Chapter 6 Point Estimation Fall 2022 - Mat 301 Fall 2022, section 97/98/99, Fall 2022 | WebAssign
11/9/22, 9:41 PM
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QUESTION
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2
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11
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POINTS
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Due Date
SAT, NOV 12, 2022
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Chapter 6 Point Estimation Fall 2022 - Mat 301 Fall 2022, section 97/98/99, Fall 2022 | WebAssign
1.
[–/10 Points]
MY NOTES
DETAILS
11/9/22, 9:41 PM
DEVORESTAT9 6.1.001.
ASK YOUR TEACHER
PRACTICE ANOTHER
Consider the accompanying data on flexural strength (MPa) for concrete beams of a certain type.
9.7
5.8
6.8
6.5
9.0
11.6
7.8
9.7
7.4
7.8
7.7
7.4
7.7
8.7
8.9
11.3
6.8
6.3
6.3
8.1
7.0
7.3
7.9
7.2
7.0
10.7
11.8
(a) Calculate a point estimate of the mean value of strength for the conceptual population of all
beams manufactured in this fashion. [Hint: Σx i = 220.2.] (Round your answer to three decimal
places.)
MPa
State which estimator you used.
s
̂p
s/x
x
(b) Calculate a point estimate of the strength value that separates the weakest 50% of all such
beams from the strongest 50%.
MPa
State which estimator you used.
x
s/x
s
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Chapter 6 Point Estimation Fall 2022 - Mat 301 Fall 2022, section 97/98/99, Fall 2022 | WebAssign
11/9/22, 9:41 PM
̂p
(c) Calculate a point estimate of the population standard deviation 𝜎. [Hint: Σx i 2 = 1868.74.]
(Round your answer to three decimal places.)
MPa
Interpret this point estimate.
This estimate describes the center of the data.
This estimate describes the linearity of the data.
This estimate describes the bias of the data.
This estimate describes the spread of the data.
Which estimator did you use?
s/x
̂p
s
x
(d) Calculate a point estimate of the proportion of all such beams whose flexural strength
exceeds 10 MPa. [Hint: Think of an observation as a "success" if it exceeds 10.] (Round your
answer to three decimal places.)
(e) Calculate a point estimate of the population coefficient of variation 𝜎/𝜇. (Round your answer
to four decimal places.)
State which estimator you used.
s
̂p
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Chapter 6 Point Estimation Fall 2022 - Mat 301 Fall 2022, section 97/98/99, Fall 2022 | WebAssign
11/9/22, 9:41 PM
x
s/x
2.
[–/9 Points]
MY NOTES
DETAILS
ASK YOUR TEACHER
PRACTICE ANOTHER
Consider the accompanying data on flexural strength (MPa) for concrete beams of a certain type.
5.6 7.2 7.3 6.3 8.1 6.8 7.0 7.2 6.8
6.5
7.0
6.3
7.9 9.0
9.0 8.7 7.8 9.7 7.4 7.7 9.7 8.0 7.7 11.6 11.3 11.8 10.7
The data below give accompanying strength observations for cylinders.
6.6 5.8 7.8 7.1 7.2 9.2 6.6
8.3
7.0
8.4
7.7 8.1 7.4 8.5 8.9 9.8 9.7 14.1 12.6 11.5
. , Y n. Suppose that the X i's constitute a random sample from a distribution with mean 𝜇 1 and standard
Prior to obtaining data, denote the beam strengths by X 1, . . . , X m and the cylinder strengths by Y 1, . .
deviation 𝜎 1 and that the Y i's form a random sample (independent of the X i's) from another distribution
with mean 𝜇 2 and standard deviation 𝜎 2.
(a) Use rules of expected value to show that X − Y is an unbiased estimator of 𝜇 1 − 𝜇 2.
E(X − Y) = E(X) − E(Y) = 𝜇 1 − 𝜇 2
E(X − Y) =
E(X) − E(Y)
nm
E(X − Y) = E(X) − E(Y)
E(X − Y) =
= 𝜇1 − 𝜇2
2
= 𝜇1 − 𝜇2
E(X) − E(Y) = 𝜇 − 𝜇
1
2
E(X − Y) = nm E(X) − E(Y) = 𝜇 1 − 𝜇 2
Calculate the estimate for the given data. (Round your answer to three decimal places.)
