prepared by dr. nafez m. barakat islamic university of...
TRANSCRIPT
1
Solution to selected problems in midterm exam in principal of statistics
PREPARED BY Dr. Nafez M. Barakat
Islamic university of Gaza
MIDTERM 2009-2010
2
3
P(1.99 < x < 2.0) = p(
36
06.0
0.299.1 < z <
36
06.0
0.20.2 ) = p(-1 < z < 0)
= p(z< 0) – p (z< -1) = 0.05 - 0.1587 = 0.3413
P( 1xX )= 0.99 , p(z >z1) = 0.99, p(z < z1) = 1-0.99 = 0.01, z1= -1.23
9767.136
06.023.10.21
nzx
p(x > 90) = 0.0668, p(z > z1 ) = 0.0668, p( z < z1) = 1.0 -0.0668=
0.9332
Z1 = 1.5 ,
xz ,
75905.1
, 10
P(x> 95) = p( z> (95-75)/10) = p(z>2.0)
= 1- p(z< 2.0) = 1 – 0.9772=0.0228
.p(-z1 < z < z1) = 0.8664, 2 p( 0< z < z1) = 0.4332
P(z< z1) – 0.5 = 0.4332, z1 = 1.5
4
zx
Lower interval xL = 75 -1.5*10 = 60
Upper interval xu = 75+ 1.5*10 = 90
05745.075
)45.01(45.0)1(
nP
P( 0.3 < p < 0.5) = P( (0.3-0.45)/0.5745 < Z < ((0.5-0.45)/0.5745)
= P(-2.61 < Z < 0.87) = 0.8033
P(Z< Z1) = 0.80, Z1 = 0.84
498.075
)45.01(45.0*84.045.0
)1(
nZp
5
5371)50(
)1000(*)96.1(*2
22
2
22
e
zn
6
5.04972.0)67.067.0()67.067.0(
)67.067.0(
ZPX
P
XP
MIDTERM 18/4/2011
SECTION I: MULTIPLE-CHOICE
For each question in this section, circle the correct answer. Each problem is worth 1
point.
7
1. The process of using sample statistics to draw conclusions about true population parameters
is called
a) statistical inference.
b) the scientific method.
c) sampling.
d) descriptive statistics.
2. A summary measure that is computed to describe a characteristic from only a sample of the
population is called
a) a parameter.
b) a census.
c) a statistic.
d) the scientific method.
3. In a right-skewed distribution
a) the median equals the arithmetic mean.
b) the median is less than the arithmetic mean.
c) the median is larger than the arithmetic mean.
d) none of the above.
1. When extreme values are present in a set of data, which of the following descriptive
summary measures are most appropriate:
a) CV and range.
b) arithmetic mean and standard deviation.
c) interquartile range and median.
d) variance and interquartile range.
2. In its standardized form, the normal distribution
a) has a mean of 0 and a standard deviation of 1.
b) has a mean of 1 and a variance of 0.
c) has an area equal to 0.5.
d) cannot be used to approximate discrete probability distributions.
1. If a particular batch of data is approximately normally distributed, we would find that
approximately
a) 2 of every 3 observations would fall between 1 standard deviation around the
mean.
b) 4 of every 5 observations would fall between 1.28 standard deviations around the
mean.
c) 19 of every 20 observations would fall between 2 standard deviations around the
mean.
d) All the above.
2. The Central Limit Theorem is important in statistics because
a) for a large n, it says the population is approximately normal.
b) for any population, it says the sampling distribution of the sample mean is
approximately normal, regardless of the sample size.
c) for a large n, it says the sampling distribution of the sample mean is
approximately normal, regardless of the shape of the population.
d) for any sized sample, it says the sampling distribution of the sample mean is
approximately normal.
8
1. Which of the following statements about the sampling distribution of the sample mean is
incorrect?
a) The sampling distribution of the sample mean is approximately normal whenever
the sample size is sufficiently large (n 30).
b) The sampling distribution of the sample mean is generated by repeatedly taking
samples of size n and computing the sample means.
c) The mean of the sampling distribution of the sample mean is equal to .
d) The standard deviation of the sampling distribution of the sample mean is
equal to .
