RuoLin Li P2022STAT217L01
Assignment 1 is due on Friday, May 13, 2022 at 11:59pm.
The number of attempts available for each question is noted beside the question. If you are having trouble figuring out your error, you
should consult the textbook, or ask a fellow student, one of the TA’s or your professor for help.
There are also other resources at your disposal, such as the Mathematics Continuous Tutorials. Don’t spend a lot of time guessing – it’s
not very efficient or effective.
Make sure to give lots of significant digits for (floating point) numerical answers. For most problems when entering numerical answers,
you can if you wish enter elementary expressions such as 2∧ 3 instead of 8, sin(3 ∗ pi/2)instead of -1, e∧ (ln(2)) instead of 2, (2+
tan(3))∗ (4− sin(5))∧6−7/8 instead of 27620.3413, etc.
Problem 1. (1 point)
Two random samples are selected from two independent popula-
tions. A summary of the samples sizes and sample means is given
below:
n1 = 45, x¯1 = 52.8
n2 = 48, x¯2 = 78.5
If the 96% confidence interval for the difference µ1− µ2 of the
means is (-29.2897, -22.1103), what is the value of the pooled
variance estimator? (You may assume equal population vari-
ances.)
Pooled Variance Estimator =
Answer(s) submitted:
• 0.300867
(incorrect)
Problem 2. (1 point)
Altough in general you cannot know the sampling distribution of
the sample mean exactly, by what distribution can you often ap-
proximate it?
• A. normal distribution
• B. uniform distribution
• C. reverse-J-shaped distribution
• D. none of the above
Answer(s) submitted:
• A
(correct)
Problem 3. (1 point)
The data below gives the mean price (in cents) of a litre of regular
gasoline at self-service filling stations at a sample of six urban
centres in Canada in May 2012.
Urban area Price/litre
Halifax 131.5
Saint John, NB 129.7
Montreal 135.8
Calgary 114.5
Vancouver 144.4
Victoria 131.8
Provide answers to the following to two decimal places.
Part (a) Find the sample mean (in cents).
Part (b) Find the sample standard deviation (in cents).
Part (c) Using the appropriate t distribution, find a 90
Part (d) Would a 95
Answer(s) submitted:
• 131.2833
• 9.766763
• 123.2488
• 139.3179
• 121.0337
• 141.5329
(correct)
1
Problem 4. (1 point)
Random samples of female and male UVA undergraduates are
asked to estimate the number of alcoholic drinks that each con-
sumes on a typical weekend. The data is below:
Females (Population 1): 2, 4, 3, 4, 6, 3, 5, 6, 4, 6
Males (Population 2): 6, 6, 8, 5, 9, 9, 8, 7, 7, 7
Give a 91% confidence interval for the difference between mean
female and male drink consumption. (Assume that the population
variances are equal.)
Confidence Interval =
Answer(s) submitted:
• (-6.731, 0.931)
(incorrect)
Problem 5. (1 point)
The weights of a random sample of cereal boxes that are supposed
to weigh 1 pound are given below. Estimate the standard devia-
tion of the entire population with 91.7% confidence. Assume the
population is normally distributed.
0.99 1.01 0.95 1.04
1.02 1.01 1 0.99
LCL =
UCL =
Answer(s) submitted:
• 0.9824
• 1.020
(incorrect)
Problem 6. (1 point)
A statistics practitioner took a random sample of 46 observations
from a population whose standard deviation is 27 and computed
the sample mean to be 96.
Note: For each confidence interval, enter your answer in the form
(LCL, UCL). You must include the parentheses and the comma
between the confidence limits.
A. Estimate the population mean with 95% confidence.
Confidence Interval =
B. Estimate the population mean with 90% confidence.
Confidence Interval =
C. Estimate the population mean with 99% confidence.
Confidence Interval =
Answer(s) submitted:
•
•
•
(incorrect)
2
Problem 7. (1 point)
An epidemiologist is worried about the prevalence of the flu in
East Vancouver and the potential shortage of vaccines for the area.
She will need to provide a recommendation for how to allocate the
vaccines appropriately across the city. She takes a simple random
sample of 332 people living in East Vancouver and finds that 38
have recently had the flu.
Part i) The proportion of the 332 people who have recently had
the flu, 38/332, is a:
• A. variable of interest.
• B. parameter.
• C. statistic.
Part ii) Use the sample data to compute a 95% confidence inter-
val for the true proportion of East Vancouver residents that have
recently had the flu.
(Please carry answers to at least six decimal places in intermediate
steps. Give your final answer to the nearest three decimal places).
95% confidence interval = ( , )
Answer(s) submitted:
•
•
(incorrect)
Problem 8. (1 point)
Holding everything else constant, which change to the sample
size will reduce the width of a confidence interval for a popula-
tion mean by half?
• A. Raise the sample size to the power 4.
• B. Quadruple the sample size.
• C. Half the sample size.
• D. Square the sample size.
• E. Double the sample size.
