Applied Statistics and Probability for Engineers, Student Workbook with Solutions

Douglas C. Montgomery, George C. Runger

Chapter 9

Tests of Hypotheses for a Single Sample - all with Video Answers

Educators

Chapter Questions

Problem 1

In each of the following situations, state whether it is a correctly stated hypothesis testing problem and why.
(a) $H_0: \mu=25, H_1: \mu \neq 25$
(b) $H_0: \sigma>10, H_1: \sigma=10$
(c) $H_0: \bar{x}=50, H_1: \bar{x} \neq 50$
(d) $H_0: p=0.1, H_1: p=0.5$
(c) $H_0: s=30, H_1: s>30$

Tyler Moulton

Tyler Moulton

Numerade Educator

Problem 2

A textile fiber manufacturer is investigating a new drapery yarn, which the company claims has a mean thread elongation of 12 kilograms with a standard deviation of 0.5 kilograms. The company wishes to test the hypothesis $H_0: \mu=12$ against $H_1: \mu<12$, using a random sample of four specimens.
(a) What is the type I error probability if the critical region is defined as $\bar{x}<11.5$ kilograms?
(b) Find $\beta$ for the case where the true mean elongation is 11.25 kilograms.

Tyler Moulton

Numerade Educator

Problem 2

if we wanted the power of the test to be at least 0.9 ?
(d) Explain how the question in part (a) could be answered by constructing a two-sided confidence interval on the mean female body temperature.
(e) Is there evidence to support the assumption that female body temperature is normally distributed?

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Problem 3

Repeat Exercise 9-2 using a sample size of $n=16$ and the same critical region.

Tyler Moulton

Numerade Educator

Problem 4

In Exercise 9-2, find the boundary of the critical region if the type I error probability is specified to be $\alpha=0.01$.

Tyler Moulton

Numerade Educator

Problem 5

In Exercise 9-2, find the boundary of the critical region if the type 1 error probability is specified to be 0.05 .

Tyler Moulton

Numerade Educator

Problem 6

The heat evolved in calories per gram of a cement mixture is approximately normally distributed. The mean is thought to be 100 and the standard deviation is 2 . We wish to test $H_0: \mu=100$ versus $H_1: \mu \neq 100$ with a sample of $n=9$ specimens.
(a) If the acceptance region is defined as $98.5 \leq \bar{x} \leq 101.5$, find the type 1 error probability $\alpha$.
(b) Find $\beta$ for the case where the true mean heat evolved is 103.
(c) Find $\beta$ for the case where the true mean heat evolved is 105. This value of $\beta$ is smaller than the one found in part (b) above. Why?

Tyler Moulton

Numerade Educator

Problem 7

A manufacturer of semiconductor devices takes a random sample of size $n$ of chips and tests them, classifying each chip as defective or nondefective. Let $X_i=0$ if the chip is nondefective and $X_i=1$ if the chip is defective. The sample fraction defective is

$$
\hat{p}_i=\frac{X_1+X_2+\cdots+X_n}{n}
$$

What are the sampling distribution, the sample mean, and sample variance estimates of $\hat{p}$ when
(a) The sample size is $n=50$ ?
(b) The sample size is $n=80$ ?
(c) The sample size is $n=100$ ?
(d) Compare your answers to parts (a)-(c) and comment on the effect of sample size on the variance of the sampling distribution.

Amany Waheeb

Numerade Educator

Problem 7

Repeat Exercise 9-6 using a sample size of $n=5$ and the same acceptance region.

Tyler Moulton

Numerade Educator

Problem 8

A consumer products company is formulating a new shampoo and is interested in foam height (in millimeters). Foam height is approximately normally distributed and has a standard deviation of 20 millimeters. The company wishes to
test $H_0: \mu=175$ millimeters versus $H_1: \mu>175$ millimeters, using the results of $n=10$ samples.
(a) Find the type I error probability $\alpha$ if the critical region is $\bar{x}>185$.
(b) What is the probability of type II error if the true mean foam height is 195 millimeters?

Tyler Moulton

Numerade Educator

Problem 9

In Exercise 9-8, suppose that the sample data result in $\bar{x}=190$ millimeters.
(a) What conclusion would you reach?
(b) How "unusual" is the sample value $\bar{x}=190$ millimeters if the true mean is really 175 millimeters? That is, what is the probability that you would observe a sample average as large as 190 millimeters (or larger), if the true mean foam height was really 175 millimeters?

Victor Salazar

Numerade Educator

Problem 10

Repeat Exercise $9-8$ assuming that the sample size is $n=16$ and the boundary of the critical region is the same.

Tyler Moulton

Numerade Educator

Problem 11

Consider Exercise 9-8, and suppose that the sample size is increased to $n=16$.
(a) Where would the boundary of the critical region be placed if the type I error probability were to remain equal to the value that it took on when $n=10$ ?
(b) Using $n=16$ and the new critical region found in part (a), find the type II error probability $\beta$ if the true mean foam height is 195 millimeters.
(c) Compare the value of $\beta$ obtained in part (b) with the value from Exercise $9-8$ (b). What conclusions can you draw?

Tyler Moulton

Numerade Educator

Problem 12

A manufacturer is interested in the output voltage of a power supply used in a PC. Output voltage is assumed to be normally distributed, with standard deviation 0.25 Volts, and the manufacturer wishes to test $H_0: \mu=5$ Volts against $H_1: \mu \neq 5$ Volts, using $n=8$ units.
(a) The acceptance region is $4.85 \leq \bar{x} \leq 5.15$. Find the value of $\alpha$.
(b) Find the power of the test for detecting a true mean output voltage of 5.1 Volts.

Tyler Moulton

Numerade Educator

Problem 13

Rework Exercise 9-12 when the sample size is 16 and the boundaries of the acceptance region do not change.

Tyler Moulton

Numerade Educator

Problem 14

Consider Exercise 9-12, and suppose that the manufacturer wants the type I error probability for the test to be $\alpha=0.05$. Where should the acceptance region be located?