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Chapter 6 Point Estimation Fall 2022 - Mat 301 Fall 2022, section 97/98/99, Fall 2022 | WebAssign
11/9/22, 9:41 PM
MPa
(b) Use rules of variance to obtain an expression for the variance and standard deviation
(standard error) of the estimator in part (a).
= 𝜎X2 + 𝜎Y2
V(X − Y) = V(X) + V(Y)
𝜎22
+
=
n1
𝜎X − Y =
V(X − Y)
𝜎22
=
+
n1
Compute the estimated standard error. (Round your answer to three decimal places.)
MPa
(c) Calculate a point estimate of the ratio 𝜎 1/𝜎 2 of the two standard deviations. (Round your
answer to three decimal places.)
(d) Suppose a single beam and a single cylinder are randomly selected. Calculate a point
estimate of the variance of the difference X − Y between beam strength and cylinder strength.
(Round your answer to two decimal places.)
MPa 2
3.
[–/6 Points]
MY NOTES
DETAILS
DEVORESTAT9 6.1.007.
ASK YOUR TEACHER
PRACTICE ANOTHER
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Chapter 6 Point Estimation Fall 2022 - Mat 301 Fall 2022, section 97/98/99, Fall 2022 | WebAssign
11/9/22, 9:41 PM
(a) A random sample of 10 houses in a particular area, each of which is heated with natural gas, is
house. The resulting observations are 138, 148, 103, 122, 109, 125, 99, 147, 118, 84. Let 𝜇 denote the
selected and the amount of gas (therms) used during the month of January is determined for each
average gas usage during January by all houses in this area. Compute a point estimate of 𝜇.
therms
(b) Suppose there are 14,000 houses in this area that use natural gas for heating. Let 𝜏 denote the total
amount of gas used by all of these houses during January. Estimate 𝜏 using the data of part (a).
therms
What estimator did you use in computing your estimate?
nx
s
̂p
s/x
(c) Use the data in part (a) to estimate p, the proportion of all houses that used at least 100 therms.
(d) Give a point estimate of the population median usage (the middle value in the population of all
houses) based on the sample of part (a).
therms
What estimator did you use?
x
s
s/x
̂p
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Page 7 of 17
Chapter 6 Point Estimation Fall 2022 - Mat 301 Fall 2022, section 97/98/99, Fall 2022 | WebAssign
4.
[–/2 Points]
MY NOTES
DETAILS
11/9/22, 9:41 PM
DEVORESTAT9 6.1.008.
ASK YOUR TEACHER
PRACTICE ANOTHER
In a random sample of 160 components of a certain type, 56 are found to be defective.
(a) Give a point estimate of the proportion of all such components that are not defective.
(b) A system is to be constructed by randomly selecting two of these components and
connecting them in series, as shown here.
The series connection implies that the system will function if and only if neither component is
defective (i.e., both components work properly). Estimate the proportion of all such systems
that work properly. [Hint: If p denotes the probability that a component works properly, how can
P(system works) be expressed in terms of p?] (Enter your answer to four decimal places.)
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Chapter 6 Point Estimation Fall 2022 - Mat 301 Fall 2022, section 97/98/99, Fall 2022 | WebAssign
5.
[–/2 Points]
DETAILS
MY NOTES
11/9/22, 9:41 PM
DEVORESTAT9 6.1.009.
ASK YOUR TEACHER
PRACTICE ANOTHER
Each of 145 newly manufactured items is examined and the number of scratches per item is recorded
(the items are supposed to be free of scratches), yielding the following data:
Number of
scratches
per item
0
1
2
3
4
5
6
7
15
33
40
29
14
6
4
4
Observed
frequency
distribution with parameter 𝜇.
Let X = the number of scratches on a randomly chosen item, and assume that X has a Poisson
(a) Find an unbiased estimator of 𝜇 and compute the estimate for the data. [Hint: E(X) = 𝜇 for X
Poisson, so E(X) = ?] (Round your answer to two decimal places.)
standard error. [Hint: 𝜎 X 2 = 𝜇 for X Poisson.] (Round your answer to three decimal places.)
