Stat 2470, Homework #6, Fall 2014
Name ______________________________________
Instructions: Show work or give calculator commands used to solve each problem. You may use Excel or
other software for any graphs. Be sure to answer all parts of each problem as completely as possible,
and attach work to this cover sheet with a staple.
1. Below is a table with data for the flexural strength in MPa for a certain type of concrete beams.
5.9
7.2
7.3
6.3
8.1
6.8
7.0
7.6
6.8
6.5
7.0
6.3
7.9
9.0
8.2
8.7
7.8
9.7
7.4
7.7
9.7
7.8
7.7
11.6
11.3
11.8
10.7
a. Calculate a point estimate of the mean value of strength for the conceptual population of all
beams of this type, and state which estimator you used and why.
b. Calculate a point estimate of the strength value that separates the weakest 50% of all such
beams from the strongest 50% and state which estimator you used.
c. Calculate and interpret a point estimate of the population standard deviation 𝜎. Which
estimator did you use?
d. Calculate a point estimate of the proportion of all such beams whose flexural strength
exceeds 10 MPa.
𝜎
e. Calculate a point estimate of the population coefficient of variation , and state which
𝜇
estimator you used.
2. A sample of 20 students who had recently taken elementary statistics yielded the following
information on the brand of calculator owned (T=Texas Instrument, H=Hewlett Packard,
C=Casio, S=Sharp).
T
T
H
T
C
T
T
S
C
H
S
S
T
H
C
T
T
T
H
T
a. Estimate the true proportion of all such students who own a Texas Instrument calculator.
b. Of the 10 students who owned a TI calculator, 4 had graphing calculators. Estimate the
proportion of students who do not own a TI graphing calculator.
3. A random sample of 10 houses in a particular area, each of which is heated with natural gas, is
selected and the amount of gas (therms) used during the month of January is determined for
each house. The resulting observations are 103, 156, 118, 89, 125, 147, 122, 109, 138, 99. Let 𝜇
denote the average gas usage during January by all houses in this area.
a. Compute a point estimate of 𝜇.
b. Suppose there are 10,000 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). What estimator did you use the computing your estimate?
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 based on the sample in part (a).
What estimator did you use?
4. Let X denote the proportion of allotted time that a randomly selected student spends working
on a certain aptitude test. Suppose the probability density function of X is 𝑓(𝑥; 𝜃) =
(𝜃 + 1)𝑥 𝜃 , 0 ≤ 𝑥 ≤ 1
, where −1 < 𝜃. A random sample of ten students yields the data in the
{
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
table.
0.92
0.79
0.90
0.65
0.86
0.47
0.73
0.97
0.94
0.77
Obtain the maximum likelihood estimate of 𝜃, and then compute the estimate for the given
data.
5. Consider a normal population distribution with the value of 𝜎 known.
a. What is the confidence level for the interval 𝑥̅ ± 2.81𝜎/√𝑛?
b. What is the confidence level for the interval 𝑥̅ ± 1.44𝜎/√𝑛?
c. What is the value of 𝑧𝛼/2 in the CI formula results in a confidence level of 99.7%?
d. What is the value of 𝑧𝛼/2 in the CI formula results in a confidence level of 75%?
6. Suppose that a random sample of 50 bottles of a particular brand of cough syrup is selected and
the alcohol content of each bottle is determined. Let 𝜇 denote the average alcohol content for
the population of all bottles of the brand under study. Suppose the resulting 95% confidence
interval is (7.8, 9.4).
a. Would a 90% confidence interval calculated from this same sample have been narrower or
wider than the given interval? Explain your reasoning.
b. Consider the following statement: There is a 95% chance that 𝜇 is between 7.8 and 9.4. Is
this statement correct? Why or why not?
c. Consider the following statement: We can be highly confident that 95% of all bottles of this
type of cough syrup have an alcohol content between 7.8 and 9.4. Is this statement correct?
Why or why not?
d. Consider the following statement: If the process of selecting a sample of size 50 and then
computing the corresponding 95% confidence interval is repeated 100 times, 95 of the
resulting intervals will include 𝜇. Is this statement correct? Why or why not?
7. By how much must the sample size n be increased if the width of the confidence interval is to be
halved? If the sample size is increased by a factor of 25, what effect will this have on the width
of the interval? Justify your assertions.
8. An article reports that for a sample of 50 kitchens with gas cooking appliances monitored during
a one-week period, the sample mean 𝐶𝑂2 level (ppm) was 654.16, and the sample standard
deviation was 164.33.
a. Calculate and interpret a 95% confidence interval for true average 𝐶𝑂2 level in the
population of all home from which the sample was selected.
b. Suppose the investigators had made a rough guess of 175 for the value of 𝑠 before collecting
data. What sample size would be necessary to obtain an interval width of 50 ppm for a
confidence level of 95%?
9. Determine the confidence level for each of the following large-sample, one-sided confidence
bounds.
a. Upper bound: 𝑥̅ + 0.83𝑠/√𝑛
b. Lower bound: 𝑥̅ − 2.05𝑠/√𝑛
c. Upper bound: 𝑥̅ + 0.67𝑠/√𝑛
10. Determine the value of the following quantities.
a. 𝑡0.1,15
b. 𝑡0.05,15
c. 𝑡0.05,25
d. 𝑡0.05,40
e. 𝑡0.005,40
11. Determine the 𝑡 critical value for a two-sided confidence interval in each of the following
situations. And then describe the confidence interval upper and lower bound for each.
a. Confidence level: 95%; df: 10
b. Confidence level: 95%; df: 15
c. Confidence level: 99%; df: 15
d. Confidence level: 99%; df: 5
e. Confidence level: 98%; df: 24
f. Confidence level: 99%; df: 38
12. A sample of 14 joint specimens of a particular type gave a sample mean proportional limit stress
of 8.48 MPa and a sample standard deviation of 0.79 MPa. Calculate and interpret a 95%
confidence interval for the true average proportional limit stress of all such joints. What, if any,
assumptions did you make about the distribution of proportional limit stress?
13. Determine the values of the following quantities.
2
2
2
2
a. 𝜒0.1,15
b. 𝜒0.1,25
c. 𝜒0.01,25
d. 𝜒0.005,25
2
e. 𝜒0.99,25
2
f. 𝜒0.995,25
14. Determine
a. The 95th percentile of the chi-squared distribution with 𝜈 = 10.
b. The 5th percentile of the chi-squared distribution with 𝜈 = 10.
c. 𝑃(10.98 ≤ 𝜒 2 ≤ 36.78) where 𝜒 2 is a chi-squared random variable with 𝜈 = 22.
d. 𝑃(𝜒 2 < 14.611 𝑜𝑟 𝜒 2 > 37652) where 𝜒 2 is a chi-squared random variable with 𝜈 = 25.