Jacquelinne S. Mejia Sandoval

Jacquelinne S. Mejia Sandoval

Numerade Educator

Problem 15

If we plot the probability of accepting $H_0: \mu=\mu_0$ versus various values of $\mu$ and connect the points with a smooth curve, we obtain the operating characteristic curve (or the OC curve) of the test procedure. These curves are used extensively in industrial applications of hypothesis testing to display the sensitivity and relative performance of the test. When the true mean is really equal to $\mu_0$, the probability of accepting $H_0$ is $1-\alpha$. Construct an OC curve for Exercise 9-8, using values of the true mean $\mu$ of $178,181,184,187,190$, 193, 196, and 199.

Jameson Kuper

Numerade Educator

Problem 16

Convert the OC curve in Exercise 9-15 into a plot of the power function of the test.

Jameson Kuper

Numerade Educator

Problem 17

A random sample of 500 registered voters in Phoenix is asked if they favor the use of oxygenated fuels year-round to reduce air pollution. If more than 400 voters respond positively, we will conclude that at least $60 \%$ of the voters favor the use of these fuels.
(a) Find the probability of type I error if exactly $60 \%$ of the voters favor the use of these fuels.
(b) What is the type II error probability $\beta$ if $75 \%$ of the voters favor this action?
Hint: use the normal approximation to the binomial.

Tyler Moulton

Numerade Educator

Problem 18

The proportion of residents in Phoenix favoring the building of toll roads to complete the freeway system is believed to be $p=0.3$. If a random sample of 10 residents shows that 1 or fewer favor this proposal, we will conclude that $p<0.3$.
(a) Find the probability of type I error if the true proportion is $p=0.3$.
(b) Find the probability of committing a type II error with this procedure if $p=0.2$.
(c) What is the power of this procedure if the true proportion is $p=0.2$ ?

Tyler Moulton

Numerade Educator

Problem 19

The proportion of adults living in Tempe, Arizona, who are college graduates is estimated to be $p=0.4$. To test this hypothesis, a random sample of 15 Tempe adults is selected. If the number of college graduates is between 4 and 8 , the hypothesis will be accepted; otherwise, we will conclude that $p \neq 0.4$.
(a) Find the type I error probability for this procedure, assuming that $p=0.4$.
(b) Find the probability of committing a type II error if the true proportion is really $p=0.2$.

Tyler Moulton

Numerade Educator

Problem 20

The mean water temperature downstream from a power plant cooling tower discharge pipe should be no more than $100^{\circ} \mathrm{F}$. Past experience has indicated that the standard deviation of temperature is $2^{\circ} \mathrm{F}$. The water temperature is measured on nine randomly chosen days, and the average temperature is found to be $98^{\circ} \mathrm{F}$.
(a) Should the water temperature be judged acceptable with $\alpha=0.05$ ?
(b) What is the $P$-value for this test?
(c) What is the probability of accepting the null hypothesis at $\alpha=0.05$ if the water has a true mean temperature of $104^{\circ} \mathrm{F}$ ?

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 21

Reconsider the chemical process yield data from Exercise 8-9. Recall that $\sigma=3$, yield is normally distributed and that $n=5$ observations on yield are $91.6 \%, 88.75 \%, 90.8 \%$, $89.95 \%$, and $91.3 \%$. Use $\alpha=0.05$.
(a) Is there evidence that the mean yield is not $90 \%$ ?
(b) What is the $P$-value for this test?
(c) What sample size would be required to detect a true mean yield of $85 \%$ with probability 0.95 ?
(d) What is the type II error probability if the true mean yield is $92 \%$ ?
(e) Compare the decision you made in part (c) with the $95 \%$ Cl on mean yield that you constructed in Exercise 8-7.

Evelyn Cunningham

Evelyn Cunningham

Numerade Educator

Problem 22

A manufacturer produces crankshafts for an automobile engine. The wear of the crankshaft after 100,000 miles ( 0.0001 inch) is of interest because it is likely to have an impact on warranty claims. A random sample of $n=15$ shafts is tested and $\bar{x}=2.78$. It is known that $\sigma=0.9$ and that wear is normally distributed.
(a) Test $H_0: \mu=3$ versus $H_0: \mu \neq 3$ using $\alpha=0.05$.
(b) What is the power of this test if $\mu=3.25$ ?
(c) What sample size would be required to detect a true mean of 3.75 if we wanted the power to be at least 0.9 ?

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 23

A melting point test of $n=10$ samples of a binder used in manufacturing a rocket propellant resulted in $\bar{x}=154.2^{\circ} \mathrm{F}$. Assume that melting point is normally distributed with $\sigma=1.5^{\circ} \mathrm{F}$.
(a) Test $H_0: \mu=155$ versus $H_0: \mu \neq 155$ using $\alpha=0.01$.
(b) What is the $P$-value for this test?
(c) What is the $\beta$-error if the true mean is $\mu=150$ ?
(d) What value of $n$ would be required if we want $\beta<0.1$ when $\mu=150$ ? Assume that $\alpha=0.01$.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 24

The life in hours of a battery is known to be approximately normally distributed, with standard deviation $\sigma=1.25$ hours. A random sample of 10 batteries has a mean life of $\bar{x}=40.5$ hours.
(a) Is there evidence to support the claim that battery life exceeds 40 hours? Use $\alpha=0.05$.
(b) What is the $P$-value for the test in part (a)?
(c) What is the $\beta$-error for the test in part (a) if the true mean life is 42 hours?
(d) What sample size would be required to ensure that $\beta$ does not exceed 0.10 if the true mean life is 44 hours?
(c) Explain how you could answer the question in part (a) by calculating an appropriate confidence bound on life.

Robin Corrigan

Numerade Educator

Problem 25

An engineer who is studying the tensile strength of a steel alloy intended for use in golf club shafts knows that tensile strength is approximately normally distributed with $\sigma=60 \mathrm{psi}$. A random sample of 12 specimens has a mean tensile strength of $\bar{x}=3250 \mathrm{psi}$.
(a) Test the hypothesis that mean strength is 3500 psi. Use $\alpha=0.01$.
(b) What is the smallest level of significance at which you would be willing to reject the null hypothesis?
(c) Explain how you could answer the question in part (a) with a two-sided confidence interval on mean tensile strength.

