number of pedestrians was then recorded to be
105.7. Assume that the population standard deviation is known to be 16.
a. State the null and alternative hypotheses.
b. Can we conclude at the 5% significance level that the site is acceptable? How do you
know?
c. Please state which type of error (Type I or Type II) may be committed here? Consider
your conclusion in part (b) of this question and explain your answer. Also, what are the
implications of committing such error?
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14. A fitness trainer wants to estimate the average weight loss of people who are in his
new workout class. In a preliminary study, he guesses that the standard deviation of the
population of weight losses is about 10 pounds.
a. How large of a sample should he take to estimate the mean weight loss to within two
pounds, with 99% confidence?
b. How large of a sample should he take to estimate the mean weight loss to within two
pounds, with 95% confidence?
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15. A statistics practitioner is in the process of testing to determine whether there is
enough evidence to infer that the population mean is different from 180. She calculated
the mean and standard deviation of a sample of 200 observations as 𝑋̅ = 175 and s = 22.
a. Calculate the value of the test statistic (you must decide whether the z or t is
appropriate) to determine whether there is enough evidence at the 5% significance level.
b. Repeat this with s = 45.
c. Discuss what happens to the test statistic when the standard deviation increases.
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16. A chain coffee shop owner buys a new espresso machine and wants to determine
whether this new machine is generating any profits. To test the profitability of the
machine, the owner collects data on the profit made (in thousands of dollars) from 16
randomly selected stores and finds that the sample mean and sample standard deviation
are $500 and $200, respectively. If the profit level is normally distributed
a. State the null and alternative hypothesis.
b. Test the hypotheses using a standardized test statistic at 5% significance level.
c. Explain the results you find.
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17. A politician claims that the average UAE resident is more than 20 pounds
overweight. To test his claim, a random sample of 20 UAE residents was weighed, and
the difference between their actual and ideal weights was calculated. The data are listed
below. Do these data allow us to infer at the 5% significance level that the politician's
claim is true?
16 23 18 41 22 18 23 19 22 15
18 35 16 15 17 19 23 15 16 26
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18. Has the building and allocating Terminal 3 to Emirates Airlines resulted in better
on-time performance? Before terminal 3 was built and allocated to Emirates Airlines,
Emirates airline claimed that 92% of its flights were on time. A random sample of 165
flights completed this year reveals that 153 were on time. Can we conclude at the 5%
significance level that the Emirates airline's on-time performance has improved after
allocating terminal 3 only to Emirates airline?
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19. A recent survey in Dubai revealed that 60% of the vehicles traveling on highways,
where speed limits are posted at 120 km per hour, were exceeding the limit. Suppose
you randomly record the speeds of ten vehicles traveling on 311 where the speed limit
is 120 km per hour. Let X denote the number of vehicles that were exceeding the limit.
a. What is the distribution of X?
b. Find P(X = 10).
c. Find the probability that the number of vehicles exceeding the speed limit is between
4 and 9(exclusive)
d. Find the probability that two of the recorded vehicles were exceeding the speed limit
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20. A security department receives an average of 10 telephone calls each afternoon
between 2 and 4 P.M. The calls occur randomly and independently of one another.
a. Find the probability that the department will receive 13 calls between 2 and 4 P.M.
on a particular afternoon.
b. Find the probability that the department will receive seven calls between 2 and 3 P.M.
on a particular afternoon.
c. Find the probability that the department will receive at least five calls between 2 and
4 P.M. on a particular afternoon.
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21. A survey among teenagers in the US found that 30% of teenage consumers receive
their spending money from part-time jobs. If five teenagers are selected at random, find
the probability that at least three of them will have part-time jobs.
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22. Statistics show that the survival rate for patients suffering from a certain type of
cancer is 25%. What is the probability that of 6 randomly selected patients, 4 will
recover?
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23. The recent average starting salary for new business college graduates is $47,500.
Assume salaries are normally distributed with a standard deviation of $4,500.
a. What is the probability of a new graduate receiving a salary between $45,000 and
$50,000?
b. What is the probability of a new graduate getting a starting salary in excess of
$55,000?
c. What percent of starting salaries are no more than $42,250?
d. What is the cutoff for the bottom 5% of the salaries?
e. What is the cutoff for the top 3% of the salaries?
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24. The lifetime of an iPhone battery that is advertised to last for 5,000 days is normally
distributed with a mean of 5,100 days and a standard deviation of 200 days.
a. What is the probability that a battery lasts longer than the advertised figure?
b. If we wanted to be sure that 98% of all iPhone batteries last longer than the advertised
figure, what figure should be advertised?
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25. A new online gaming website has an average random hit rate of 2.9 unique visitors
every 4 minutes. What is the probability of getting exactly 50 unique visitors every
hour?