2. The width of a confidence interval estimate for a proportion will be
a) narrower for 99% confidence than for 95% confidence.
b) wider for a sample size of 100 than for a sample size of 50.
c) narrower for 90% confidence than for 95% confidence.
d) narrower when the sample proportion is 0.50 than when the sample proportion is
0.20.
3. A 99% confidence interval estimate can be interpreted to mean that
a) if all possible samples are taken and confidence interval estimates are developed,
99% of them would include the true population mean somewhere within their
interval.
b) we have 99% confidence that we have selected a sample whose interval does
include the population mean.
c) Both of the above.
d) None of the above.
9
SECTION II: TRUE OR FALSE
For each question in this section, indicate whether the sentence is TRUE or False. Each
problem is worth 1 point.
1. ( F ) The possible responses to the question “How many times in the past three months
have you visited a city park?” are values from a discrete variable.
2. ( T ) Other things being equal, as the confidence level for a confidence interval
increases, the width of the interval increases.
3. ( F ) The t distribution is used to develop a confidence interval estimate of the
population proportion when the population standard deviation is unknown.
4. ( F ) The confidence interval obtained will always correctly estimate the population
parameter.
5. ( F ) In estimating the population mean with the population standard deviation unknown,
if the sample size is 12, there will be 6 degrees of freedom.
6. ( T ) As the sample size increases, the effect of an extreme value on the sample mean
becomes smaller.
7. ( T ) A sampling distribution is a distribution for a statistic.
8. ( F ) The probability that a standard normal random variable, Z, is less than 5.0 is
approximately 0.
9. ( F ) The "middle spread," that is the middle 50% of the normal distribution, is equal to
one standard deviation.
10. ( F ) In a set of numerical data, the value for Q2 is always halfway between Q1 and Q3.
11. ( F ) The standard error of the sample mean is affected by the confidence level
12. ( T )A point estimator is a function of the random sample used to make
inferences about the value of an unknown population parameter
SECTION III: FREE RESPONSE QUESTIONS
(i) (2 Points)
The assets in billions of dollars of the five largest bond funds are 19.5, 16.8, 13.7,
12.8, and 10.9. Compute the standard deviation for this population of the five largest bond
funds.
Descriptive Statistics
N Mean Std. Deviation
VAR00001 5 14.7400 3.40925
Valid N (listwise) 5
11
(ii) (4 Points)
You were told that the mean score on a statistics exam is 75 with the scores
normally distributed. In addition, you know the probability of a score between 55 and 60 is
4.41% and that the probability of a score greater than 90 is 6.68%.
a. (2 Points) What is the probability of a score greater than 95?
p(x > 90) = 0.0668, p(z > z1 ) = 0.0668, p( z < z1) = 1.0 -0.0668= 0.9332
Z1 = 1.5 ,
xz ,
75905.1
, 10
P(x> 95) = p( z> (95-75)/10) = p(z>2.0) = 1- p(z< 2.0) = 1 – 0.9772=0.0228
b. (2 Points)
The middle 86.64% of the students will score between which two scores?
.p(-z1 < z < z1) = 0.8664, 2 p( 0< z < z1) = 0.4332
P(z< z1) – 0.5 = 0.4332, z1 = 1.5
zx
Lower interval xL = 75 -1.5*10 = 60 Upper interval xu = 75+ 1.5*10 = 90
(iii) (2 Points)
The head of a computer science department is interested in estimating the
proportion of students entering the department who will choose the new computer engineering
option. Suppose there is no information about the proportion of students who might choose the
option. What size sample should the department head take if she wants to be 95% confident that
the estimate is within 0.10 of the true proportion?
e=0.1 , 96.1,05.0 2/ z
97)1.0(
)5.01(5.0*96.1*96.1)1(*22
2
e
zn
(iv) (2 Points)
A hotel chain wants to estimate the average number of rooms rented daily in each
month. The population of rooms rented daily is assumed to be normally distributed for each month
with a standard deviation of 24 rooms. during February, a sample of 25 days has a sample mean of
37 rooms. Use this information to calculate a 92% confidence interval for the population mean.