Robin Corrigan

Numerade Educator

Problem 26

Suppose that in Exercise 9-25 we wanted to reject the null hypothesis with probability at least 0.8 if mean strength $\mu=3500$. What sample size should be used?

Willis James

Numerade Educator

Problem 27

Supercavitation is a propulsion technology for undersea vehicles that can greatly increase their speed. It occurs above approximately 50 meters per second, when pressure drops sufficiently to allow the water to dissociate into water vapor, forming a gas bubble behind the vehicle. When the gas bubble completely encloses the vehicle, supercavitation is said to occur. Eight tests were conducted on a scale model of an undersea vehicle in a towing basin with the average observed speed $\bar{x}=102.2$ meters per second. Assume that speed is normally distributed with known standard deviation $\sigma=$ 4 meters per second.
(a) Test the hypotheses $H_0: \mu=100$ versus $H_1: \mu<100$ using $\alpha=0.05$.
(b) Compute the power of the test if the true mean speed is as low as 95 meters per second.
(c) What sample size would be required to detect a true mean speed as low as 95 meters per second if we wanted the power of the test to be at least 0.85 ?
(d) Explain how the question in part (a) could be answered by constructing a one-sided confidence bound on the mean speed.

Robin Corrigan

Numerade Educator

Problem 28

A bearing used in an automotive application is suppose to have a nominal inside diameter of 1.5 inches. A random sample of 25 bearings is selected and the average inside diameter of these bearings is 1.4975 inches. Bearing diameter is known to be normally distributed with standard deviation $\sigma=0.01$ inch.
(a) Test the hypotheses $H_0: \mu=1.5$ versus $H_1: \mu \neq 1.5$ using $\alpha=0.01$.
(b) Compute the power of the test if the true mean diameter is 1.495 inches.
(c) What sample size would be required to detect a true mean diameter as low as 1.495 inches if we wanted the power of the test to be at least 0.9 ?
(d) Explain how the question in part (a) could be answered by constructing a two-sided confidence interval on the mean diameter.

Robin Corrigan

Numerade Educator

Problem 29

Medical researchers have developed a new artificial heart constructed primarily of titanium and plastic. The heart will last and operate almost indefinitely once it is implanted in the patient's body, but the battery pack needs to be recharged about every four hours. A random sample of 50 battery packs is selected and subjected to a life test. The average life of these batteries is 4.05 hours. Assume that battery life is normally distributed with standard deviation $\sigma=0.2$ hour.
(a) Is there evidence to support the claim that mean battery life exceeds 4 hours? Use $\alpha=0.05$.
(b) Compute the power of the test if the true mean battery life is 4.5 hours.
(c) What sample size would be required to detect a true mean battery life of 4.5 hours if we wanted the power of the test to be at least 0.9 ?
(d) Explain how the question in part (a) could be answered by constructing a one-sided confidence bound on the mean life.

Robin Corrigan

Numerade Educator

Problem 30

An article in the ASCE Journal of Energy Engineering (1999, Vol. 125, pp. 59-75) describes a study of the thermal inertia properties of autoclaved aerated concrete used as a building material. Five samples of the material were tested in a structure, and the average interior temperature ( ${ }^{\circ} \mathrm{C}$ ) reported was as follows: $23.01,22.22,22.04,22.62$, and 22.59 .
(a) Test the hypotheses $H_0: \mu=22.5$ versus $H_1: \mu \neq 22.5$, using $\alpha=0.05$. Find the $P$-value.
(b) Is there evidence to support the assumption that interior temperature is normally distributed?
(c) Compute the power of the test if the true mean interior temperature is as high as 22.75 .
(d) What sample size would be required to detect a true mean interior temperature as high as 22.75 if we wanted the power of the test to be at least 0.9 ?
(e) Explain how the question in part (a) could be answered by constructing a two-sided confidence interval on the mean interior temperature.

Amany Waheeb

Numerade Educator

Problem 31

A 1992 article in the Journal of the American Medical Association ("A Critical Appraisal of 98.6 Degrees F, the Upper Limit of the Normal Body Temperature, and Other Legacies of Carl Reinhold August Wundrlich") reported body temperature, gender, and heart rate for a number of subjects. The body temperatures for 25 female subjects follow: $97.8,97.2,97.4,97.6$, $97.8,97.9,98.0,98.0,98.0,98.1,98.2,98.3,98.3,98.4,98.4$, $98.4,98.5,98.6,98.6,98.7,98.8,98.8,98.9,98.9$, and 99.0 .
(a) Test the hypotheses $H_0: \mu=98.6$ versus $H_1: \mu \neq 98.6$, using $\alpha=0.05$. Find the $P$-value.
(b) Compute the power of the test if the true mean female body temperature is as low as 98.0.
(c) What sample size would be required to detect a true mean female body temperature as low as if we wanted the power of the test to be at least 0.9 ?
(d) Explain how the question in part (a) could be answered by constructing a two-sided confidence interval on the mean female body temperature.
(e) Is there evidence to support the assumption that female body temperature is normally distributed?

Robin Corrigan

Numerade Educator

Problem 32

Cloud scoding has boen studied for many decades as a weather modification procedure (for an interesting study of this subject, see the article in Technometrics by Simpson, Alsen, and Eden, "A Bayesian Analysis of a Multiplicative Treatment Effect in Weather Modification", Vol. 17, pp. 161166). The rainfall in acre-feet from 20 clouds that were selected at random and seeded with silver nitrate follows: 18.0 , $30.7,19.8,27.1,22.3,18.8,31.8,23.4,21.2,27.9$, 31.9, 27.1, $25.0,24.7,26.9,21.8,29.2,34.8,26.7$, and 31.6 .
(a) Can you support a claim that mean rainfall from seeded clouds exceeds 25 acre-feet? Use $\alpha=0.01$.
(b) Is there evidence that rainfall is normally distributed?
(c) Compute the power of the test if the true mean rainfall is 27 acre-feet.
(d) What sample size would be required to detect a true mean rainfall of 27.5 acre-feet if we wanted the power of the test to be at least 0.9 ?
(e) Explain how the question in part (a) could be answered by constructing a one-sided confidence bound on the mean diameter.

