CHAPTER 8 RESERVE PROBLEMS
Chapter 8 Section 1 Problem 1
For a normal population with the known variance
2
, answer the following questions.
a) What value of confidence level gives
/2
z
equal to 2.10?
b) What value of confidence level gives
/2
z
equal to 2.37?
c) What value of confidence level gives
/2
z
equal to 1.08?
SOLUTION
a) The confidence level for
2.10 / 2.10 /x n x n
− +
is determined by the value of
/2
z
/2
z
/2
z
Reserve Problems Chapter 8 Section 1 Problem 2
Consider the one-sided confidence interval expressions for a mean of normal population.
a) What confidence level will cause
z
be equal to 1.93?
b) What confidence level will cause
z
be equal to 1.18?
c) What confidence level will cause
z
be equal to 1.45?
SOLUTION
a) One-sided confidence interval with
1.93z
=
. Therefore,
( 1.93)PZ
=
and confidence
Reserve Problems Chapter 8 Section 1 Problem 3
Nate Diaz, a UFC fighter, has landed 1044 significant strikes in his previous 12 UFC fights.
Assume that the number of significant strikes is normally distributed with
10
=
.
Find the 95% two-sided confidence interval on the mean number of strikes.
SOLUTION
/2
1044 1 0.95
87; 0.025; 1.96
12 2 2
xz
−
= = = = =
Reserve Problems Chapter 8 Section 1 Problem 4
During the independent research 30 women were chosen to measure their weight. The mean
value of weight is 66 kg and it is known from the previous experience that the weight is normally
distributed with
10 kg
=
.
a) Find a 95% two-sided confidence interval on the mean weight.
b) Find a 90% two-sided confidence interval on the mean weight.
c) Which interval is wider?
SOLUTION
a)
/2
0.025, 1.96, 10, 10
2zn
= = = =
Reserve Problems Chapter 8 Section 1 Problem 5
If
x
is the sample mean of a random sample of size n from a normal population with known
variance
2
, a
( )
100 1 %
−
CI on
is given by
/2 /2
//x z n x z n
− +
, where
/2
z
is the upper
100 / 2
percentage point of the standard normal distribution.
(a) What value of
/2
z
gives 98% confidence?
(b) What value of
/2
z
gives 80% confidence?
(c) What value of
/2
z
gives 75% confidence?
SOLUTION
(a)
(c)
Reserve Problems Chapter 8 Section 1 Problem 6
A random sample has been taken from a normal distribution and the following confidence
intervals constructed using the same data: (25.53, 37.87) and (23.59, 39.81).
(a) What is the value of the sample mean?
(b) One of these intervals is a 99% CI and the other is a 95% CI. Which one is the 95% CI?
SOLUTION
Reserve Problems Chapter 8 Section 1 Problem 7
Following are two confidence interval estimates of the mean
of the cycles to failure of an
automotive door latch mechanism (the test was conducted at an elevated stress level to accelerate
the failure).
3062.9 3153.7, 3048.5 3168.1
(a) What is the value of the sample mean cycles to failure?
(b) One of these intervals is a 95% CI and the other is a 99% CI. Which one is the 95% CI?
SOLUTION
(a)
( )
13062.9 3153.7 / 2 3108.3x= + =
Reserve Problems Chapter 8 Section 1 Problem 8
The life in hours of a 75-watt light bulb is known to be normally distributed with
25
=
hours.
A random sample of 21 bulbs has a mean life of
1014x=
hours.
(a) Construct a 95% two-sided confidence interval on the mean life.
(b) Construct a 95% lower-confidence bound on the mean life. Compare the lower bound of this
confidence interval with the one in part (a).
SOLUTION
(a) 95% two-sided CI on the true mean life of a 75-watt light bulb.
25
=
(b) 95% one-sided CI on the true mean life of a 75-watt light bulb.
25
=
Reserve Problems Chapter 8 Section 1 Problem 9
Dairy cows at large commercial farms often receive injections of bST (Bovine Somatotropin), a
hormone used to spur milk production. Bauman et al. (Journal of Dairy Science, 1989) reported
that 12 cows given bST produced an average of 28.0 kg/d of milk. Assume that the standard
deviation of milk production is 2.25 kg/d.
(a) Find a 99% confidence interval for the true mean milk production.
(b) If the farms want the confidence interval to be no wider than ±0.95 kg/d, what level of
confidence would they need to use?
SOLUTION
(a) 99% CI for
,
12n=
,
2.25
=
,
28.0x=
,
/2 0.005 2.576zz
==
Reserve Problems Chapter 8 Section 2 Problem 1
Consider results of 20 randomly chosen people who have run a marathon. Their times, in
minutes, are as follows: 137, 146, 150, 163, 166, 179, 186, 193, 196, 213, 218, 225, 236, 239, 254, 269, 271,
279, 282, 295.
Calculate a 90% upper confidence bound on the mean time of the race. Assume distribution to be
normal.
SOLUTION
Consider results of 20 randomly chosen people who have run a marathon. Their times, in
SOLUTION
, 1 0.1,19
20, 214.85, 49.6538, 1.328
n
n x s t t
−
= = = = =
Reserve Problems Chapter 8 Section 2 Problem 2
Calculate a 90% upper confidence bound on the mean time of the ultra-marathon
race. Assume distribution to be normal.
Women:
1674, 1622, 1637, 1838, 1964, 1836, 1728, 1913, 1885,
1932;
Men: 1852, 2403, 2383, 2336, 2597, 2581, 2169, 2120.