Answer(s) submitted:
•
(incorrect)
Problem 9. (1 point)
The study by Schlaich et al. (1998) investigated lung function in
patients with spinal osteoporosis. The volume of forced expira-
tion in one second was recorded on each of 34 patients, this figure
being adjusted for associated variables such as age and gender
and then recorded as a percentage of some standardised figure.
Suppose the mean and standard deviation of the data were 96.5
and 14.9 respectively, both in percentage points. It is assumed
the data are from a Normal distribution with unknown mean µ
and standard deviation σ, and the researchers were interested in
estimating µ.
Provide answers to the following to two decimal places.
Part a) Compute a 90on a Student’s t-distribution. Provide the
width of the interval.
Part b) Before Student derived the t-distribution, it was common
to use the standard Normal distribution for estimating confidence
intervals as above. Recompute the confidence interval you found
in (a) but using the standard Normal rather than the t-distribution.
Provide the width of the interval.
Part c) Taking the normal width as the base, compare the relative
widths of the intervals you found in (a) and (b) rounded to two
decimal places. By what percentage of the width of the interval in
(b) is the width of the interval in (a) larger or smaller?
Schlaich, C., Minne, H. W., Bruckner, T., Wagner, G., Gebest, H.
J., Grunze, M., Ziegler, R., and Leidig-Bruckner, G. (1998): ”Re-
duced Pulmonary Function in Patients with Spinal Osteoporotic
Fractures.” Osteoporosis International 8, 261-267.
Answer(s) submitted:
•
•
•
(incorrect)
3
Problem 10. (1 point)
The time in hours for a worker to repair an electrical instrument
is a Normally distributed random variable with a mean of µ and a
standard deviation of 50. The repair times for 12 such instruments
chosen at random are as follows:
183 222 303 262 178 232 268 201 244 183 201 140
Part a) Find a 95( , ).
Part b) Find the least number of repair times needed to be sam-
pled in order to reduce the width of the confidence interval to be-
low 29 hours.
Answer(s) submitted:
•
•
•
(incorrect)
Problem 11. (1 point)
Statistics calculated from a sample of 22 observations are:
∑22i=1 xi = 1517
∑22i=1 x2i = 111843
Provide the following, rounding your answers to two decimal
places.
(a) Find the sample mean.
(b) What is the sample standard deviation?
(c) Assume that the population distribution is normal. Find a
95confidence interval for the population mean. ( , )
Answer(s) submitted:
• 68.9545
• 23.8984
• 61.3744
• 76.5346
(score 0.25)
Problem 12. (1 point)
Explain why increasing the sample size tends to result in a smaller
sampling error when a sample mean is used to estimate a popula-
tion mean.
• A. The above statement is incorrect, the sample size has
no effect on the sampling error.
• B. The larger the sample size, the more closely the possi-
ble values of x¯ cluster around the mean of x¯
• C. If the sample size is larger, the possible values of x¯ are
farther from the mean of x¯
Answer(s) submitted:
• B
(correct)
Problem 13. (1 point)
Find the critical values χ2L = χ21−α/2 and χ
2
R = χ2α/2 that corre-
spond to 90% degree of confidence and the sample size n= 26.
χ2L = χ2R =
Answer(s) submitted:
•
•
(incorrect)
Problem 14. (1 point)
In a very large population, the distribution of annual income is
skewed, with a long right tail. We take a simple random sample
of n people from this population and record the n incomes. We
expect a histogram of the n incomes in the sample
• A. will resemble a Uniform distribution provided n is
large.
• B. will resemble a Normal distribution provided n is large.
• C. will resemble a Normal distribution for all values of n.
• D. will resemble a Uniform distribution for all values of
n.
• E. will not resemble a Normal distribution whatever the
value of n.
Answer(s) submitted:
• E
(correct)
4
Problem 15. (1 point)
The weights of cans of Ocean brand tuna are supposed to have a
net weight of 6 ounces. The manufacturer tells you that the net
weight is actually a Normal random variable with a mean of 5.95
ounces and a standard deviation of 0.2 ounces. Suppose that you
draw a random sample of 42 cans.
Part i) Suppose the number of cans drawn is doubled. How will
the standard deviation of sample mean weight change?
• A. It will decrease by a factor of 2.
• B. It will increase by a factor of 2.
• C. It will increase by a factor of√2.
• D. It will decrease by a factor of√2.
• E. It will remain unchanged.
Part ii) Suppose the number of cans drawn is doubled. How will
the mean of the sample mean weight change?
• A. It will decrease by a factor of 2.
• B. It will increase by a factor of 2.
• C. It will increase by a factor of√2.
• D. It will decrease by a factor of√2.
• E. It will remain unchanged.
Part iii) Consider the statement: ’The distribution of the mean
weight of the sampled cans of Ocean brand tuna is Normal.’
• A. It is a correct statement, but it is not a result of the
Central Limit Theorem.
• B. It is an incorrect statement. The distribution of the
mean weight of the sample is not Normal.
• C. It is a correct statement, and it is a result of the Central
Limit Theorem.
Answer(s) submitted:
•
•
•
(incorrect)
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