Akhil Choudhary

Akhil Choudhary

Numerade Educator

Problem 33

The sodium content of thirty 300 -gram boxes of organic corn flakes was determined. The data (in milligrams) are as follows: $131.15,130.69,130.91,129.54,129.64,128.77,130.72$, $128.33,128.24,129.65,130.14,129.29,128.71,129.00,129.39$, $130.42,129.53,130.12,129.78,130.92,131.15,130.69,130.91$, $129.54,129.64,128.77,130.72,128.33,128.24$, and 129.65 .
(a) Can you support a claim that mean sodium content of this brand of cornflakes is 130 milligrams? Use $\alpha=0.05$.
(b) Is there evidence that sodium content is normally distributed?
(c) Compute the power of the test if the true mean sodium content is 130.5 miligrams.
(d) What sample size would be required to detect a true mean sodium content of 130.1 milligrams if we wanted the power of the test to be at least 0.75 ?
(e) Explain how the question in part (a) could be answered by constructing a two-sided confidence interval on the mean sodium content.

Harsh Gadhiya

Numerade Educator

Problem 34

Reconsider the tire testing experiment described in Exercise 8-22.
(a) The engineer would like to demonstrate that the mean life of this new tire is in excess of 60,000 kilometers. Formulate and test appropriate hypotheses, and draw conclusions using $\alpha=0.05$.
(b) Suppose that if the mean life is as long as 61,000 kilometers, the engineer would like to detect this difference with probability at least 0.90 . Was the sample size $n=16$ used in part (a) adequate? Use the sample standard deviation $s$ as an estimate of $\sigma$ in reaching your decision.

Amany Waheeb

Numerade Educator

Problem 35

Reconsider the Izod impact test on PVC pipe described in Exercise 8-23. Suppose that you want to use the data from this experiment to support a claim that the mean impact strength exceeds the ASTM standard (foot-pounds per inch). Formulate and test the appropriate hypotheses using $\alpha=0.05$.

Amany Waheeb

Numerade Educator

Problem 36

Reconsider the television tube brightness experiment in Exercise 8-24. Suppose that the design engineer believes that this tube will require 300 microamps of current to produce the desired brightness level. Formulate and test an appropriate hypothesis using $\alpha=0.05$. Find the $P$-value for this test. State any necessary assumptions about the underlying distribution of the data.

Amany Waheeb

Numerade Educator

Problem 37

Consider the baseball coefficient of restitution data first presented in Exercise 8-79.
(a) Does the data support the claim that the mean coefficient of restitution of baseballs exceeds 0.635 ? Use $\alpha=0.05$.
(b) What is the $P$-value of the test statistic computed in part (a)?
(c) Compute the power of the test if the true mean coefficient of restitution is as high as 0.64 .
(d) What sample size would be required to detect a true mean coefficient of restitution as high as 0.64 if we wanted the power of the test to be at least 0.75 ?

Amany Waheeb

Numerade Educator

Problem 38

Consider the dissolved oxygen concentration at TVA dams first presented in Exercise 8-81.
(a) Test the hypotheses $H_0: \mu=4$ versus $H_1: \mu \neq 4$. Use $\alpha=0.01$.
(b) What is the $P$-value of the test statistic computed in part (a)?
(c) Compute the power of the test if the true mean dissolved oxygen concentration is as low as 3 .
(d) What sample size would be required to detect a true mean dissolved oxygen concentration as low as 2.5 if we wanted the power of the test to be at least 0.9 ?

Rashmi Sinha

Numerade Educator

Problem 39

. Consider the cigar tar content data first presented in Exercise 8-82.
(a) Can you support a claim that mean tar content exceeds 1.5 ? Use $\alpha=0.05$
(b) What is the $P$-value of the test statistic computed in part (a)?
(c) Compute the power of the test if the true mean tar content is 1.6 .
(d) What sample size would be required to detect a true mean tar content of 1.6 if we wanted the power of the test to be at least 0.8 ?

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Problem 40

Exercise 6-22 gave data on the heights of female engineering students at ASU.
(a) Can you support a claim that mean height of female engineering students at ASU is 65 inches? Use $\alpha=0.05$
(b) What is the $P$-value of the test statistic computed in part (a)?
(c) Compute the power of the test if the true mean height is 62 inches.
(d) What sample size would be required to detect a true mean height of 64 inches if we wanted the power of the test to be at least 0.8 ?

Cerys Evans

Numerade Educator

Problem 41

Exercise 6-24 presented data on the concentration of suspended solids in lake water.
(a) Test the hypotheses $H_0: \mu=55$ versus $H_1: \mu \neq 55$, use $\alpha=0.05$.
(b) What is the $P$-value of the test statistic computed in part (a)?
(c) Compute the power of the test if the true mean concentration is as low as 50 .
(d) What sample size would be required to detect a true mean concentration as low as 50 if we wanted the power of the test to be at least 0.9 ?

Rashmi Sinha

Numerade Educator

Problem 42

Exercise 6-25 describes testing golf balls for an overall distance standard.
(a) Can you support a claim that mean distance achieved by this particular golf ball exceeds 280 yards? Use $\alpha=0.05$.
(b) What is the $P$-value of the test statistic computed in part (a)?
(c) Compute the power of the test if the true mean distance is 290 yards.
(e) What sample size would be required to detect a true mean distance of 290 yards if we wanted the power of the test to be at least 0.8 ?

Amany Waheeb

Numerade Educator

Problem 43

Consider the rivet holes from Exercise 8-35. If the standard deviation of hole diameter exceeds 0.01 millimeters, there is an unacceptably high probability that the rivet will not fit. Recall that $n=15$ and $s=0.008$ millimeters.
(a) Is there strong evidence to indicate that the standard deviation of hole diameter exceeds 0.01 millimeters? Use $\alpha=$ 0.01 . State any necessary assumptions about the underlying distribution of the data.
(b) Find the $P$-value for this test.
(c) If $\sigma$ is really as large as 0.0125 millimeters, what sample size will be required to defect this with power of at least 0.8 ?