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26. Several students and faculty at AUS play football. A survey of 12 students revealed
the following information. Estimate with 95% confidence the mean number of football
matches played. Assume that the number of matches is normally distributed with a
standard deviation of 12.
3 41 17 1 33 37 18 15 17 12 29 51
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27. The average person loses an average of 15 days per year to colds and flu. The natural
remedy echinacea reputedly boosts the immune system. One manufacturer of echinacea
pills claims that consumers of its product will reduce the number of days lost to colds
and flu by one-third. To test the claim, a random sample of 50 people was drawn. Half
took echinacea, and the other half took placebos. If we assume that the standard
deviation of the number of days lost to colds and flu with and without echinacea is 3
days, find the probability that the mean number of days lost for echinacea users is less
than that for nonusers.
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28. A sample of 50 retirees is drawn at random from a normal population whose mean
age and standard deviation are 75 and 6 years, respectively.
a. Describe the shape of the sampling distribution of the sample mean in this case.
b. Find the mean and standard error of the sampling distribution of the sample mean.
c. What is the probability that the mean age exceeds 73 years?
d. What is the probability that the mean age is at most 73 years?
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29. A survey of 100 businesses revealed that the mean after-tax profit was $80,000.
Assume the population standard deviation is $15,000:
a. Determine the 95% probability interval estimate of the mean after-tax profit.
b. Explain why you can use the probability interval formula here, even though the
population is not necessarily normal.
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30. A fitness trainer wants to estimate the average weight loss of people who are in his
new workout class. In a preliminary study, he guesses that the standard deviation of the
population of weight losses is about 10 pounds.
a. How large of a sample should he take to estimate the mean weight loss to within two
pounds, with 99% confidence?
b. How large of a sample should he take to estimate the mean weight loss to within two
pounds, with 95% confidence?
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31
31. A fast-food company is considering building a restaurant at a certain location. Based
on financial analyses, a site is acceptable only if the number of pedestrians passing the
location averages more than 100 per hour. The number of pedestrians observed for each
of 40 different one-hour periods. The number of pedestrians was then recorded to be
105.7. Assume that the population standard deviation is known to be 16.
a. State the null and alternative hypotheses.
b. Can we conclude at the 5% significance level that the site is acceptable? How do you
know?
c. Define Type I and Type II errors. Then, describe the consequences of Type I and
Type II errors.
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32. Suppose that 15% of all invoices are for amounts greater than $1,000. A random
sample of 60 invoices is taken. What is the mean and standard error of the sample
proportion of invoices with amounts in excess of $1,000? What is the probability that
the proportion of invoices in the sample is greater than 18%?
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33. An advertisement claims that four out of five doctors recommend a particular
product. A consumer group wants to test that claim, and takes a random sample of 30
doctors. The consumer group finds that of this group of doctors, only 0.75 would
recommend the product. What is the probability that the proportion of doctors in this
sample who recommend the product is 0.75 or less? Do you have reason to doubt the
manufacturer’s claim? Explain.
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34. Examine the data and the graphic below to answer the following questions.
a. Is the price of oil interval, ordinal, or nominal data?
b. Which variable is independent and which one is dependent?
c. What is the slope of the line?
d. What is the y-intercept value?
e. Which number shown in the graphic below tells you if there is a positive or negative
relationship between these two variables? What is that number? Is the relationship
positive or negative?
f. What is the correlation between the price of oil and the price of gasoline?
g. What percentage of variation in the price of gasoline is explained by the price of oil?
How do you know?
h. If the price of oil increases by $1 per barrel, how much does the price of gasoline
increase?
i. If the price of oil is $110, what is the corresponding market price of gasoline?
j. Is the price of oil a good variable to use to predict the price of gasoline (assuming you
are in a country without price controls on gasoline)? Why or why not?
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35. Answer the following questions based on the regression output where house size is
the dependent variable.
Regression Statistics
Multiple R
0.865
R Square
0.748
Adjusted R Square 0.726
Standard Error
5.195
Observations
50
a. What percentage of the variability in house size is explained by this model?
b. Which of the independent variables in the model are significant at the 1% level?
c. What is the predicted house size for an individual earning an annual income of
$40,000, having a family size of 4, and having 13 years of education?(Annual income
is in thousand $US)
d. Suppose the builder wants to test whether the coefficient on income is significantly
different from 0. What is the value of the relevant t-statistic?
e. At the 0.01 level of significance, what conclusion should the builder draw regarding
the inclusion of income in the regression model?
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36. A random sample of 15 hourly fees for car washers (including tips) was drawn from
a normal population. The sample mean and sample standard deviation were = $14.9 and
s = $6.75. Can we infer at the 5% significance level that the mean fee for car washers
(including tips) is greater than 12?