08.0,37,25,24 xn
4.83725
24*75.137
nZXIC
5.456.8.2
11
n= 16, s=400 from interval (4739.8, 5260.2) 5000x
15,602.216
4002.260* 2/2/2/ dftt
n
ste
From table T ,02.0
confidence level = (1-0.02)*100%=98%
Questions 5-6 refer to the following information:
5- A 95% confidence interval for the mean reading achievement score for a population of
third grades is (40 , 50). The margin of error of this interval is
a) 95% b) 10 c) 5 d) 2.5
12
e) The answer cannot be determined from the information given
6- The sample mean is
a) 0.95 b) 45 c) 42.5 d) 47.5
e) The answer cannot be determined from the information given
13
7- Using the same set of data, you compute a 95% confidence interval and a 99%
confidence interval. Which of the following statement is correct?
a) The intervals have the same width b) The 99% interval is wider
c) The 95 % interval is wider d) You cannot be determined which interval is
wider unless you know n and s
9 Suppose you take a simple random sample from a population known to be normally
distributed, but the value of is unknown. Your sample size is 10. Which formula
below should be used
n
to find the 90% confidence interval for the mean?
a) 1.645 b) 1.645 c) 1.833 d) 1.83310 10 10 1
0
s S
x x x x
ANSWER C Questions 13-14 refer to the following information:
13- The height of Palestinian men aged 18 to 24 are approximately normally
distribution with mean 170 cm and standard deviation 6 cm. Half of all young men
are shorter than
a) 164 cm b) 170 cm c) 176 cm
d) Can't tell, because the median height is not given.
14- Only about 5% of young men have heights outside the range
cmtocmC 18215862170.%95.02
a) 164 cm to 176 cm b) 158 cm to 182 cm
c) 152 cm to 188 cm d) 146 cm to 194 cm
15- An airplane is only allowed a gross passenger weight of 6,885 kg. If the weights of
passengers traveling by air between two cities have a mean of 80 kg and a standard
deviation of 18 kg, the approximate probability that the combined weight of 81 passengers
will exceed 6,885 kg is:
a) 0.3906 b) 0.9938 c) 0.0062 d) 0.0000
16. Given a normally distributed population with a mean of 80 and a variance of l00, we
know that the distribution of sample means computed from samples of size 25 from that
population will have a mean of _____ and a standard error of _____.
a) 80, l0
b) 80, 2 c) l00, 25
d) 80, l0
Question #6: [20 Points]
(a) [10 Points]
An article reports that (4.0, 5.6) is a 95% confidence interval for the mean
length of stay, in days, of patients in hospital for a particular operation. Suppose the sample
size is 50, find the sample mean and the standard deviation.
14
From the interval
009.2,05.0,8.0,8.4 2/ tex
,2.8150
009.28.0*2/ ss
n
ste
(b) [10 Points] How large a sample size is needed to estimate the mean annual income of CCC
company to be within $2000 with probability 0.99? Suppose there is no prior information
about the standard deviation of annual income of the CCC company, but we guess that
about 68% of their incomes are between $10000 and $40,000 and that this distribution of
incomes is approximately bell shaped
At 2000,15000,58.2,01.0 2/ ez
375374.4225)2000(
)15000(*)58.2(*2
22
2
22
e
zn
15
The Islamic University of Gaza
Faculty of Commerce
Department of Economics and Political Sciences
An Introduction to Statistics Course (ECOE 1302)
Spring Semester 2014 - 8/4/2014 Midterm Exam
Name:___________________________________________ ID:___________
Instructors: □Dr. Nafez Barakat □ Mr. Ibrahim Abed
SECTION I: MULTIPLE-CHOICE (Each problem is worth 1 point)
For each question in this section, circle the correct answer.
4. A summary measure that is computed to describe a characteristic of an entire
population is called
a) a parameter.
b) a census.
c) a statistic.
d) the scientific method.