Amany Waheeb

Numerade Educator

Problem 44

Recall the sugar content of the syrup in canned peaches from Exercise 8-36. Suppose that the variance is thought to be $\sigma^2=18$ (milligrams) ${ }^2$. A random sample of $n=10$ cans yields a sample standard deviation of $s=4.8$ milligrams.
(a) Test the hypothesis $H_0: \sigma^2=18$ versus $H_1: \sigma^2 \neq 18$ using $\alpha=0.05$.
(b) What is the $P$-value for this test?
(c) Discuss how part (a) could be answered by constructing a $95 \%$ two-sided confidence interval for $\sigma$.

Amany Waheeb

Numerade Educator

Problem 45

Consider the tire life data in Exercise 8-22.
(a) Can you conclude, using $\alpha=0.05$, that the standard deviation of tire life exceeds 200 kilometers? State any necessary assumptions about the underlying distribution of the data.
(b) Find the $P$-value for this test.

Amany Waheeb

Numerade Educator

Problem 46

Consider the Izod impact test data in Exercise 8-23.
(a) Test the hypothesis that $\sigma=0.10$ against an alternative specifying that $\sigma \neq 0.10$, using $\alpha=0.01$, and draw a conclusion. State any necessary assumptions about the underlying distribution of the data.
(b) What is the $P$-value for this test?
(c) Could the question in part (a) have been answered by constructing a $99 \%$ two-sided confidence interval for $\sigma^2$ ?

Amany Waheeb

Numerade Educator

Problem 47

Reconsider the percentage of titanium in an alloy used in aerospace castings from Exercise 8-39. Recall that $s=0.37$ and $n=51$.
(a) Test the hypothesis $H_0: \sigma=0.25$ versus $H_1: \sigma \neq 0.25$ using $\alpha=0.05$. State any necessary assumptions about the underlying distribution of the data.
(b) Explain how you could answer the question in part (a) by constructing a $95 \%$ two-sided confidence interval for $\sigma$.

Amany Waheeb

Numerade Educator

Problem 48

Consider the hole diameter data in Exercise 8-35. Suppose that the actual standard deviation of hole diameter exceeds the hypothesized value by $50 \%$. What is the probability that this difference will be detected by the test described in Exercise 9-43?

Amany Waheeb

Numerade Educator

Problem 49

Consider the sugar content in Exercise 9-44. Suppose that the true variance is $\sigma^2=40$. How large a sample would be required to detect this difference with probability at least 0.90 ?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 50

In a random sample of 85 automobile engine crankshaft bearings, 10 have a surface finish roughness that exceeds the specifications. Does this data present strong evidence that the proportion of crankshaft bearings exhibiting excess surface roughness exceeds 0.10 ? State and test the appropriate hypotheses using $\alpha=0.05$.

Rashmi Sinha

Numerade Educator

Problem 51

Continuation of Exercise 9-50. If it is really the situation that $p=0.15$, how likely is it that the test procedure in Exercise 9-50 will not reject the null hypothesis? If
$p=0.15$, how large would the sample size have to be for us to have a probability of correctly rejecting the null hypothesis of 0.9 ?

Amany Waheeb

Numerade Educator

Problem 52

Reconsider the integrated circuits described in Exercise 8-48.
(a) Use the data to test $H_0: p=0.05$ versus $H_1: p \neq 0.05$. Use $\alpha=0.05$.
(b) Find the $P$-value for the test.

Joshua Argo

Numerade Educator

Problem 53

Consider the defective circuit data in Exercise 8-48.
(a) Do the data support the claim that the fraction of defective units produced is less than 0.05 , using $\alpha=0.05$ ?
(b) Find the $P$-value for the test.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 54

An article in Fortune (September 21, 1992) claimed that nearly one-half of all engineers continue academic studies beyond the B.S. degree, ultimately receiving either an M.S. or a Ph.D. degree. Data from an article in Engineering Horizons (Spring 1990) indicated that 117 of 484 new engineering graduates were planning graduate study.
(a) Are the data from Engineering Horizons consistent with the claim reported by Fortune? Use $\alpha=0.05$ in reaching your conclusions.
(b) Find the $P$-value for this test.
(c) Discuss how you could have answered the question in part
(a) by constructing a two-sided confidence interval on $p$.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 55

A manufacturer of interocular lenses is qualifying a new grinding machine and will qualify the machine if the percentage of polished lenses that contain surface defects does not exceed $2 \%$. A random sample of 250 lenses contains six defective lenses.
(a) Formulate and test an appropriate set of hypotheses to determine if the machine can be qualified. Use $\alpha=0.05$.
(b) Find the $P$-value for the test in part (a).

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 56

A researcher claims that at least $10 \%$ of all football helmets have manufacturing flaws that could potentially cause injury to the wearer A sample of 200 helmets revealed that 16 helmets contained such defects.
(a) Does this finding support the researcher's claim? Use $\alpha=0.01$.
(b) Find the $P$-value for this test.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 57

A random sample of 500 registered voters in Phoenix is asked if they favor the use of oxygenated fuels year-round to reduce air pollution. If more than 315 voters respond positively, we will conclude that at least $60 \%$ of the voters favor the use of these fuels.
(a) Find the probability of type 1 error if exactly $60 \%$ of the voters favor the use of these fuels.
(b) What is the type II error probability $\beta$ if $75 \%$ of the voters favor this action?

Tyler Moulton

Numerade Educator

Problem 58

The advertized claim for batteries for cell phones is set at 48 operating hours, with proper charging procedures. A study of 5000 batteries is carried out and 15 stop operating prior to 48 hours. Do these experimental results support the claim that less than 0.2 percent of the company's batteries will fail during the advertized time period, with proper charging procedures? Use a hypothesis-testing procedure with $\alpha=0.01$.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 59

Consider the following frequency table of observations on the random variable $X$.
(a) Based on these 100 observations, is a Poisson distribution with a mean of 1.2 an appropriate model? Perform a good-ness-of-fit procedure with $\alpha=0.05$.
(b) Calculate the $P$-value for this test.