(b) What is the standard deviation (standard error) of your estimator? Compute the estimated
6.
[–/8 Points]
MY NOTES
DETAILS
DEVORESTAT9 6.1.011.
ASK YOUR TEACHER
PRACTICE ANOTHER
Of n 1 randomly selected male smokers, X 1 smoked filter cigarettes, whereas of n 2 randomly selected
female smokers, X 2 smoked filter cigarettes. Let p 1 and p 2 denote the probabilities that a randomly
selected male and female, respectively, smoke filter cigarettes.
(a) Show that
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Chapter 6 Point Estimation Fall 2022 - Mat 301 Fall 2022, section 97/98/99, Fall 2022 | WebAssign
11/9/22, 9:41 PM
(X 1/n 1) − (X 2/n 2) is an unbiased estimator for p 1 − p 2. [Hint: E(X i) = n ip i for i = 1, 2.]
1
n1
E
X1
n1
−
X2
n2
=
E
−
1
n2
E
1
n1
=
−
1
n2
= p1 − p2
(b) What is the standard error of the estimator in part (a)?
(c) How would you use the observed values x 1 and x 2 to estimate the standard error of your
estimator?
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Chapter 6 Point Estimation Fall 2022 - Mat 301 Fall 2022, section 97/98/99, Fall 2022 | WebAssign
11/9/22, 9:41 PM
You would put them directly into the formula to obtain the estimate of the standard error.
You would use them to estimate the necessary quantities and then put these into the
formula.
There is not enough information.
You can not use the observed values to estimate the standard error of your estimator.
(d) If n 1 = n 2 = 210, x 1 = 138, and x 2 = 165, use the estimator of part (a) to obtain an
estimate of p 1 − p 2. (Round your answer to three decimal places.)
(e) Use the result of part (c) and the data of part (d) to estimate the standard error of the
estimator. (Round your answer to three decimal places.)
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Chapter 6 Point Estimation Fall 2022 - Mat 301 Fall 2022, section 97/98/99, Fall 2022 | WebAssign
7.
[–/4 Points]
MY NOTES
DETAILS
11/9/22, 9:41 PM
DEVORESTAT9 6.2.020.
ASK YOUR TEACHER
PRACTICE ANOTHER
A diagnostic test for a certain disease is applied to n individuals known to not have the disease. Let
X = the number among the n test results that are positive (indicating presence of the disease, so X is
the number of false positives) and p = the probability that a disease-free individual's test result is
positive (i.e., p is the true proportion of test results from disease-free individuals that are positive).
Assume that only X is available rather than the actual sequence of test results.
(a) Derive the maximum likelihood estimator of p.
̂p =
If n = 20 and x = 7, what is the estimate?
̂p =
(b) Is the estimator of part (a) unbiased?
Yes
No
(c) If n = 20 and x = 7, what is the mle of the probability (1 − p) 5 that none of the next five
tests done on disease-free individuals are positive? (Round your answer to four decimal places.)
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Chapter 6 Point Estimation Fall 2022 - Mat 301 Fall 2022, section 97/98/99, Fall 2022 | WebAssign
8.
[–/4 Points]
DETAILS
MY NOTES
11/9/22, 9:41 PM
DEVORESTAT9 6.2.022.MI.
ASK YOUR TEACHER
PRACTICE ANOTHER
Let X denote the proportion of allotted time that a randomly selected student spends working on a
certain aptitude test. Suppose the pdf of X is
f(x; 𝜃) =
(𝜃 + 1)x 𝜃
0
0≤x≤1
otherwise
where −1 < 𝜃. A random sample of ten students yields data x 1 = 0.86, x 2 = 0.90, x 3 = 0.49, x 4 = 0.95,
x 5 = 0.94, x 6 = 0.73, x 7 = 0.79, x 8 = 0.65, x 9 = 0.79, x 10 = 0.92.