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37. A company claims that 10% of the users of a certain drug experience depression as
a side effect. In clinical studies of this drug, 81 of the 900 subjects experienced
depression. We want to test their claim and find out whether the actual percentage is
not 10%.
a. State the appropriate null and hypotheses.
b. Is there enough evidence at the 5% significance level to infer that the this claim is
correct?
c. Compute the p-value of the test.
d. Construct a 95% confidence interval estimate of the population proportion of the
users of this drug who experience depression.
e. Explain how to use this confidence interval to test the hypotheses
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38. The admissions officer of a university is trying to develop a formal system to decide
which students to admit to the university. She believes that determinants of success
include the standard variables— high school grades and SAT scores. To investigate the
issue, she randomly sampled 100 fourth year students and recorded the following
variables:
GPA for the first 3 years at the university (range: 0 to 12)
GPA from high school (range: 0 to 12)
SAT score (range: 400 to 1600)
The regression output is summarized below.
What is the dependent variable?
b. What are the independent variables?
c. What percentage of variation in the dependent variable is explained by the
independent variables?
d. What is the coefficient of determination? Interpret its value.
e. What is the regression equation?
f. Test to determine whether each of the independent variables is linearly related to the
dependent variable in this model
g. If HS GPA increases by 1 point, how is university GPA affected?
h. If a student has a 3.7 has GPA and a 27 SAT, what is the expected university GPA
using this model?
Practice worksheet
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39. A researcher believes that energy drink consumption may increase heart rate.
Suppose it is known that heart rate (in beats per minute) is normally distributed with an
average of 70 bpm for adults. A random sample of 25 adults was selected and it was
found that their average heartbeat was 73 bpm after energy drink consumption, with a
standard deviation of 7 bpm.
a. Formulate the null and alternative hypotheses.
b. Test the hypotheses in the previous question at the 10% significance level to
determine if we can infer that energy drink consumption increases heart rate.
c. If you were to estimate the p-value of the test statistic in part (b), would it be more
than or less than 10%? Do not compute the p-value of the test statistic: only state your
reason based on your result in part (b).
Practice worksheet
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40. Alaa and Sara are friends. Let X denote the number of cats that Alaa may have in
the next two years, and let Y denote the number of cats Sara may have during the same
period. The marginal probability distributions of X and Y are shown below.
x
0
1
2
y
0
1
2
P(x)
0.5
0.3
0.2
P(y)
0.4
0.5
0.1
a. Compute the mean and variance of X.
b. Compute the mean and variance of Y.
c. Assume that X and Y are independent and find their bivariate distribution.
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41
41. Alaa and Sara are friends. Let X denote the number of cats that Alaa may have in
the next two years, and let Y denote the number of cats Sara may have during the same
period. The marginal probability distributions of X and Y are shown below.
x
0
1
2
y
0
1
2
P(x)
0.5
0.3
0.2
P(y)
0.4
0.5
0.1
a. Compute the mean and variance of X.
b. Compute the mean and variance of Y.
c. Assume that X and Y are independent and find their bivariate distribution.
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42
42. The joint probability distribution of variables X and Y is shown in the table below.
Ahmed and Hassan are car salespeople. Let X denote the number of cars that Ahmed
will sell in a month, and let Y denote the number of cars Hassan will sell in a month.
X
Y
1
2
3
1
0.3
0.18
0.12
2
0.15
0.09
0.06
3
0.05
0.03
0.02
a. Determine the marginal probability distribution of X.
b. Determine the marginal probability distribution of Y.
c. Calculate E(X) and E(Y).
d. Calculate V(X) and V(Y).
e. Develop the probability distribution of X + Y.
f. Calculate E(X + Y) directly by using the probability distribution of X + Y.
g. Calculate V(X + Y) directly by using the probability distribution of X + Y.
h. Verify that E(X + Y) = E(X) + E(Y).
i. Verify that V(X + Y) = V(X) + V(Y)? Why did you get this result?
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43
43. A standard certification test was given at three locations. 1,000 candidates took the
test at location A, 600 candidates at location B, and 400 candidates at location C. The
percentages of candidates from locations A, B, and C who passed the test were 70%,
68%, and 77%, respectively. One candidate is selected at random from among those
who took the test.
a. What is the probability that the selected candidate passed the test?
b. If the selected candidate passed the test, what is the probability that the candidate
took the test at location B?
c. What is the probability that the selected candidate took the test at location C and
failed?
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44
44. The final grades in a statistics course are normally distributed with a mean of 75
and a standard deviation of 10. The professor must convert all marks to letter grades.
She decides that she wants 10% A's, 35% B's, 40% C's, 10% D's, and 5% F's. Determine
the cutoffs for each letter grade.
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45
45. Assume that a there is a random system of police patrol in Dubai and further assume
that a police officer may visit a given location Y=0,1,2,3,… times per half-hour period,
with each location being visited an average of once per time period. Also assume that
Y is Poisson distributed.
a. What is the probability that the police officer will miss a given location during a half
hour period?
b. What is the probability that it will be visited once?
c. Twice?
d. At least once?
