Sec$on 09
Single Sample:
Es/ma/ng the Mean
A pump powered by a 25 kW electric motor is opera/ng at a constant speed.
Maintenance staff have randomly sampled the power usage of the electric motor 6
/mes. The following sta/s/c was then computed:
`x = 25
Assume that the power usage follows a normal distribu/on with a true standard
devia/on of 0.4 kW, what is the 95% confidence interval for the mean power usage
of the 25 kW electric motor?
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Single Sample:
Es/ma/ng the Mean
(Concept of Error)
Case 1
For the data given in Example #1 and from the solu/on obtained in part 1:
How large a sample is needed if we wish to be 95% confident that our sample
mean will be within 0.20 kW of the true mean?
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Single Sample:
Es/ma/ng the Mean
(Concept of Error)
Case 2
A plas/c extrusion machine produces sheets with a nominal thickness of 7.5 mm.
A sample of 11 measurements has produces a mean thickness of 7.8 mm with a
standard devia/on of 0.8 mm.
What is the 99% confidence interval for the mean sheet thickness?
Assume a normal distribu/on.
**Random Sample from a
normal popula/on with
unknown Variance.**
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Single Sample:
Es/ma/ng the Mean
(Concept of Error)
Case 3
**Random Sample of
size n³30 from a normal
popula/on with
unknown Variance.**
Suppose a PC manufacturer wants to evaluate the performance of its hard disk
memory system. One measure of performance is the average /me between
failures of the disk drive. To es/mate this value, a quality control engineer
recorded the /me between failures from a random sample of 45 disk-drive
failures. The following sample sta/s/cs were computed:
`x = 1762 hours and S = 215 hours Es/mate the true mean /me between
failures with a 90% confidence interval.
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We want to es/mate at 95% CI the difference between the mean star/ng salaries for
recent graduates with civil engineering and oil&gas engineering degrees in Alberta.
The following informa/on is available (June 2006 © APEGGA):
Two Sample:
Es/ma/ng the
Difference of the Means
1.A random sample of 46 star/ng salaries for AB oil&gas engineering graduates
(Case 1)
produced a sample mean of $54,788 with a normal popula/on standard devia/on of
From Popula/ons with $5,120.
Known variances but
different
2.A random sample of 37 star/ng salaries for AB civil engineering graduates
produced a sample mean of $61,442 with a normal popula/on standard devia/on of
$3,600.
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Two Sample:
Es/ma/ng the Difference
of the Means
(Case 2)
During the first week of May 2003, 25 of Palm M515 PDA were auc/oned off on
eBay, 7 of which had the “buy-it-now” op/on.
*“Buy-it-now” op/on:
235 225 225 240 250 250 210
*“Bidding only”:
250 249 255 200 199 240 228 255 232 246 210 178 246 240 245 225 246 225
Find the 95% CI for the difference of the means. Assume both popula/ons are
normal with equal variances.
From Normal
Popula/ons with
Unknown but equal
Variances
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Two Sample:
Es/ma/ng the Difference
of the Means
(Case 3)
From Normal
Popula/ons with
Unknown and unequal
Variances
A taxi company is trying to decide whether to purchase brand A or brand B /res for
its fleet of taxis. To es/mate the difference in the two brands, an experiment is
conducted using twelve of each brand. The /res are run un/l they wear out. The
results are:
Brand A: mean=36,300 km & Stdev = 5,000 km
Brand B: mean=38,100 km & Stdev = 6,100 km
Compute the 95% CI for
assuming the popula/ons to be approximately normal.
You may not assume that the variances are equal.
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The following table provides data on the modulus of elas/city obtained 1 min amer
loading in a certain configura/on. The values were also obtained 4 weeks amer
loading for the same lumber specimens.
Two Sample:
Es/ma/ng the Difference
of the Means for Paired
Observa/ons
(Case 4)
What is the 95% CI for the difference in modulus of elas/city?
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Single Sample:
Es/ma/ng a Popula/on
Propor/on
In 2000, the Americans were asked: “Are you willing to pay much higher prices in
order to protect the environment?” Of 1154 respondents, 518 were willing to do
so.
Find and interpret a 95% confidence interval for the popula/on propor/on of
adult Americans willing to do so at the /me of the survey.
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Single Sample:
Es/ma/ng the Variance
A quality control supervisor in a cannery knows that the exact amount each can
contains will vary, since there are certain uncontrollable factors that affect the
amount of fill. The mean fill per can is important, but equally important is the
varia/on, S2, of the amount of fill. If S2 is large, some cans will contain too linle
and others too much. To es/mate the varia/on of fill at the cannery, the
supervisor randomly selects 10 cans and weights the content of each.
The weights (in ounces) are:
7.96 7.90 7.98 8.01 7.97 7.96 8.03 8.02 8.04 8.02
Construct a 90% confidence interval for the true standard devia/on of the can
weights.
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Two Samples:
Es/ma/ng the Ra/o of 2
Variances
A firm has been experimen/ng with 2 different physical arrangements of its
assembly line. It has been determined that both arrangements yield
approximately the same average number of finished units per day. To obtain an
arrangement that produces greater process control, you suggest that the
arrangement with the smaller variance in the number of finished units produced
per day be permanently adopted. 2 independent random samples yield the
following data:
Construct a 95% CI for
. Based on the result, which of the
2 arrangements would you recommend?
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