(a) Use the method of moments to obtain an estimator of 𝜃
1
1+X
−1
1
X−1
−2
1
1+X
1
X−1
−1
1
1−X
−2
Compute the estimate for this data. (Round your answer to two decimal places.)
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Chapter 6 Point Estimation Fall 2022 - Mat 301 Fall 2022, section 97/98/99, Fall 2022 | WebAssign
(b) Obtain the maximum likelihood estimator of 𝜃.
Σln(X i)
n
−n
Σln(X i)
11/9/22, 9:41 PM
−1
−1
Σln(X i)
n
n
Σln(X i)
Σln(X i)
−n
Compute the estimate for the given data. (Round your answer to two decimal places.)
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Chapter 6 Point Estimation Fall 2022 - Mat 301 Fall 2022, section 97/98/99, Fall 2022 | WebAssign
9.
[–/4 Points]
MY NOTES
DETAILS
11/9/22, 9:41 PM
DEVORESTAT9 6.2.025.
ASK YOUR TEACHER
PRACTICE ANOTHER
The shear strength of each of ten test spot welds is determined, yielding the following data (psi).
400
385
373
389
362
375
358
367
409
415
(a) Assuming that shear strength is normally distributed, estimate the true average shear
strength and standard deviation of shear strength using the method of maximum likelihood.
(Round your answers to two decimal places.)
average
psi
standard deviation
psi
welds will have their strengths. [Hint: What is the 95th percentile in terms of 𝜇 and 𝜎? Now use
(b) Again assuming a normal distribution, estimate the strength value below which 95% of all
the invariance principle.] (Round your answer to two decimal places.)
psi
Use the given data to obtain the mle of P(X ≤ 400). [Hint: P(X ≤ 400) = Φ((400 − 𝜇)/𝜎).]
(c) Suppose we decide to examine another test spot weld. Let X = shear strength of the weld.
(Round your answer to four decimal places.)
You may need to use the appropriate table in the Appendix of Tables to answer this question.
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Chapter 6 Point Estimation Fall 2022 - Mat 301 Fall 2022, section 97/98/99, Fall 2022 | WebAssign
10.
[–/1 Points]
MY NOTES
DETAILS
11/9/22, 9:41 PM
DEVORESTAT9 6.2.030.
ASK YOUR TEACHER
PRACTICE ANOTHER
parameter 𝜆. The experimenter then leaves the test facility unmonitored. On his return 24 hours later,
At time t = 0, 22 identical components are tested. The lifetime distribution of each is exponential with
still in operation (so 8 have failed). Derive the mle of 𝜆. [Hint: Let Y = the number that survive 24
the experimenter immediately terminates the test after noticing that y = 14 of the 22 components are
exponentially distributed. This relates 𝜆 to p, so the former can be estimated once the latter has been.]
hours. Then Y ~ Bin(n, p). What is the mle of p? Now notice that p = P(X i ≥ 24), where X i is
(Round your answer to four decimal places.)
=
11.
[–/1 Points]
MY NOTES
DETAILS
DEVORESTAT9 6.SE.037.
ASK YOUR TEACHER
PRACTICE ANOTHER
When the sample standard deviation S is based on a random sample from a normal population
distribution, it can be shown that
2/(n − 1) Γ(n/2)𝜎/Γ((n − 1)/2)
Use this to obtain an unbiased estimator for 𝜎 of the form cS. What is c when n = 22? (Round your
E(S) =
answer to four decimal places.)
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Chapter 6 Point Estimation Fall 2022 - Mat 301 Fall 2022, section 97/98/99, Fall 2022 | WebAssign
12.
[–/2 Points]
MY NOTES
DETAILS
11/9/22, 9:41 PM
DEVORESTAT9 6.SE.502.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
A sample of 20 students who had recently taken elementary statistics yielded the following information
on brand of calculator owned. (T = Texas Instruments, H = Hewlett Packard, C = Casio, S = Sharp):
C S
T S T H C C S T
S S H T T H H
T C S
(a) Estimate the true proportion of all such students who own a Texas Instruments calculator.
(b) Of the 6 students who owned a TI calculator, 2 had graphing calculators. Estimate the
proportion of students who do not own a TI graphing calculator.
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