5. Which descriptive summery measures are considered to be resistant statistics?
a) The arithmetic mean and standard deviation.
b) The interquartile range and range.
c) The mode and variance.
d) The median and interquartile range.
6. Which of the following is a continuous quantitative variable?
a) The color of a student’s eyes
b) The number of employees of an insurance company
c) The amount of milk produced by a cow in one 24-hour period
d) The number of gallons of milk sold at the local grocery store yesterday
7. The possible responses to the question "How would you rate the quality of your
purchase experience with 1 = excellent, 2 = good, 3 = decent, 4 = poor, 5 =
terrible?" are values from a
a) discrete numerical random variable.
b) continuous numerical random variable.
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c) categorical random variable.
d) parameter
8. . In sampling from a large population with = 20, the standard error of the mean is
found to be 2. The size of the sample used is:
a) 100 b) 40
c) 10
d) 20
9. The smaller the spread of scores around the arithmetic mean,
a) the smaller the interquartile range.
b) the smaller the standard deviation.
c) the smaller the coefficient of variation .
d) All the above
10. For sample size 16, the sampling distribution of the sample mean will be approximately
normally distributed
e) regardless of the shape of the population.
f) if the shape of the population is symmetrical.
g) if the sample standard deviation is known.
h) if the sample is normally distributed
13. In left-skewed distributions, which of the following is the correct statement?
a) The distance from Q1 to Q2 is smaller than the distance from Q2 to Q3.
b) The distance from the smallest observation to Q1 is larger than the distance
from Q3 to the largest observation.
c) The distance from the smallest observation to Q2 is smaller than the distance from
Q2 to the largest observation.
d) The distance from Q1 to Q3 is twice the distance from the Q1 to Q2.
12.According to the Chebyshev rule, at least what percentage of the observations in any data set
are contained within a distance of 3 standard deviations around the mean?
a) 67%
b) 75%
c) 88.89%
d) 99.7%
13.For some positive value of Z, the probability that a standard normal variable is between
0 and Z is 0.3340. The value of Z is
a) 0.07
b) 0.37
c) 0.97
d) 1.06
14. A confidence interval was used to estimate the proportion of statistics students that are
female. A random sample of 72 statistics students generated the following 90% confidence
interval: (0.438, 0.642). Using the information above, what total size sample would be
necessary if we wanted to estimate the true proportion to within ±0.08 using 95%
confidence?
a) 105
b) 150
c) 420
d) 597
17
2
2
z
n p 1 pe
=( 1.96/0.08)2* 0.54* (1-0.54) = 150
15.When extreme values are present in a set of data, which of the following descriptive
summary measures are most appropriate?
a) CV and range
b) arithmetic mean and standard deviation
c) interquartile range and median
d) variance and interquartile rangs
Section II: TRUE Or FALSE (Each problem is worth 1 point). For each question in this section, indicate whether the sentence is TRUE or False.
2. ( F ) A statistic is usually used to provide an estimate for a usually
unobserved parameter .
3. ( T ) As a general rule, an observation is considered an extreme value if its absolute
value of Z score is greater than3 .
4. ( F ) The answer to the question “How do you rate the quality of your
business statistics course” is an example of an ordinal scaled variable .
5. ( T ) The coefficient of variation is a measure of relative variation .
6. ( F ) The t distribution is used to develop a confidence interval estimate of the
population proportion when the population standard deviation is unknown .
7. ( T ) The width of a confidence interval equals twice the sampling error
8. ( T ) A point estimate consists of a single sample statistic that is used to estimate the
true population parameter
8. ( F ) The amount of bleach a machine pours into bottles has a mean of 36 oz . with a
standard deviation of 0.15 oz . Suppose we take a random sample of 36 bottles
filled by this machine . The sampling distribution of the sample mean has a
standard error of 0.15.
