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ST361 HW8: Practice Problems of M2
Notice: (1) No need to turn in
(2) You may use one sheet of notes (8.5 by 11 inches) and a calculator. You may not share a
calculator, pencil, paper or anything else during the exam.
The following notation may be used without further specification:
n : sample size
X : sample mean
s : sample standard deviation (SD)
p : sample proportion
Also
 : population mean
 : population standard deviation (SD)
 : population proportion
statistic  mean of the particular statistic listed in the subscript
 statistic  standard error of the particular statistic listed in the subscript :
Ex.  p 
 1   
n
 the standard error of the sample proportion p
Part I: Multiple Choice Questions
___________ 1. Which of the following properties are not true regarding the sampling distribution of
X?
(a)  X   no matter how large n is
(b)  X  
n
(c) By the central limit theorem, the distribution of X is normal no matter how large
n is
(d) When the population being sampled follows a normal distribution, the
distribution of X is normal no matter how large n is
___________ 2. Which of the following is not a statistic?
(a) The sample mean
(b) The sample median
(c) The population proportion
(d) The sample IQR (interquartile range)
(e) All of the above are statistics
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Questions 3 to 6. A random sample x1 , x2 , , xn with size n  9 is drawn from a population that follows
normal distribution with mean   10 and standard deviation   6.
___________ 3. What is  X , the mean of the sampling distribution of X ?
(a) 15
(b) 10
(c) 9
(d) 6
(e) 2
___________ 4. What is  X , the standard error of X ?
(a) 15
(b) 10
(c) 9
(d) 6
(e) 2
___________ 5. What is the probability that X falls within 1 unit of the population mean 
i.e.,10  1  X  10  1 ?
(a) 100 %
(b) 68 %
(c) 38 %
(d) 18 %
(e) 13 %


___________ 6. If the sample size increase form n = 9 to n = 100, then comparing to the case of n = 9, the
probability that X falls within 1 unit of the population mean
i.e.,10  1  X  10  1 becomes
(a) higher
(b) lower
(c) same
(d) cannot tell


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___________ 7. X follows a certain distribution with mean  and SD  . Among the following
estimators, which is the best point estimator for  ?
(a) Sample median on a sample of size 40
(b) Sample mean based on a sample of size 40
(c) Sample mean based on a sample of size 100
(d) Sample standard deviation based on a sample of size 100
___________ 8. X is normally distributed with known SD  . The margin of error associated with a 90%
confidence interval for  is
(a) 1.28  n
(b) 1.64  n
(c) 1.96  n
(d) 2.58  n
(e) 2.81  n
___________ 9. Which of the following is a property of the t distribution?
(a) The t distribution has degrees of freedom n-1
(b) The t curve is bell-shaped and centered at 0
(c) The t curve is more spread out than a z curve
(d) As the degrees of freedom increases, the spread of the corresponding t curve
decreases, and the sequence of t curves approaches the z curve.
(e) All of the above
___________ 10.
In developing a 95% confidence interval for the population mean  , a sample of 100
observations was used, and the confidence interval was 70  2. Had the sample size
been 400 instead of 100, the confidence interval would have been
(a) 70  4
(b) 70  2
(c) 70  1
___________ 11.
(Continuous from Question 10.) If the confidence level is increased to 99% (and
still use a sample of 100 observations), then comparing with the 95% interval 70 
2, the 99% confidence interval for 
(a) becomes wider
(b) becomes narrower
(c) remains unchanged
(d) cannot tell
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___________ 12.
If a 95% confidence interval for  is (23.9, 30.1), then
(a) 95% of all of the possible samples produce intervals that do cover 
(b) There is a 95% chance that  is between 23.9 and 30.1
(c) 95% of all the possible values of  fall within the interval (23.9, 30.1)
___________ 13.
The Central Limit Theorem predicts that
(a) The sampling distribution of  will be approximately normal for n >30
(b) the sampling distribution of x will be approximately normal for n > 30
(c) the sampling distribution of p will be approximately normal for n > 30
(d) the sampling distribution of  will be approximately normal for n > 30
___________ 14. Which of the following statements is true?
A) A statistical hypothesis is a claim or assertion about the values of parameters.
B) A test of hypothesis is a method for using sample data to decide whether the null
hypothesis should be rejected.
C) A type II error consists of not rejecting the null hypothesis H o when H o is false.
D) All of the above statements are true.
Questions 15 to 16. The Rockwell hardness of certain metal pins is known to have a mean   50 and a
standard deviation   4. Assume that the distribution of all such pin hardness measurements is known
to be normal.
___________ 15.
If we randomly select 1 pin from the population, what is the probability that the
hardness is less than 46, i.e., Pr( X < 46)?
(a) 0.3085
(b) 0.1587
(c) 0.0668
(d) 0.0228
(e) 0.0000
___________ 16.
If now a random sample of 16 pins is obtained, what is the probability that the
average hardness is less than 46, i.e., Pr( X < 46)?
(a) 0.3085
(b) 0.1587
(c) 0.0668
(d) 0.0228
(e) 0.0000
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___________ 17.
A)
B)
C)
D)
In hypothesis-testing analysis, a type I error occurs only if the null hypothesis H o is
Rejected when it is true
Rejected when it is false
Not rejected when it is false
Not rejected when it is true
___________ 18. The test statistic for a z test for  is
A) x
B) x  hypothesized value
C)
x  hypothesized value