Manik Pulyani

Numerade Educator

Problem 60

Let $X$ denote the number of flaws observed on a large coil of galvanized steel. Seventy-five coils are inspected and the following data were observed for the values of $X$ :
(a) Does the assumption of the Poisson distribution seem appropriate as a probability model for this data? Use $\alpha=0.01$.
(b) Calculate the $P$-value for this test.

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Problem 61

The number of calls arriving at a switchboard from noon to 1 PM during the business days Monday through Friday is monitored for six weeks (i.e., 30 days). Let $X$ be defined as the number of calls during that one-hour period. The relative frequency of calls was recorded and reported as
(a) Does the assumption of a Poisson distribution seem appropriate as a probability model for this data? Use $\alpha=0.05$.
(b) Calculate the $P$-value for this test.

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Problem 62

Consider the following frequency table of observations on the random variable $X$ :
(a) Based on these 50 observations, is a binomial distribution with $n=6$ and $p=0.25$ an appropriate model? Perform a goodness-of-fit procedure with $\alpha=0.05$.
(b) Calculate the $P$-value for this test.

Amany Waheeb

Numerade Educator

Problem 63

Define $X$ as the number of underfilled bottles from a filling operation in a carton of 24 bottles. Sixty cartons are inspected and the following observations on $X$ are recorded:
(a) Based on these 75 observations, is a binomial distribution an appropriate model? Perform a goodness-of-fit procedure with $\alpha=0.05$.
(b) Calculate the $P$-value for this test.

Brandon Cleary

Numerade Educator

Problem 64

The number of cars passing eastbound through the intersection of Mill and University Avenues has been tabulated by a group of civil engineering students. They have obtained the data in the adjacent table:
(a) Does the assumption of a Poisson distribution seem appropriate as a probability model for this process? Use $\alpha=0.05$.
(b) Calculate the $P$-value for this test.

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Problem 65

A company operates four machines three shifts each day. From production records, the following data on the number of breakdowns are collected:
Test the hypothesis (using $\alpha=0.05$ ) that breakdowns are independent of the shift. Find the $P$-value for this test.

Sheryl Ezze

Numerade Educator

Problem 66

Patients in a hospital are classified as surgical or medical. A record is kept of the number of times patients require nursing service during the night and whether or not these patients are on Medicare. The data are presented here:
Test the hypothesis (using $\alpha=0.01$ ) that calls by surgicalmedical patients are independent of whether the patients are receiving Medicare. Find the $P$-value for this test.

Harsh Gadhiya

Numerade Educator

Problem 67

Grades in a statistics course and an operations research course taken simultaneously were as follows for a group of students.
Are the grades in statistics and operations research related? Use $\alpha=0.01$ in reaching your conclusion. What is the $P$-value for this test?

Raymond Matshanda

Raymond Matshanda

Numerade Educator

Problem 68

An experiment with artillery shells yields the following data on the characteristics of lateral deflections and ranges. Would you conclude that deflection and range are independent? Use $\alpha=0.05$. What is the $P$-value for this test?

Harsh Gadhiya

Numerade Educator

Problem 69

A study is being made of the failures of an electronic component. There are four types of failures possible and two mounting positions for the device. The following data have been taken:
Would you conclude that the type of failure is independent of the mounting position? Use $\alpha=0.01$. Find the $P$-value for this test.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 70

A random sample of students is asked their opinions on a proposed core curriculum change. The results are as follows.
Test the hypothesis that opinion on the change is independent of class standing. Use $\alpha=0.05$. What is the $P$-value for this test?

Nick Johnson

Numerade Educator

Problem 72

Consider the situation of Exercise 9-76. After collecting a sample, we are interested in testing $H_0: p=0.10$ versus $H_1: p \neq 0.10$ with $\alpha=0.05$. For each of the following situations, compute the $p$-value for this test:
(a) $n=50, \hat{p}=0.095$
(b) $n=100, \hat{p}=0.095$
(c) $n=500, \hat{p}=0.095$
(d) $n=1000, \hat{p}=0.095$
(e) Comment on the effect of sample size on the observed $P$-value of the test.

Amany Waheeb

Numerade Educator

Problem 73

. An inspector of flow metering devices used to administer fluid intravenously will perform a hypothesis test to determine whether the mean flow rate is different from the flow rate setting of 200 milliliters per hour. Based on prior information the standard deviation of the flow rate is assumed to be known and equal to 12 milliliters per hour. For each of the following sample sizes, and a fixed $\alpha=0.05$, find the probability of a type II error if the true mean is 205 milliliters per hour.
(a) $n=20$
(b) $n=50$
(c) $n=100$
(d) Does the probability of a type II error increase or decrease as the sample size increases? Explain your answer.

Amany Waheeb

Numerade Educator

Problem 74

Suppose that in Exercise 9-73, the experimenter had believed that $\sigma=14$. For each of the following sample sizes, and a fixed $\alpha=0.05$, find the probability of a type II error if the true mean is 205 milliliters per hour.
(a) $n=20$
(b) $n=50$
(c) $n=100$
(d) Comparing your answers to those in Exercise 9-73, does the probability of a type II error increase or decrease with the increase in standard deviation? Explain your answer.

Amany Waheeb

Numerade Educator

Problem 75

The marketers of shampoo products know that customers like their product to have a lot of foam. A manufacturer of shampoo claims that the foam height of his product exceeds 200 millimeters. It is known from prior experience that the standard deviation of foam height is 8 millimeters. For each of the following sample sizes, and a fixed $\alpha=0.05$, find the power of the test if the true mean is 204 millimeters.
(a) $n=20$
(b) $n=50$
(c) $n=100$
(d) Does the power of the test increase or decrease as the sample size increases? Explain your answer.