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46
46. Elizabeth has decided to form a portfolio by putting 30% of her money into stock 1
and 70% into stock 2. She assumes that the expected returns will be 10% and 18%,
respectively, and that the standard deviations will be 15% and 24%, respectively.
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47
47. Let X be a normally distributed random variable with a mean of 12 and a standard
deviation of 1.5. What proportions of the values of X are:
a. less than 14
b. more than 8
c. between 10 and 13
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48
48. The number of midterm exams taken per semester by AUS students is normally
distributed with a mean of 11 and a standard deviation of 3.
a. What proportion of students takes more than 13 exams per semester?
b. What is the probability that in a random sample of 20 students more than 240
midterms are taken? (Hint: What is the mean number of midterms taken by the sample
of 20 students?)
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49
49. The time it takes for a QBA professor to grade his midterm exam is normally
distributed with a mean of 26 minutes and a standard deviation of 10 minutes. There
are a total of 200 students in the professor's classes. What is the probability that he
needs more than 80 hours to mark all the midterm exams? (The 200 midterm exams of
the students in this year's classes can be considered a random sample of the many
thousands of midterm exams the professor has marked and will mark.)
Practice worksheet
50
50. A car dealer in Dubai claims in an advertisement that 3% of all its cars require a
service in the first year. An SBM student who takes the QBA class wants to check the
claim by surveying 400 people who recently purchased one of the dealer's cars. What
is the probability that more than 4.7% require a service within the first year? What
would you say about the advertisement's honesty if in a random sample of 400 people
4.7% report at least one service call?
Practice worksheet
51
51. Among the most exciting aspects of a student's life are the important calls they make
where such critical issues as the make and model of the car he will purchase and whether
she will get a new designer handbag are discussed.
A sample of 20 students was asked how many hours per month are devoted to these
calls. The responses are listed here. Assuming that the variable is normally distributed
with a standard deviation of 8 hours, estimate the mean number of hours spent on those
calls by all students. Use a confidence level of 90%.
14
17
3
6
17
3
8
4
20
15
7
9
0
5
11
15
18
13
8
4
Practice worksheet
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52. A statistics professor wants to compare today's students with those 5 years ago. All
his current students’ grades are stored on his computer so that he can easily find the
population mean. However, the grades 5 years ago are in a random file in his computer.
He does not want to retrieve all the grades and will be satisfied with a 95% confidence
interval estimate of the mean grade 5 years ago. If he assumes that the population
standard deviation is 12, how large a sample should he take to estimate the mean to
within 2 grades?
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53. A random sample of 12 second-year AUS students enrolled in a business statistics
course was drawn. At the course's completion, each student was asked how many hours
he or she spent doing the homework in statistics. The data are listed here. It is known
that the population standard deviation is σ =16. The instructor has recommended that
students devote 6 hours per week for the duration of the 12-week semester, for a total
of 72 hours. Test to determine whether there is evidence that the average student spent
less than the recommended amount of time.
Compute p-value of the test.
62
80
52
60
72
76
58
80
76
60
70
76
Practice worksheet
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54. A smartphone manufacturer advertises that, on average, its cell phone batteries will
last more than 50000 hours. To test the claim, an AUS student took a random sample
of 100 smartphones and measured the amount of time until each battery lasted. If we
assume that the lifetime of this type of battery has a standard deviation of 4000 hours,
can we conclude at the 5% significance level that the claim is true? (Mean for the
sample is 50650)
Practice worksheet
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55. A restaurant manager claims that, on average, a chef is required to prepare more
than five cakes per week. To examine the claim, the manager asks a random sample of
10 chefs to report the number of cakes they prepare weekly. The results are exhibited
here (below). Can the manager conclude at the 5% significance level that the claim is
true, assuming that the number of cakes is normally distributed with a standard
deviation of 1.5?
2
7
4
8
9
5
11
3
7
4
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56
56. A travel agent believes that the average number of days traveled per year among
international travelers is less than 10 days. From past experience, he knows that the
population standard deviation is 3 days. In testing to determine whether his belief is
true at α = .01, he uses the sample of 100 travelers. What is the probability of a Type II
error, given that the true population average is 9 days?
Practice worksheet
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57. A company that produces Internet services wanted to determine the number of
devices connected to the Internet American homes contain. The company surveyed 240
randomly selected homes and determined the number of devices connected to the
Internet (the mean for the sample is 4.66 and the standard deviation is 2.37). If there
are 100 million households, estimate with 99% confidence the total number of devices
connected to the Internet in the United States.
