Applied Statistics and Probability for Engineers

Douglas C. Montgomery

Chapter 8

Statistical Intervals for a Single Sample - all with Video Answers

Educators

Section 1

Confidence Interval on the Mean of a Normal Distribution, Variance Known

Problem 1

For a normal population with known variance $\sigma^{2}$, answer the following questions:
(a) What is the confidence level for the interval $\bar{x}-2.14 \sigma / \sqrt{n}$ $\leq \mu \leq \bar{x}+2.14 \sigma / \sqrt{n} ?$
(b) What is the confidence level for the interval $\bar{x}-2.49 \sigma / \sqrt{n}$ $\bar{x}-2.49 \sigma / \sqrt{n} \leq \mu \leq \bar{x}+2.49 \sigma / \sqrt{n} ?$
(c) What is the confidence level for the interval $\bar{x}-1.85 \sigma / \sqrt{n}$ $\leq \mu \leq \bar{x}+1.85 \sigma / \sqrt{n} ?$
(d) What is the confidence level for the interval $\mu \leq \bar{x}+$ $2.00 \sigma / \sqrt{n} ?$
(e) What is the confidence level for the interval $\bar{x}-1.96 \sigma / \sqrt{n} \leq \mu ?$

Willis James

Willis James

Numerade Educator

Problem 2

For a normal population with known variance $\sigma^{2}$ :
(a) What value of $z_{\alpha / 2}$ in Equation $8-5$ gives $98 \%$ confidence?
(b) What value of $z_{\alpha / 2}$ in Equation $8-5$ gives $80 \%$ confidence?
(c) What value of $z_{\alpha / 2}$ in Equation $8-5$ gives $75 \%$ confidence?

Willis James

Numerade Educator

Problem 3

Consider the one-sided confidence interval expressions for a mean of a normal population.
(a) What value of $z_{\alpha}$ would result in a $90 \%$ CI?
(b) What value of $z_{\alpha}$ would result in a $95 \%$ CI?
(c) What value of $z_{\alpha}$ would result in a $99 \%$ CI?

Willis James

Numerade Educator

Problem 4

A confidence interval estimate is desired for the gain in a circuit on a semiconductor device. Assume that gain is normally distributed with standard deviation $\mathrm{s}=20 .$
(a) Find a $95 \% \mathrm{CI}$ for $\mathrm{m}$ when $n=10$ and $\bar{x}=1000$.
(b) Find a $95 \%$ CI for $m$ when $n=25$ and $\bar{x}=1000$.
(c) Find a $99 \% \mathrm{CI}$ for $\mathrm{m}$ when $n=10$ and $\bar{x}=1000$.
(d) Find a $99 \%$ CI for $m$ when $n=25$ and $\bar{x}=1000$.
(e) How does the length of the CIs computed change with the changes in sample size and confidence level?

Willis James

Numerade Educator

Problem 5

A random sample has been taken from a normal distribution and the following confidence intervals constructed using the same data: (38.02,61.98) and (39.95,60.05)
(a) What is the value of the sample mean?
(b) One of these intervals is a $95 \% \mathrm{CI}$ and the other is a $90 \%$ CI. Which one is the $95 \%$ CI and why?

Willis James

Numerade Educator

Problem 6

A random sample has been taken from a normal distribution and the following confidence intervals constructed using the same data: (37.53,49.87) and (35.59,51.81)
(a) What is the value of the sample mean?
(b) One of these intervals is a $99 \% \mathrm{CI}$ and the other is a $95 \%$
CI. Which one is the $95 \%$ CI and why?

Willis James

Numerade Educator

Problem 7

Consider the gain estimation problem in Exercise $8-4$
(a) How large must $n$ be if the length of the $95 \% \mathrm{CI}$ is to be $40 ?$
(b) How large must $n$ be if the length of the $99 \%$ CI is to be $40 ?$

Willis James

Numerade Educator

Problem 8

Following are two confidence interval estimates of the mean $\mathrm{m}$ of the cycles to failure of an automotive door latch mechanism (the test was conducted at an elevated stress level to accelerate the failure).
$$
3124.9 \leq \mu \leq 3215.7 \quad 3110.5 \leq \mu \leq 3230.1
$$
(a) What is the value of the sample mean cycles to failure?
(b) The confidence level for one of these CIs is $95 \%$ and for the other is $99 \%$. Both CIs are calculated from the same sample data. Which is the $95 \%$ CI? Explain why.

Willis James

Numerade Educator

Problem 9

Suppose that $n=100$ random samples of water from a freshwater lake were taken and the calcium concentration (milligrams per liter) measured. A $95 \%$ CI on the mean calcium concentration is $0.49 \leq \mu \leq 0.82$.
(a) Would a $99 \%$ CI calculated from the same sample data be longer or shorter?
(b) Consider the following statement: There is a $95 \%$ chance that $\mu$ is between 0.49 and $0.82 .$ Is this statement correct? Explain your answer.
(c) Consider the following statement: If $n=100$ random samples of water from the lake were taken and the $95 \% \mathrm{CI}$ on $\mu$ computed, and this process were repeated 1000 times, 950 of the CIs would contain the true value of $\mu$. Is this statement correct? Explain your answer.

Willis James

Numerade Educator

Problem 10

Past experience has indicated that the breaking strength of yarn used in manufacturing drapery material is normally distributed and that $\sigma=2$ psi. A random sample of nine specimens is tested, and the average breaking strength is found to be 98 psi. Find a $95 \%$ two-sided confidence interval on the true mean breaking strength.