Amany Waheeb

Numerade Educator

Problem 76

Suppose we wish to test the hypothesis $H_0: \mu=85$ versus the alternative $H_1: \mu>85$ where $\sigma=16$. Suppose that the true mean is $\mu=86$ and that in the practical context of the problem this is not a departure from $\mu_0=85$ that has practical significance.
(a) For a test with $\alpha=0.01$, compute $\beta$ for the sample sizes $n=25,100,400$, and 2500 assuming that $\mu=86$.
(b) Suppose the sample average is $\bar{x}=86$. Find the $P$-value for the test statistic for the different sample sizes specified in part (a). Would the data be statistically significant at $\alpha=0.01$ ?
(c) Comment on the use of a large sample size in this problem.

Amany Waheeb

Numerade Educator

Problem 77

The cooling system in a nuclear submarine consists of an assembly of welded pipes through which a coolant is circulated. Specifications require that weld strength must meet or exceed 150 psi.
(a) Suppose that the design engineers decide to test the hypothesis $H_0: \mu=150$ versus $H_1: \mu>150$. Explain why this choice of alternative hypothesis is better than $H_1: \mu<150$.
(b) A random sample of 20 welds results in $\bar{x}=153.7$ psi and $s=11.3$ psi. What conclusions can you draw about the hypothesis in part (a)? State any necessary assumptions about the underlying distribution of the data.

Amany Waheeb

Numerade Educator

Problem 78

Suppose we are testing $H_0: p=0.5$ versus $H_0: p \neq 0.5$. Suppose that $p$ is the true value of the population proportion.
(a) Using $\alpha=0.05$, find the power of the test for $n=100$, 150 , and 300 assuming that $p=0.6$. Comment on the effect of sample size on the power of the test.
(b) Using $\alpha=0.01$, find the power of the test for $n=100$, 150 , and 300 assuming that $p=0.6$. Compare your answers to those from part (a) and comment on the effect of $\alpha$ on the power of the test for different sample sizes.
(c) Using $\alpha=0.05$, find the power of the test for $n=100$, assuming $p=0.08$. Compare your answer to part (a) and comment on the effect of the true value of $p$ on the power of the test for the same sample size and $\alpha$ level.
(d) Using $\alpha=0.01$, what sample size is required if $p=0.6$ and we want $\beta=0.05$ ? What sample is required if $p=0.8$ and we want $\beta=0.05$ ? Compare the two sample sizes and comment on the effect of the true value of $p$ on sample size required when $\beta$ is held approximately constant.

Amany Waheeb

Numerade Educator

Problem 79

Consider the television picture tube brightness experiment described in Exercise 8-24.
(a) For the sample size $n=10$, do the data support the claim that the standard deviation of current is less than 20 microamps?
(b) Suppose instead of $n=10$, the sample size was 51 . Repeat the analysis performed in part (a) using $n=51$.
(c) Compare your answers and comment on how sample size affects your conclusions drawn in parts (a) and (b).

Amany Waheeb

Numerade Educator

Problem 80

Consider the fatty acid measurements for the diet margarine described in Exercise 8-25.
(a) For the sample size $n=6$, using a two-sided alternative hypothesis and $\alpha=0.01$, test $H_0 ; \sigma^2=1.0$.
(b) Suppose instead of $n=6$, the sample size was $n=51$. Repeat the analysis performed in part (a) using $n=51$.
(c) Compare your answers and comment on how sample size affects your conclusions drawn in parts (a) and (b).

Amany Waheeb

Numerade Educator

Problem 81

A manufacturer of precision measuring instruments claims that the standard deviation in the use of the instruments is at most 0.00002 millimeter. An analyst, who is unaware of the claim, uses the instrument eight times and obtains a sample standard deviation of 0.00001 millimeter.
(a) Confirm using a test procedure and an $\alpha$ level of 0.01 that there is insufficient evidence to support the claim that the standard deviation of the instruments is at most 0.00002 . State any necessary assumptions about the underlying distribution of the data.
(b) Explain why the sample standard deviation, $s=0.00001$, is less than 0.00002 , yet the statistical test procedure results do not support the claim.

Sheryl Ezze

Numerade Educator

Problem 82

A biotechnology company produces a therapeutic drug whose concentration has a standard deviation of 4 grams per liter. A new method of producing this drug has been proposed, although some additional cost is involved. Management will authorize a change in production technique only if the standard deviation of the concentration in the new process is less than 4 grams per liter. The researchers chose $n=10$ and obtained the following data in grams per liter. Perform the necessary analysis to determine whether a change in production technique should be implemented.

Narayan Hari

Numerade Educator

Problem 83

Consider the 40 observations collected on the number of nonconforming coil springs in production batches of size 50 given in Exercise 6-79.
(a) Based on the description of the random variable and these 40 observations, is a binomial distribution an appropriate model? Perform a goodness-of-fit procedure with $\alpha=0.05$.
(b) Calculate the $P$-value for this test.

Amany Waheeb

Numerade Educator

Problem 84

Consider the 20 observations collected on the number of errors in a string of 1000 bits of a communication channel given in Exercise 6-80.
(a) Based on the description of the random variable and these 20 observations, is a binomial distribution an appropriate model? Perform a goodness-of-fit procedure with $\alpha=0.05$.
(b) Calculate the $P$-value for this test.

Amany Waheeb

Numerade Educator

Problem 85

Consider the spot weld shear strength data in Exercise 6-23. Does the normal distribution seem to be a reasonable model for these data? Perform an appropriate goodness-of-fit test to answer this question.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 86

Consider the water quality data in Exercise 6-24.
(a) Do these data support the claim that mean concentration of suspended solids does not exceed 50 parts per million? Use $\alpha=0.05$.
(b) What is the $P$-value for the test in part (a)?
(c) Does the normal distribution seem to be a reasonable model for these data? Perform an appropriate goodness-of-fit test to answer this question.