Practice worksheet
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58. Amazon sells ground and whole bean coffee over the Internet. Buyers enter their
orders, pay by credit card, and receive delivery by postal services. Amazon analyzed
and determined that the average order would have to exceed $85 if the selling coffee
on Internet were to be profitable. To determine whether selling coffee on Internet would
be profitable in one large city, Amazon offered the service and recorded the size of the
order for a random sample of buyers. (Sample mean is 89.27, sample standard deviation
is 17.30 and sample size is 85). Can we infer from these data that an e-grocery will be
profitable in this city at alpha 5% significance level?
Practice worksheet
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59. Use data file “data”. A Dubai Honda dealer manager has been inspecting weekly
advertisement expenditures. They have been giving advertisements to the national
newspaper for the past 6 months and the number of ads per week has varied from one
to seven. The dealer’s sales staff has been tracking the number of customers who enter
the store each week. The number of ads and the number of customers per week for the
past 26 weeks are recorded.
a. What is the dependent variable? What is the independent variable?
b. What is the coefficient of determination? Interpret its value.
c. What is the regression equation?
Practice worksheet
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Practice worksheet
61
61. In 2014, the average annual salary of a business school graduate was $50,000 with
a standard deviation of $6125. If a simple random sample of 50 business graduates was
taken, what is the probability that the sample mean will exceed $52,500?
Practice worksheet
62
62. Bill Clinton, the former President of the United States, believes that the proportion
of voters who will vote for Hillary Clinton, the former Secretary of State, in the year
2016 presidential elections is 0.52. A sample of 500 voters is selected at random. What
is the probability that the number of voters in the sample who will vote for Hillary
Clinton in the year 2016 is between 265 and 280?
Practice worksheet
63
63. Suppose that the starting salaries of female CFO’s have a positively skewed
distribution with mean of $156,000 and a standard deviation of $32,000. The starting
salaries of male CFO’s are positively skewed with a mean of $150,000 and a standard
deviation of $30,000. A random sample of 50 male CFO’s and a random sample of 40
female CFO’s are selected.
a. What is the sampling distribution of the sample mean difference 𝑋̅1−𝑋̅2? Explain.
b. What is the probability that the sample mean salary of female CFO’s will not exceed
that of the male CFO’s?
Practice worksheet
64
64. 30% of all children affected by an infectious disease recover from it without any
intervention. A big pharmaceutical company develops a new drug for that infectious
disease and ten children who have the disease were randomly selected and received the
medication. Eight of these children recovered from the disease shortly thereafter.
Assume that the medication was ineffective and absolutely worthless. What is the
probability that at least nine of ten children receiving the new drug will recover? Based
on this probability, explain what your finding would mean.
Practice worksheet
65
65. Assume that you just uploaded a video you made on Youtube and further assume
that you have observed that the number of hits you get for your video occur at a rate of
5 in one hour.
a. What is the probability that your video will get no hits in a one hour period after you
uploaded it?
b. What is the probability that it will be viewed once in a one hour period after you
uploaded your video?
c. What is the probability that it will be viewed at least once in a one hour period after
you uploaded your video?
Practice worksheet
66
66. The length of time patients must wait to see a doctor at an emergency room in a
Dubai hospital has a uniform distribution between 20 minutes and 1.5 hours.
If X is the waiting time
a. What is the probability that a patient would have to wait no more than 50 minutes?
b. What is the probability that a patient would have to wait exactly one hour?
c. What is the probability that a patient would have to wait between one and two hours?
Practice worksheet
67
67. The final course grades of a mathematics class at AUS are normally distributed with
a mean of 82 and a standard deviation of 4. (Total possible points = 100.)
a. What is the probability that a randomly selected student will have a score of 86 or
higher?
b. What is the probability that a randomly selected student will have a score between
80 and 90?
c. If four students are selected randomly, what is the probability that they will have an
average score of 85 or higher?
Practice worksheet
68. Find the following probabilities
68
Practice worksheet
69
69. “Genius” is an organization whose members possess IQs that are in the top 2% of
the population. It is known that IQs are normally distributed with a mean of 110 and a
standard deviation of 20. Find the minimum IQ needed to be a “Genius” member.
Practice worksheet
70
70. The final grades in a statistics course are normally distributed with a mean of 75
and a standard deviation of 10. The professor must convert all marks to letter grades.
She decides that she wants 10% A's, 35% B's, 40% C's, 10% D's, and 5% F's. Determine
the cutoffs for each letter grade.
Practice worksheet
71
71. The assembly line that produces an electronic component of a missile system has
historically resulted in a 3% defective rate. A random sample of 500 components is
drawn. What is the probability that the defective rate is greater than 5%? Suppose that
in the random sample the defective rate is 5%. Explain what the result you found
suggests about the defective rate on the assembly line?