Willis James

Numerade Educator

Problem 11

The yield of a chemical process is being studied. From previous experience, yield is known to be normally distributed and $\sigma=3$. The past five days of plant operation have resulted in the following percent yields: $91.6,88.75,90.8,89.95,$ and $91.3 .$ Find a $95 \%$ two-sided confidence interval on the true mean yield.

Willis James

Numerade Educator

Problem 12

The diameter of holes for a cable harness is known to have a normal distribution with $\sigma=0.01$ inch. A random sample of size 10 yields an average diameter of 1.5045 inch. Find a $99 \%$ two-sided confidence interval on the mean hole diameter.

Willis James

Numerade Educator

Problem 13

A manufacturer produces piston rings for an automobile engine. It is known that ring diameter is normally distributed with $\sigma=0.001$ millimeters. A random sample of 15 rings has a mean diameter of $\bar{x}=74.036$ millimeters.
(a) Construct a $99 \%$ two-sided confidence interval on the mean piston ring diameter.
(b) Construct a $99 \%$ lower-confidence bound on the mean piston ring diameter. Compare the lower bound of this confidence interval with the one in part (a).

Willis James

Numerade Educator

Problem 14

The life in hours of a 75 -watt light bulb is known to be normally distributed with $\sigma=25$ hours. A random sample of 20 bulbs has a mean life of $\bar{x}=1014$ 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).

Willis James

Numerade Educator

Problem 15

A civil engineer is analyzing the compressive strength of concrete. Compressive strength is normally distributed with $\sigma^{2}=1000(\mathrm{psi})^{2} .$ A random sample of 12 specimens has a mean compressive strength of $\bar{x}=3250$ psi.
(a) Construct a $95 \%$ two-sided confidence interval on mean compressive strength.
(b) Construct a $99 \%$ two-sided confidence interval on mean compressive strength. Compare the width of this confidence interval with the width of the one found in part (a).

Willis James

Numerade Educator

Problem 16

Suppose that in Exercise $8-14$ we wanted the error in estimating the mean life from the two-sided confidence interval to be five hours at $95 \%$ confidence. What sample size should be used?

Willis James

Numerade Educator

Problem 17

Suppose that in Exercise $8-14$ you wanted the total width of the two-sided confidence interval on mean life to be six hours at $95 \%$ confidence. What sample size should be used?

Willis James

Numerade Educator

Problem 18

Suppose that in Exercise $8-15$ it is desired to estimate the compressive strength with an error that is less than 15 psi at $99 \%$ confidence. What sample size is required?

Willis James

Numerade Educator

Problem 19

By how much must the sample size $n$ be increased if the length of the $\mathrm{CI}$ on $\mu$ in Equation $8-5$ is to be halved?

Willis James

Numerade Educator

Problem 20

If the sample size $n$ is doubled, by how much is the length of the CI on $\mu$ in Equation $8-5$ reduced? What happens to the length of the interval if the sample size is increased by a factor of four?

Willis James

Numerade Educator

Problem 21

An article in the Journal of Agricultural Science ["The Use of Residual Maximum Likelihood to Model Grain Quality Characteristics of Wheat with Variety, Climatic and Nitrogen Fertilizer Effects" (1997, Vol. 128, pp. $135-142$ ) ] investigated means of wheat grain crude protein content (CP) and Hagberg falling number (HFN) surveyed in the United Kingdom. The analysis used a variety of nitrogen fertilizer applications $(\mathrm{kg} \mathrm{N} / \mathrm{ha}),$ temperature $\left({ }^{\circ} \mathrm{C}\right),$ and total monthly rainfall $(\mathrm{mm})$. The following data below describe temperatures for wheat grown at Harper Adams Agricultural College between 1982 and $1993 .$ The temperatures measured in June were obtained as follows:
$$
\begin{array}{llllll}
15.2 & 14.2 & 14.0 & 12.2 & 14.4 & 12.5 \\
14.3 & 14.2 & 13.5 & 11.8 & 15.2 &
\end{array}
$$
Assume that the standard deviation is known to be $\sigma=0.5$
(a) Construct a $99 \%$ two-sided confidence interval on the mean
temperature.
(b) Construct a $95 \%$ lower-confidence bound on the mean temperature.
(c) Suppose that you wanted to be $95 \%$ confident that the error in estimating the mean temperature is less than 2 degrees Celsius. What sample size should be used?
(d) Suppose that you wanted the total width of the two-sided confidence interval on mean temperature to be 1.5 degrees Celsius at $95 \%$ confidence. What sample size should be used?

Willis James

Numerade Educator

Problem 22

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 $\mathrm{A}_{590} .$ Suppose that for the bacterial strain Acinetobacter, five measurements gave readings of 2.69,5.76,2.67,1.62 and 4.12 dyne-cm $^{2}$. Assume that the standard deviation is known to be 0.66 dyne-cm $^{2}$.
(a) Find a $95 \%$ confidence interval for the mean adhesion.
(b) If the scientists want the confidence interval to be no wider than 0.55 dyne-cm $^{2}$, how many observations should they take?

Willis James

Numerade Educator

Problem 23

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 $28.0 \mathrm{~kg} / \mathrm{d}$ of milk. Assume that the standard deviation of milk production is $2.25 \mathrm{~kg} / \mathrm{d}$
(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 $\pm 1.25 \mathrm{~kg} / \mathrm{d},$ what level of confidence would they need to use?

Willis James

Numerade Educator