9. ( F ) A university dean is interested in determining the proportion of students who
receive some sort of financial aid. Rather than examine the records for all
students, the dean randomly selects 200 students and finds that 118 of them are
receiving financial aid. Use a 90% confidence interval to estimate the true
proportion of students who receive financial aid. The answer is 0.59 0.068165
P=118/200 = 0.59,
057.059.0200
)59.01(*59.0*645.159.0
)1(
n
PPZP
10. ( F ) As an aid to the establishment of personnel requirements, the director of a
hospital wishes to estimate the mean number of people who are admitted to the
18
emergency room during a 24-hour period. The director randomly selects 64
different 24-hour periods and determines the number of admissions for each.
For this sample, 8.19X and s2 = 25. The 95% confidence interval for the
population mean is 19.8 4.96
SECTION III: FREE RESPONSE QUESTIONS
Question #3: (5 Points) The owner of a fish market determined that the average weight for a catfish is 3.2 pounds. He also knew that the probability of a randomly selected catfish that would weigh more than 3.8 pounds is 20% and the probability that a randomly selected catfish that would weigh less than 2.8 pounds is 30%.
a) Find the probability that a randomly selected catfish will weigh less than 3.6 pounds ?
20.0)8.3( wp , 80.0)
6.0(,20.0)
2.38.3(
ZPZp
714.0,84.06.0
7123.0)5602.0()714.0
2.36.3()6.3(
ZPZPwP
b) The middle 40% of the catfish will weigh between which two numbers ?
ZZZP ,70.0)( 1 1 = 0.525
19
574.3714.0*525.02.3 ZXU
825.2714.0*525.02.3 ZX L
Question #4: (5 Points) A study at a college in the west coast reveals that, historically, 45% of their students are
minority students . If a random sample of size 75 is selected
e) Find the probability that the sample proportion of minority students lies between 30% and 50% ?
75,45.0 n
8058.0
0197.08078.0)611.2()87.0(
)87.0611.2(
)
75
)45.01(45.0
45.05.0
75
)45.01(45.0
45.03.0()5.03.0(
ZPZP
ZP
ZPPP
f) 95% of the samples proportions of minority students will be greater than
21
what value ? P(Z>Z1) = 0.95, P(Z< Z1) = 0.05 Z1 = -1.645
355.075
)45.01(45.0*645.145.0
)1(
nZp
The Islamic University of Gaza
Faculty of Commerce
Department of Economics and Political Sciences
An Introduction to Statistics Course (ECOE 1302) Spring Semester 2014
Final Examination Date : 31/5/2014
Name:_____________________________________________
ID:_____________ Time: Two hour’s
Instructor's: Mr. Ibrahim Abed and D. Nafez Barakat DON'T WRITE ON THIS TABLE
QUESTION #1 #2 #3 #4 #5 #6 #7 TOTAL
POINTS
Question #1: [15 Points] For each question in this section, circle the correct answer. Each problem is worth 1
point.
21
1. A summary measure that is computed to describe a characteristic from only a sample of
the
population is called
a) a parameter b) a census.
c) a statistic d) the scientific method
2. Which of the following statements about the median is not true?
a) It is more affected by
extreme values than the
arithmetic mean
b) It is a measure of
central tendency c) It is equal to Q2.
d) It is equal to the mode
in bell-shaped "normal"
distributions
3. According to the Chebyshev rule, at least what percentage of the observations in any data set
are contained
within a distance of 3 standard deviations around the mean?
a) 67%
b) 75%
c) 88.89%
d) 99.7%
4. For some positive value of Z, the probability that a standard normal variable is between 0 and Z
is 0.3770. The value of Z is
a) 0.18
b) 0.81
c) 1.47
d) 1.16
5. If we know that the length of time it takes a college student to find a parking spot in the library
parking lot follows a normal distribution with a mean of 3.5 minutes and a standard deviation
of 1 minute, find the point in the distribution in which 75.8% of the college students exceed
when trying to find a parking spot in the library parking lot.
a) 4.2 minutes b) 3.2 minutes
c) 3.4 minutes
d) 2.8 minutes
6. The standard error of the mean
a) is never larger than
the standard deviation
of the population
b) decreases as the
sample size increases c) measures the variability
of the mean from
sample to sample.