n
x  hypothesized value
D)
s
n
Part II: True or False
___________ 19.
The level of significance of a test is the probability of making a Type I Error
___________ 20.
A small p-value indicates that the observed sample is inconsistent with the null
hypothesis
___________ 21.
The p-value of an upper tail t test is the area to the left of the calculated Z-test
statistic under the Z curve
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Part II: Computational Problems
22. A simple random sample of 10 students is selected from College A. For the 10 students, the number
of days each was absent during the last semester was to be
3, 7, 2, 1, 0, 2, 4, 2, 6, and 2.
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Note that
x
i 1
i
 29. and the sample SD is s = 2.18.
(a) Calculate X , the sample mean number of absent days, and use it as a point estimate for  , the mean
number of absent days for the college’s students.
(b) Report the S.E. of your point estimate in (a)
(c) Is your point estimate in (a) an unbiased estimator? Why or why not?
(d) What assumptions are required here to have X normally distributed? (circle one)
i. The number of absent days follows a normal distribution
ii. The sample SD follows a normal distribution
iii. No assumption is needed as X follows a normal distribution by the central limit theorem
(e) Calculate the 95% confidence interval for  . Report the final answer in the format of  a, b  .
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(f) Assume the population mean absent days for College B is known to be 4.1 days. Based on the fact that
falls above the value 4.1

contains the value 4.1
the your 95% CI for  in (e) 
, it suggests that
falls
below
the
value
4.1

(circle one)
College A has higher mean absent days

 the two colleges have similar mean absent days
College A has lower mean absent days

(circle one)
(g) Let   the proportion of students being absent more than 5 days. Report an unbiased point estimate
of  . (5 points)
(h) Report the S.E. of your point estimate. (Hint: if the population proportion  is known, then
p 
 1   
n
) (5 points)
(i) Your estimator in (g) is unbiased because (circle one) (3 points)
i. its distribution is normal
ii. its mean is equal to 
iii. its SE is equal to
 1   
n
iv. it is based on a sample with size greater than 30
(j) The confidence interval formula for estimating  should only be used when your point estimator in
(g) follows a normal distribution. This normality can be achieved when _____________________
(a)
(b)
(c)
(d)
n ≥ 30
np > 30
np > 10
np > 10 and n(1-p) > 10
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23. A researcher wishes to test the claim that for a particular manufacturer of cereal, the mean weight in
its boxes of cereal is less than 18 ounces. A sample of 36 boxes yields a sample mean weight of
17.88 ounces. Assume that the population standard deviation is .28 ounces. Let=.05.
a. Conduct a 5-step test of hypotheses:
i. H0:
Ha:
ii. =_______.
iii. Test Statistic (if a t test is used, also report the df):
iv. P-value:
v. Conclusion (both statistical conclusion and conclusion in the context of the problem) :
b. Find a 95% confidence interval for = true mean weight.
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24. A study was done on 31 female patients following a new treatment for cardio-vascular disease (CVD).
Doctors measured the increase in exercise capacity (in minutes) over a 6-week period. The
conventional treatment had produced an average increase of =2 minutes. Researchers wish to claim
that the new treatment will increase the mean exercise capacity more than the conventional treatment.
The data yielded y  2.17 and s=1.05. Let=.05.
a. Conduct a 5-step test of hypotheses:\
i. H0:
Ha:
ii. =_______.
iii. Test Statistic (if a t test is used, also report the df):
iv. P-value:
v. Conclusion (both statistical conclusion and conclusion in the context of the problem) :
b. Find a 95% confidence interval for .
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Answers:
CC | BECA | CBECA | ABDBE | AC
TTF
22. 2.9; 0.689; yes, because the mean of x-bar is equal to ; i; (1.34, 4.46)
Contains 4.1, have similar mean absent days
0.2; 0.126; ii; d
23. (a) i. H0: ≥18 Ha: <18; ii. =0.05; iii z*= -2.57; iv. P-val = 0.0051
v. Reject H0 and conclude that the mean weight in a box is less than 18 ounces.
(b) (17.788, 17.971)
24. (a) i. H0: ≤ 2 Ha: ; ii. =0.05; iii. t*= 0.901 with df=30; iv. p-val = 0.188
v. Do not reject H0 and conclude that the new treatment does not increase the mean exercise
capacity more than the conventional treatment.
(b) (1.785, 2.555)
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