Practice worksheet
72
72. A business student claims that, on average, business students have to take more
classes than students in other colleges. She claims that business students take more than
4 classes per semester. To examine this claim, a statistics professor asks a random
sample of 10 business students to report the number of classes that they take. The results
are exhibited below. Can the professor conclude, at the 5% level of significance, that
the claim is true? Assume that the number of classes is normally distributed with a
standard deviation of 1.5.
3
6
4
5
5
6
4
3
4
5
a. State the null and alternative hypotheses.
b. Test the student’s claim by calculating the p-value. Show all intermediate steps.
c. How do you know whether the student’s claim is supported or not?
Practice worksheet
73
73. A federal agency responsible for enforcing laws governing weights and measures
routinely inspects packages to determine whether the weight of the contents is at least
as great as that advertised on the package. A random sample of 18 containers whose
packaging states that the contents weigh 8 ounces was drawn. The contents were
weighed and the results follow.
7.8
7.91
7.93
7.99
7.94
7.75
7.97
7.95
7.79
8.06
7.82
7.89
7.92
7.87
7.92
7.98
8.05
7.91
a. Can we conclude at the 5% significance level that on average the containers are
mislabeled?
b. How do you know? Explain.
Practice worksheet
74
74. Several students and faculty at AUS make international calls during the academic
year. A survey of 12 students and faculty revealed the following information about the
number of times an international call is made during the spring semester. Assume that
the number of calls made is normally distributed with a standard deviation of 12 calls.
3
41
17
1
33
37
18
15
17
12
29
51
a. Estimate with 90% confidence the average number of calls during the spring semester.
b. How large a sample should we take so that the mean number of calls is within 2 calls
with 95% confidence?
Practice worksheet
75
75. A fast-food company is considering building a restaurant at a certain location. Based
on financial analyses, a site is acceptable only if the number of pedestrians passing the
location averages more than 100 per hour. The number of pedestrians observed for each
of 40 different one-hour periods. The number of pedestrians was then recorded to be
105.7. Assume that the population standard deviation is known to be 16.
a. State the null and alternative hypotheses.
b. Can we conclude at the 5% significance level that the site is acceptable? How do you
know?
c. Please state which type of error (Type I or Type II) may be committed here? Consider
your conclusion in part (b) of this question and explain your answer. Also, what are the
implications of committing such error?
Practice worksheet
76
76. A fitness trainer wants to estimate the average weight loss of people who are in his
new workout class. In a preliminary study, he guesses that the standard deviation of the
population of weight losses is about 10 pounds.
a. How large of a sample should he take to estimate the mean weight loss to within two
pounds, with 99% confidence?
b. How large of a sample should he take to estimate the mean weight loss to within two
pounds, with 95% confidence?
Practice worksheet
77
77. A statistics practitioner is in the process of testing to determine whether there is
enough evidence to infer that the population mean is different from 180. She calculated
the mean and standard deviation of a sample of 200 observations as 𝑋̅ = 175 and s = 22.
a. Calculate the value of the test statistic (you must decide whether the z or t is
appropriate) to determine whether there is enough evidence at the 5% significance level.
b. Repeat this with s = 45.
c. Discuss what happens to the test statistic when the standard deviation increases.
Practice worksheet
78
78. A chain coffee shop owner buys a new espresso machine and wants to determine
whether this new machine is generating any profits. To test the profitability of the
machine, the owner collects data on the profit made (in thousands of dollars) from 16
randomly selected stores and finds that the sample mean and sample standard deviation
are $500 and $200, respectively. If the profit level is normally distributed
a. State the null and alternative hypothesis.
b. Test the hypotheses using a standardized test statistic at 5% significance level
c. Explain the results you find
Practice worksheet
79
79. A politician claims that the average UAE resident is more than 20 pounds
overweight. To test his claim, a random sample of 20 UAE residents was weighed, and
the difference between their actual and ideal weights was calculated. The data are listed
below. Do these data allow us to infer at the 5% significance level that the politician's
claim is true?
16
23
18
41
22
18
23
19
22
15
18
35
16
15
17
19
23
15
16
26
a. State the null and alternative hypothesis.
b. Test the hypotheses using a standardized test statistic at 5% significance level
c. Explain the results you find
Practice worksheet
80
80. Has the building and allocating Terminal 3 to Emirates Airlines resulted in better
on-time performance? Before terminal 3 was built and allocated to Emirates Airlines,
Emirates airline claimed that 92% of its flights were on time. A random sample of 165
flights completed this year reveals that 153 were on time. Can we conclude at the 5%
significance level that the Emirates airline's on-time performance has improved after
allocating terminal 3 only to Emirates airline?
Practice worksheet
81
81. In the United States, voters who are neither Democrat nor Republican are called
Independents. It is believed that 10% of all voters are Independents. A survey asked 25
people to identify themselves as Democrat, Republican, or Independent.
a. What is the probability that none of the people are Independent?
b. What is the probability that fewer than five people are Independent?