d) All of the
above
7. The Dean of Students mailed a survey to a total of 400 students. The sample included 100
students randomly selected from each of the freshman, sophomore, junior, and senior classes on
campus last term. What sampling method was used?
a) Simple random sample b) Systematic sample c) Stratified
sample d) Cluster sample
8. In the construction of confidence intervals, if all other quantities are unchanged, an increase in
the sample size will lead to a interval.
a) narrower
b) wider
c) less significant
d) Biased
9. A university dean is interested in determining the proportion of students who receive
some sort of financial aid. Rather than examine the records for all students , the dean
randomly selects 200 students and finds that 118 of them are receiving financial aid . If
the dean wanted to estimate the proportion of all students receiving financial aid to
within 3% with 99% reliability .how many students would need to be sampled
P=118/200 = 0.59
31782.1646503.0*03.0
)59.01(*59.0*575.2*575.2
e
)(1Zn
2
2
pp
a) 1844 b) 1784 c) 1503 d) 1435
10. Which of the following is most likely a parameter as opposed to a statistic?
22
a) The average score
of the first five
students
completing an
assignment
b) The proportion
of females
registered to
vote in a county
c) The average
height of people
randomly
selected from a
database.
d) The proportion of
trucks stopped
yesterday that were
cited for bad
brakes
11. In aright _ skewed distribution
a) The arithmetic mean equals the median.
b) The arithmetic mean is less than the median.
c) The arithmetic mean is larger than the median.
d) None of the above
12.The width of a confidence interval estimate for a proportion will be
A) narrower when the sample proportion is 0.10 than when the sample
proportion is 0.45.
B) wider for 90% confidence than for 95% confidence.
C) narrowest when the sample proportion is 0.5.
D) narrower for a sample size of 50 than for a sample size of 100.
E) wider when the sample proportion is 0.95 than when the sample proportion is
0.55.
Question #2: [15 Points]
For each question in this section , indicate whether the sentence
is true or false. Each problem is worth 1 point.
1- ( T ) The possible responses to the question “How long have you been living at your
current
residence?” are values from a continuous variable
2- ( F ) The answer to the question “What is your favorite color?” is an example of an ordinal
scaled
variable
3- ( T ) In a sample of size 40, the sample mean is 15. In this case, the sum of all observations
in the
sample is 600iX .
4- ( T ) The coefficient of variation measures variability in a data set relative to the size of the
arithmetic
mean
5- ( F ) The probability that a standard normal random variable, Z, is between 1.00 and 3.00
is 0.2574.
6- ( T ) As the sample size increases, the effect of an extreme value on the sample mean becomes
smaller
7- ( F ) The standard error of the sampling distribution of a sample proportion is 1p p
n
23
8- ( T ) Other things being equal, as the confidence level for a confidence interval increases, the
width of the interval increases
9- ( T ) A point estimate consists of a single sample statistic that is used to estimate the true
population parameter
10- ( F ) For a given level of significance, if the sample size is increased, the probability of
committing a Type I error will increase
11- ( T ) For a given sample size, the probability of committing a Type II error will increase
when the
probability of committing a Type I error is reduced
Bonus
Question #7: [5 Points]
Suppose that the incomes in a population have mean $22000 and standard deviation $4000
. A sample of size 40 is selected
a) What is the probability that the sample mean will be within $2000 of the population
mean ?
µ=22000 , 4000 n=40
9984.00008.09992.0
)162.3()1625.3()162.3162.3(
)162.3(()40/4000
2000()2000(
ZPZPZP
ZPPZPXp
b) Suppose that we want the probability of X being within $2000 of the population mean
to be 0.95 . How large a sample do we need?