Amany Waheeb

Numerade Educator

Problem 87

Consider the golf ball overall distance data in Exercise 6-25.
(a) Do these data support the claim that the mean overall distance for this brand of ball does not exceed 270 yards? Use $\alpha=0.05$.
(b) What is the $P$-value for the test in part (a)?
(c) Do these data appear to be well modeled by a normal distribution? Use a formal goodness-of-fit test in answering this question.

Amany Waheeb

Numerade Educator

Problem 88

Consider the baseball coefficient of restitution data in Exercise 8-79. If the mean coefficient of restitution exceeds 0.635 , the population of balls from which the sample has been taken will be too "lively" and considered unacceptable for play.
(a) Formulate an appropriate hypothesis testing procedure to answer this question.
(b) Test these hypotheses using the data in Exercise 8-79 and draw conclusions, using $\alpha=0.01$.
(c) Find the $P$-value for this test.
(d) In Exercise 8-79(b), you found a 99\% confidence interval on the mean coefficient of restitution. Does this interval, or a one-sided CL, provide additional useful information to the decision maker? Explain why or why not.

Amany Waheeb

Numerade Educator

Problem 89

Consider the dissolved oxygen data in Exercise 8-81. Water quality engineers are interested in knowing whether these data support a claim that mean dissolved oxygen concentration is 2.5 milligrams per liter.
(a) Formulate an appropriate hypothesis testing procedure to investigate this claim.
(b) Test these hypotheses, using $\alpha=0.05$, and the data from Exercise 8-81.
(c) Find the $P$-value for this test.
(d) In Exercise 8-81(b) you found a $95 \% \mathrm{CI}$ on the mean dissolved oxygen concentration. Does this interval provide useful additional information beyond that of the hypothesis testing results? Explain your answer.

Amany Waheeb

Numerade Educator

Problem 90

The mean pull-off force of an adhesive used in manufacturing a connector for an automotive engine application should be at least 75 pounds. This adhesive will be used unless there is strong evidence that the pull-off force does not meet this requirement. A test of an appropriate hypothesis is to be conducted with sample size $n=10$ and $\alpha=0.05$. Assume that the pull-off foree is normally distributed, and $\sigma$ is not known.
(a) If the true standard deviation is $\sigma=1$, what is the risk that the adhesive will be judged acceptable when the true mean pull-off force is only 73 pounds? Only 72 pounds?
(b) What sample size is required to give a $90 \%$ chance of detecting that the true mean is only 72 pounds when $\sigma=1$ ?
(c) Rework parts (a) and (b) assuming that $\sigma=2$. How much impact does increasing the value of $\sigma$ have on the answers you obtain?

Amany Waheeb

Numerade Educator

Problem 91

. Suppose that we wish to test $H_0: \mu=\mu_0$ versus $H_1: \mu \neq \mu_0$, where the population is normal with known $\sigma$. Let $0<\epsilon<\alpha$, and define the critical region so that we will reject $H_0$ if $z_0>z_6$ or if $z_0<-z_{a-6}$, where $z_0$ is the value of the usual test statistic for these hypotheses.
(a) Show that the probability of type I error for this test is $\alpha$.
(b) Suppose that the true mean is $\mu_1=\mu_0+\delta$. Derive an expression for $\beta$ for the above test.

Amany Waheeb

Numerade Educator

Problem 92

Derive an expression for $\beta$ for the test on the variance of a normal distribution. Assume that the twosided alternative is specified.

Amany Waheeb

Numerade Educator

Problem 93

. When $X_1, X_2, \ldots, X_n$ are independent Poisson random variables, each with parameter $\lambda$, and $n$ is large, the sample mean $\bar{X}$ has an approximate normal distribution with mean $\lambda$ and variance $\lambda / n$. Therefore,

$$
Z=\frac{\bar{x}-\lambda}{\sqrt{\lambda / n}}
$$

has approximately a standard normal distribution. Thus we can test $H_0: \lambda=\lambda_0$ by replacing $\lambda$ in $Z$ by $\lambda_0$. When $X_i$ are Poisson variables, this test is preferable to the largesample test of Section 9-2.5, which would use $S / \sqrt{n}$ in the denominator, because it is designed just for the Poisson distribution. Suppose that the number of open circuits on a semiconductor wafer has a Poisson distribution. Test data for 500 wafers indicate a total of 1038 opens. Using $\alpha=0.05$, does this suggest that the mean number of open circuits per wafer exceeds 2.0?

Amany Waheeb

Numerade Educator

Problem 94

When $X_1, X_2, \ldots, X_n$ is a random sample from a normal distribution and $n$ is large, the sample standard deviation has approximately a normal distribution with mean $\sigma$ and variance $\sigma^2 /(2 n)$. Therefore, a large-sample test for $H_0: \sigma=\sigma_0$ can be based on the statistic

$$
Z=\frac{S-\sigma_0}{\sqrt{\sigma_0^2 /(2 n)}}
$$

Use this result to test $H_0$ : $\sigma=10$ versus $H_1: \sigma<10$ for the golf ball overall distance data in Exercise 6-25.

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Problem 95

Continuation of Exercise 9-94. Using the results of the previous exercise, find an approximately unbiased estimator of the 95 percentile $\theta=\mu+1.645 \sigma$. From the fact that $\bar{X}$ and $S$ are independent random variables, find the standard error of $\theta$. How would you estimate the standard error?

Manik Pulyani

Numerade Educator

Problem 96

Continuation of Exercises 9-94 and 9-95. Consider the golf ball overall distance data in Exercise $6-25$. We wish to investigate a claim that the 95 percentile of overall distance does not exceed 285 yards. Construct a test statistic that can be used for testing the appropriate hypotheses. Apply this procedure to the data from Exercise $6-25$. What are your conclusions?

Amany Waheeb

Numerade Educator

Problem 97

Let $X_1, X_2, \ldots, X_n$ be a sample from an exponential distribution with parameter $\lambda$. It can be shown that $2 \lambda y_{i-1}^n X_i$ has a chi-square distribution with $2 n$ degrees of freedom. Use this fact to devise a test statistic and critical region for $H_0: \lambda=\lambda_0$ versus the three usual alternatives.

Manik Pulyani

Numerade Educator