Practice worksheet
82
82. After analyzing several months of sales data, the owner of an appliance store
produced the following joint probability distribution of the number of refrigerators and
stoves sold daily.
Refrigerators
Stoves
0
1
2
0
0.10
0.16
0.14
1
0.07
0.15
0.11
2
0.05
0.14
0.08
a. Find the marginal probability distribution of the number of refrigerators sold daily.
b. Find the marginal probability distribution of the number of stoves sold daily.
c. Compute the mean and variance of the number of stoves sold daily.
Practice worksheet
83
83. Three messenger services deliver to Sharjah. Service A has 50% of all the scheduled
deliveries, service B has 30%, and service C has the remaining 20%. Their on-time rates
are 65%, 50%, and 45% respectively. Define event O as a service delivers a package
on time.
a. Calculate P (A and O).
b. You are given that P (Ac and O) = 0.24. If a package was delivered on time, what is
the probability that it was service A?
Practice worksheet
84
84. A math tutor claims that the proportion of students who gets an A after just taking
two sessions with him is 55%. What is the probability that in a random sample of 300
students who hires him as a tutor, less than 49% get A? If 49% of the sample actually
got an A, what does this suggest about the tutor’s claim?
Practice worksheet
85
85. The length of a volleyball game is normally distributed with a mean of 110 minutes
and a standard deviation of 12.5 minutes. What is the probability that a randomly
selected game is longer than 120 minutes?
Practice worksheet
86
86. A uniformly distributed random variable has minimum and maximum values of 25
and 75, respectively.
a. Draw the density function.
b. Determine P (39 < X < 54)
c. Draw the density function including the calculation of the probability in part (b).
Practice worksheet
87
87. The following are the ages of a random sample of 8 employees in a bank:
50 66 21 33 31 57 45 47
It is known that the ages are normally distributed with a standard deviation of 10.
Determine the 90% confidence interval estimate of the population mean. Interpret the
interval estimate.
Practice worksheet
88
88. Many hotels that serve the surfers make their forecasts of incomes on the
assumption that the average surfer surfs four times per year. To examine the validity of
this assumption, a random sample of 96 surfers is drawn and each is asked to report the
number of times he or she surfed the previous year (the sample mean is 4.70). If we
assume that the standard deviation is 3, can we infer at the 10% significance level that
the assumption is wrong?
Practice worksheet
89
89. Before the recent changes in the educational system, one private high school
claimed that 91% of its graduates were placed in prestigious universities. A random
sample of 495 students graduated this year reveals that 459 were placed in prestigious
universities. Can we conclude at the 5% significance level that this high school's
performance has improved?
Practice worksheet
90
90. Use the graph below to answer the following questions.
a. When you look at the scatter plot, you can see three numbers. Describe what each
number represents (how do you name them) in this graph and how they are interpreted
(what do they tell us).
b. Write down the deterministic model for the above relationship between SUV sales
and price of gas.
c. Based on the graph above, if the price is ten, what would be SUV sales?
Practice worksheet
91
91. After analyzing several months of sales data, the owner of an appliance store
produced the following joint probability distribution of the number of refrigerators and
stoves sold daily.
Refrigerators
Stoves
0
1
2
0
0.06
0.14
0.12
1
0.09
0.17
0.13
2
0.05
0.18
0.04
a. Find the marginal probability distribution of the number of refrigerators sold daily.
b. Find the marginal probability distribution of the number of stoves sold daily
c. Compute the mean and variance of the number of refrigerators sold daily.
d. Compute the mean and variance of the number of stoves sold daily.
e. Compute the covariance and the coefficient of correlation.
Practice worksheet
92
92. Suppose X is a binomial random variable with n=25 and p=0.7. Use Table 1 to find
the following
a.P(X=18)
b.P(X=15)
c.P(X ≤20)
d.P(X≥16)
Practice worksheet
93
93. A student majoring in accounting is trying to decide on the number of firms to which
he should apply. Given his work experience and grades, he can expect to receive a job
offer from 70% of the firms to which he applies. The student decides to apply to only
four firms. What is the probability that he receives no job offers?
Practice worksheet
94
94. In the United States, voters who are neither Democrat nor Republican are called
Independents. It is believed 20% of all voters are independents. A survey asked 25
people to identify themselves as Democrat, Republican, or Independent.
a. What is the probability that none of the people are Independent?
b. What is the probability that fewer than five people are Independent?
c. What is the probability that more than two people are Independent?
Practice worksheet
95
95. Define/discuss/interpret the following terms: population, sample, parameter,
statistic, descriptive statistics(graphical and numeric), inferential statistics, random
variable, data, interval data, nominal data, ordinal data, time series data, cross sectional
data, variance, standard deviation, covariance, correlation coefficient, coefficient of
determination.