α=0.05 Zα/2 = 1.96 e = 2000
An Introduction to Statistics Course (ECOE 1302) Spring Semester 2012
Final Examination Date : 28/5/2012
3-Asummary measure that is computed to describe a characteristic of an entire population
is called
162000*2000
4000*4000*1.96*1.96
e
σZn
2
22
α/2
24
a) a parameter b) a census c) a statistics d) The scientific method
4- According to the chebyshev rule , at least 93.75% of all observations in any data set are
contained within a distance of how many standard deviations around the mean
%75.93%100)4
11(%100
11
2
k
a) 1 b) 2 c) 3 d) 4
5- The mean age of five people in a room is 30 years . One of the people whose age is 50
years leaves the room . the mean age of the remaining four people in the room is
a) 40 b) 30 c) 25 d) Not able to be determined
from the information given
6- For some value of Z , the probability that a standard variable is below Z is 0.791.The
value of z is
a) 0.81 b) -0.31 c) 0.31 d) 1.96
7- Sampling distributions describe the distribution of
a) parameters b) Statistics c) Both a and b d) Neither a nor b
9- The symbol for the power of a statistical test is
a) b) 1- c) d) 1-
10- A type II error is committed when
a) we reject 0H that is true. b) we reject 0H that is false.
c) we don’t reject 0H that is true. d) we don’t reject 0H that is false
11- A university dean is interested in determining the proportion of students who receive
some sort of financial aid. Rather than examine the records for all students , the dean
randomly selects 200 students and finds that 118 of them are receiving financial aid . If the
dean wanted to estimate the proportion of all students receiving financial aid to within 3%
with 99% reliability .how many students would need to be sampled
sampled?
P=118/200 = 0.59
31782.1646503.0*03.0
)59.01(*59.0*575.2*575.2
e
)(1Zn
2
2
pp
a) 1844 b) 1784 c) 1503 d) 1435
12) In a survey of 3200 T.V. viewers, 20% said they watch network news programs.
Find standard error for the sample proportion.
A) 0.0071 B) 0.0865 C) 0.0721 D) 0.0142 E) 0.0649
25
For each question in this section , indicate whether the sentence
is true or false. Each problem is worth 1 point.
1- ( F ) A statistic is usually unobservable while a parameter is usually observable
2- ( F ) As the sample size increases , the standard error of the mean increases. n
SS
X
3- ( T ) Other things being equal , as the confidence level for a confidence interval increases
, the width of the interval increases
4- ( T ) The t distribution is used to develop a confidence interval estimate of the population
mean when the population standard deviation is unknown
7- ( T ) The mean of the sampling distribution of a sample mean is the population mean
8- ( F ) The quality ("terrible" , "poor" , "fair" , " acceptable" , "very good" , and " excellent"
) of a day care center is an example of a numerical variable
9- ( T ) The coefficient of variation measures variability in a data set relative to the size of
the arithmetic mean
10- ( T ) A box plot is a graphical representation of a 5 _ number summary
11- ( F ) The probability that a standard normal random variable Z is below 1.96 is 0.4750
13- ( T ) A 95% confidence interval for will be wider than a 96% confidence interval for
15- ( T ) The a mount of water consumed by a person per week is an example of a
continuous variable
An Introduction to Statistics Course (ECOE 1302)
Spring Semester 2015 - 8/4/2015 Midterm Exam
Name:___________________________________________ ID:___________
Instructors: □Dr. Nafez Barakat □ Mr. Ibrahim Abed
SECTION I: MULTIPLE-CHOICE (Each problem is worth 1 point)
For each question in this section, circle the correct answer.
9. The collection and summarization of the socioeconomic and physical
characteristics of the employees of a particular firm is an example of
a) inferential statistics.
b) descriptive statistics.
c) a parameter.
d) a statistic.
10. Researchers are concerned that the weight of the average American school child is
increasing implying, among other things, that children’s clothing should be
manufactured and marketed in larger sizes. If X is the weight of school children
sampled in a nationwide study, then X is an example of
a) a categorical random variable.
b) a discrete random variable.
26
c) a continuous random variable.
d) a parameter.
3 .Which of the following is the easiest to compute?
a) The arithmetic mean.
b) The median.
c) The mode.
d) The standard deviation.