Calculate a 95% confidence interval on the mean for women and men separately. Assume
distribution to be normal.
SOLUTION
Reserve Problems Chapter 8 Section 2 Problem 3
Twenty-five wooden beams were ordered for a construction project. The sample mean and the
sample standard deviation were measured:
310 cmx=
,
6 cms=
. Calculated confidence
interval for the mean is
307.95;312.05
.
Which confidence level was chosen? Assume distribution to be normal.
SOLUTION
25, 310, 6n x s= = =
Reserve Problems Chapter 8 Section 2 Problem 4
The number of steps to cover the same distance was measured for 41 randomly chosen people.
The sample mean is
2220x=
steps and the sample standard deviation is
444s=
steps. Assume
distribution to be normal.
a) Construct a 90% two-sided confidence interval on the mean.
b) Construct a 90% lower confidence bound on the mean.
c) Construct a 90% upper confidence bound on the mean.
SOLUTION
41, 2220, 444n x s= = =
Reserve Problems Chapter 8 Section 2 Problem 5
A random sample has been taken from a normal distribution. Output from a software package
follows:
Variable
N
Mean
SE Mean
StDev
Variance
Sum
x
?
?
1.55
6.02
?
757.90
(a) Fill in the missing quantities.
(b) Find a 95% CI on the population mean.
SOLUTION
(b) 95% confidence interval on mean
Reserve Problems Chapter 8 Section 2 Problem 6
A particular brand of diet margarine was analysed to determine the level of polyunsaturated fatty
acid (in percentages). A sample of six packages resulted in the following data: 16.8, 17.2, 17.4,
16.9, 16.5, 17.1.
(a) Choose the correct normal probability plot for these data.
A
B
C
D
Is it reasonable to assume that the level of polyunsaturated fatty acid is normally distributed?
(b) Calculate a 99% confidence interval on the mean
.
SOLUTION
(a) The data appear to be normally distributed based on the normal probability plot below.
(b) 99% CI on the mean level of polyunsaturated fatty acid.
Reserve Problems Chapter 8 Section 2 Problem 7
Ishikawa et al. (Journal of Bioscience and Bioengineering, 2012) studied the adhesion of various
biofilms to solid surfaces for possible use in environmental technologies. Adhesion assay is
conducted by measuring absorbance at
590
A
. Suppose that for bacterial strain Acinetobacter, five
measurements gave readings of 2.69, 5.76, 2.67, 1.62 and 4.12 dyne-cm2.
(a) Choose the correct normal probability plot for these data.
A
B
C
D
Is it reasonable to assume that the data are normally distributed?
(b) Find a 95% confidence interval for the mean adhesion.
SOLUTION
(a) The following normal probability plot is used to evaluate the distribution.
(b) 95% CI for
,
5n=
,
3.372x=
,
1.604s=
,
Reserve Problems Chapter 8 Section 3 Problem 1
Determine the confidence level for a one-sided limit that requires the following percentiles.
a)
218.55
=
, degrees of freedom = 12, lower confidence limit.
b)
26.84
=
, degrees of freedom = 19, lower confidence limit.
c)
238.08
=
, degrees of freedom = 23, lower confidence limit.
SOLUTION
Reserve Problems Chapter 8 Section 3 Problem 2
A lady on a diet measures her weight every week. The sample standard deviation is 0.7 kg, the
number of measurements is 7. Construct a 95% two-sided confidence interval for
2
.
SOLUTION
7, 0.7ns==
Reserve Problems Chapter 8 Section 3 Problem 3
In the 16/17 season, Valencia CF (Spanish soccer club) had 19 home games. The number of fans
visited 10 football matches at the Mestalla stadium (home stadium of Valencia CF) is shown in
the table below.
43602
34456
42070
20512
30823
27446
43333
20570
25694
32221
Enter your answers in scientific notation using three significant figures ( e. g. 9.87E+ 6).
a) Construct a 90% two-sided confidence interval for
2
.
b) Calculate 90% lower confidence limit for
2
.
c) Calculate 90% upper confidence limit for
2
.
SOLUTION
10, 8787.52ns==
Reserve Problems Chapter 8 Section 3 Problem 4
The percentage of alcohol in the cough syrup was measured in 101 randomly selected samples.
The sample standard deviation is
0.5s=
.
Construct a 90% two-sided confidence interval for
.
SOLUTION
2 2 2 2
/2, 1 0.05,100 1 /2, 1 0.95,100
0.1, 124.34, 77.93
nn
− − −
= = = = =
Reserve Problems Chapter 8 Section 3 Problem 5
Determine the
2
percentile that is required to construct each of the following CIs.
(a) Confidence level = 95%, degrees of freedom = 23, one-sided (upper)
(b) Confidence level = 99%, degrees of freedom = 13, one-sided (lower)
(c) Confidence level = 90%, degrees of freedom = 24, two-sided.
SOLUTION
(a) 95% upper CI and degrees of freedom = 23
Reserve Problems Chapter 8 Section 3 Problem 6
A rivet is to be inserted into a hole. A random sample of
15n=
parts is selected, and the hole
diameter is measured. The sample standard deviation of the hole diameter measurements is
0.005s=
millimeters.
Find a 99% lower confidence bound on the standard deviation.
SOLUTION
99% lower confidence bound for
2
.
Reserve Problems Chapter 8 Section 3 Problem 7