Practice worksheet
96
96. Create a sample of three numbers whose mean is 10 and standard deviation is 0
Practice worksheet
97
97. Michelin, a tire manufacturer, wants to advertise a mileage interval that excludes
no more than 11% of the mileage on tires they sell. Suppose that the average tire
mileage is 25000 and the standard deviation is 4000.
a. What interval would you suggest for the advertisement? (Use Chebyshev's theorem).
b. What would be the advertisement mileage interval that excludes no more than 5% of
the mileage on tires they sell if it is known that the mileage follows the normal
distribution?
Practice worksheet
98
98. The U.S. mint produces dimes with an average diameter of 0.5 and a standard
deviation of 0.01.
a. Using Chebyshev's theorem, find a lower bound for the number of coins in a lot of
400 coins having diameter between 0.48 and 0.52.
b. What would be the number of coins with a diameter between 0.48 and 0.52 if the
diameter is normally distributed?
Practice worksheet
99
99. The mean grade point average (gpa) for AUS students is 2.5 with a standard
deviation of 0.5
a. If the histogram for gpa’s is approximately mounded, what percent of the gpa’s would
you expect between 1.5 and 3.5?
b. If the histogram for gpa’s is approximately mounded, what percent of the gpa’s
would you expect greater than 3.5?
c. If the histogram for gpa’s is NOT mounded, what percent of the gpa’s would you
expect between 1.5 and 3.5?
Practice worksheet
100
100. An investment of $1,000 you made 4 years ago was worth $1,200 after the first
year, $1,200 after the second year, $1,500 after the third year, and $2,000 today.
a. Compute the annual rates of return.
b. Compute the average annual and median of the rates of return.
c. Compute the overall (compound) return.
d. Discuss which of the above is the best measure of the performance of the investment.
Practice worksheet
101
101.In 2007 (the latest year reported) 134,543,000 tax returns were filed in the United
States. The Internal Revenue Service (IRS) examined 0.5% of them to determine if they
were correctly done. To determine how well the auditors are performing, a random
sample of these returns was drawn and the additional tax was reported. The mean return
from this sample is $11,000 with a standard deviation of $4,000.
a. What is the population of interest?
b. What is the sample size?
c. What is the main idea to be tested?
Practice worksheet
102
102. Are the marks one receives in a course related to the amount of time spent studying
the subject? To analyze this mysterious possibility, a student took a random sample of
5 students who had enrolled in an accounting class last semester. He asked each to
report his or her mark in the course and the total number of hours spent studying
accounting. These data are listed here.
a. Calculate the covariance.
b. Calculate the coefficient of correlation.
c. Calculate the coefficient of determination
Practice worksheet
103
103. The number of males and females enrolled at AUS are listed per major in the table
below. Use this table to answer questions
a. If a student is chosen at random, what is the probability that the student is a female?
b. If a student is chosen at random, what is the probability that the student is a male
majoring in engineering?
c. If one student is chosen to represent the student body, what are the odds in favor of
selecting a female?
d. If one student is chosen from economics major, which is more likely, selecting a
male or selecting a female?
Practice worksheet
104
104. Suppose that five good fuses and two defective ones have been mixed up. To find
the defective fuses, we test them one-by-one, at random and without replacement. What
is the probability that we are lucky and find both of the defective fuses in the first two
tests?
Practice worksheet
105
105. Let A and B be independent events with P(A) = and P(A ∪ B) = 2P(B) − P(A).
Find
a) P(B)
b) P(A|B)
c) P(B’|A)
Practice worksheet
106
106. Your favorite football team is in the final playoffs of the UEFA Champions League.
You have assigned a probability of 60% that it will win the championship. Past records
indicate that when teams win the championship, they win the first game of the series
70% of the time. When they lose the series, they win the first game 25% of the time.
The first game is over; your team has lost. What is the probability that it will win the
championship?
Practice worksheet
107
107. Assume that you have a new gun’s test results. A sample of 4 shots with the
following deviations from the target is reported: 25cm, 8cm, 16cm and 19cm.
a. Calculate the sample mean, variance, and standard deviation.
b. Calculate the coefficient of variation. The old gun’s coefficient of variation was 0.48.
Which gun produces more consistent shuts?
Practice worksheet
108
108. Consider the following data:
X 11 17 18
Y 3 5 11
a. Calculate the covariance for the sample.
b. Calculate the coefficient of correlation given the fact that standard deviations for X
and Y are 3.78 and 4.16 respectively.
c. What does the coefficient of correlation tell you about the relationship between X
and Y?
Practice worksheet
109
109. A survey of a magazine's subscribers indicates that 40% own a house, 70% own a
car, and 65% of the homeowners also own a car. What proportion of subscribers:
a. own both a car and a house?
b. own a car or a house, or both?
c. own neither a car nor a house?