4.According to the Chebyshev rule, at least 93.75% of all observations in any data set are
contained within a distance of how many standard deviations around the mean?
a) 1
b) 2
c) 3
d) 4 %75.93%100)4
11(%100
11
2
k
5. For some positive value of X, the probability that a standard normal variable is between
0 and +1.5X is 0.4332. The value of X is
e) 0.10
f) 0.50
g) 1.00
h) 1.50
6.. In its standardized form, the normal distribution
a) has a mean of 0 and a variance of 1.
b) has a mean of 1 and a variance of 0.
c) has an area equal to 0.5.
d) cannot be used to approximate discrete probability distributions.
7. At a computer manufacturing company, the actual size of computer chips is normally
distributed with a mean of 1 centimeter and a standard deviation of 0.1 centimeter. A
random sample of 12 computer chips is taken. What is the standard error for the sample
mean?
a. 0.029
b. 0.050
c. 0.091
d. 0.120
8. If you were constructing a 99% confidence interval of the population mean based on a
sample of n=25 where the standard deviation of the sample s = 0.05, the critical value
of t will be
a. 2.7969
b. 2.7874
c. 2.4922
d. 2.4851
9. In the construction of confidence intervals, if all other quantities are unchanged, an
increase in the sample size will lead to a interval.
a. narrower
b. wider
c. less significant
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d. biased
10. For sample size 16, the sampling distribution of the mean will be approximately
normally distributed
a. regardless of the shape of the population.
b. if the shape of the population is symmetrical.
c. if the sample standard deviation is known.
d. if the sample is normally distributed.
Section II: TRUE Or FALSE (Each problem is worth 1 point).For each question in this section, indicate whether
the sentence is TRUE or False.
1.( F ) A statistic is usually unobservable while a parameter is usually observable.
2. ( F ) Marital status is an example of an ordinal scaled variable.
3. ( F ) The line drawn within the box of the boxplot always represents the arithmetic
mean.
4. ( T ) The Z score of an observation measures how many standard deviations is the value
from the mean
5. ( T ) The probability that a standard normal random variable, Z, is between 1.50 and
2.10 is the same as the probability Z is between – 2.10 and – 1.50.
6. ( T ) If the population distribution is skewed, in most cases the sampling distribution of
the mean can be approximated by the normal distribution if the samples contain at
least 30 observations
7.( F ) In forming a 90% confidence interval for a population mean from a sample size of
22, the number of degrees of freedom from the t distribution equals 22
8.( T ) Other things being equal, as the confidence level for a confidence interval
increases, the width of the interval increases.
9. ( T ) The sample mean is a point estimate of the population mean.
10. ( F ) The confidence interval obtained will always correctly estimate the population
parameter
SECTION III: FREE RESPONSE QUESTIONS
Question #3: (5 Points) A university dean is interested in determining the proportion of students who receive some sort of
financial aid. Rather than examine the records for all students, the dean randomly selects 200
students and finds that 118 of them are receiving financial aid.
a) Use a 90% confidence interval to estimate the true proportion of students who receive
financial aid?
28
4)0.59797312,60.48202687(200
0.460.541.6450.59
n
p)p(1
α/2ZP90%C.I
b) An economist is interested in studying the incomes of consumers in a particular region. The
population standard deviation is known to be $1,000. A random sample of 50 individuals
resulted in an average income of $15,000. What total sample size would the economist need to
use for a 95% confidence interval if the width of the interval should not be more than $100?
Question #4: (5 Points) The amount of time required for an oil and filter change on an automobile is normally
distributed with a mean of 45 minutes and a standard deviation of 10 minutes. A random
sample of 16 cars is selected.
1. Find the probability that the sample mean is between 45 and 52 minutes?
)16/10
4552
16/10
4545()5245(
zpxp
4974.05.09974.0)0()8.2()8.20( zpzpzp
2. 95% of all sample means will fall between what two values?
95.0)( 11 zzzp
475.0)0( 1 zzp 96.1975.0)( 11 zzzp
6.641096.145 zxu
4.251096.145 zxl
You may use the following formulae:
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13671536.6450*50
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e
σZn
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2
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