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Introduction To The Practice Of Statistics 6th Edition David S. Moore, George P. McCabe, Bruce A. Craig - Solutions
More on computing the P-value. A test of the null hypothesis H0:μ = μ0 gives test statistic z = −1.73.(a) What is the P-value if the alternative is Ha:μ > μ0?(b) What is the P-value if the alternative is Ha:μ < μ0?(c) What is the P-value if the alternative is Ha:μ = μ0?
Computing the P-value. A test of the null hypothesis H0:μ = μ0 gives test statistic z = 1.34.(a) What is the P-value if the alternative is Ha:μ > μ0?(b) What is the P-value if the alternative is Ha:μ < μ0?(c) What is the P-value if the alternative is Ha:μ = μ0?
Translating research questions into hypotheses.Translate each of the following research questions into appropriate H0 and Ha.(a) Census Bureau data show that the mean household income in the area served by a shopping mall is $62,500 per year. A market research firm questions shoppers at the mall to
Even more on determining hypotheses. In each of the following situations, state an appropriate null hypothesis H0 and alternative hypothesis Ha. Be sure to identify the parameters that you use to state the hypotheses. (We have not yet learned how to test these hypotheses.)(a) A sociologist asks a
More on determining hypotheses. State the null hypothesis H0 and the alternative hypothesis Ha in each case. Be sure to identify the parameters that you use to state the hypotheses.(a) A university gives credit in French language courses to students who pass a placement test. The language
Determining hypotheses. State the appropriate null hypothesis H0 and alternative hypothesis Ha in each of the following cases.(a) A 2002 study reported that 70% of students owned a cell phone. You plan to take an SRS of students to see if the percent has increased.(b) The examinations in a large
What’s wrong? Here are several situations where there is an incorrect application of the ideas presented in this section. Write a short paragraph explaining what is wrong in each situation and why it is wrong.(a) A significance test rejected the null hypothesis that the sample mean is equal to
What’s wrong? Here are several situations where there is an incorrect application of the ideas presented in this section. Write a short paragraph explaining what is wrong in each situation and why it is wrong.(a) A random sample of size 20 is taken from a population that is assumed to have a
One-sided and two-sided P-values. The P-value for a two-sided significance test is 0.06.(a) State the P-values for the one-sided tests.(b) What additional information do you need to properly assign these P-values to the > and < (one-sided) alternatives?
More on the P-value and the significance level. The P-value for a significance test is 0.074.(a) Do you reject the null hypothesis at level α = 0.05?(b) Do you reject the null hypothesis at level α = 0.01?(c) Explain your answers.
P-value and the significance level. The P-value for a significance test is 0.026.(a) Do you reject the null hypothesis at level α = 0.05?(b) Do you reject the null hypothesis at level α = 0.01?(c) Explain your answers.
More on two-sided tests and confidence intervals. A 95% confidence interval for a population mean is (57, 65).(a) Can you reject the null hypothesis that μ = 68 at the 5% significance level? Explain.(b) Can you reject the null hypothesis that μ = 62 at the 5% significance level? Explain.
Two-sided significance tests and confidence intervals. The P-value for a two-sided test of the null hypothesis H0:μ = 30 is 0.08.(a) Does the 95% confidence interval include the value 30? Explain.(b) Does the 90% confidence interval include the value 30? Explain.
Testing a random number generator. Statistical software has a“random number generator” that is supposed to produce numbers uniformly distributed between 0 to 1. If this is true, the numbers generated come from a population with μ = 0.5. A command to generate 100 random numbers gives outcomes
Computing the test statistic and P-value. You will perform a significance test of H0:μ = 25 based on an SRS of n = 25. Assume σ = 5.(a) If x = 27, what is the test statistic z?(b) What is the P-value if Ha: μ > 25?(c) What is the P-value if Ha: μ = 25?
The Supreme Court speaks. The Supreme Court has said that zscores beyond z∗ = 2 or 3 are generally convincing statistical evidence.For a two-sided test, what significance level corresponds to z∗ = 2? To z∗ = 3?
More on finding significant z-scores. Consider a significance test of the true mean based on an SRS of 30 observations from a Normal population.The alternative hypothesis is that the true mean is larger than 1000. What values of the z statistic are statistically significant at theα = 0.05 level?
Finding significant z-scores. Consider a significance test of the true mean based on an SRS of 30 observations from a Normal population.The alternative hypothesis is that the true mean is different from 1000.What values of the z statistic are statistically significant at the α = 0.05 level?
More on the Normal curve and the P-value. A test statistic for a two-sided significance test for a population mean is z = −1.2. Sketch a standard Normal curve and mark this value of z on it. Find the P-value and shade the appropriate areas under the curve to illustrate your calculations.
Normal curve and the P-value. A test statistic for a two-sided significance test for a population mean is z = 2.7. Sketch a standard Normal curve and mark this value of z on it. Find the P-value and shade the appropriate areas under the curve to illustrate your calculations.
DXA scanners. A dual-energy X-ray absorptiometry (DXA) scanner is used to measure bone mineral density for people who may be at risk for osteoporosis. To ensure its accuracy, the company uses an object called a “phantom” that has known mineral density μ = 1.4 grams per square centimeter. Once
Food court survey. The food court at your dormitory has been redesigned.A survey is planned to determine whether or not students think that the new design is an improvement. Sampled students will respond on a seven-point scale with scores less than 4 favoring the old design and scores greater than
Like your job? A Gallup Poll asked working adults about their job satisfaction. One question was “All in all, which best describes how you feel about your job?” The possible answers were “love job,” “like job,” “dislike job,” and “hate job.” Fifty-nine percent of the sample
Telemarketing wages. An advertisement in the student newspaper asks you to consider working for a telemarketing company. The ad states, “Earn between $500 and $1000 per week.” Do you think that the ad is describing a confidence interval?Explain your answer.
CHALLENGE More than one confidence interval. As we prepare to take a sample and compute a 95% confidence interval, we know that the probability that the interval we compute will cover the parameter is 0.95. That’s the meaning of 95%confidence. If we use several such intervals, however, our
Accuracy of a laboratory scale. To assess the accuracy of a laboratory scale, a standard weight known to weigh 10 grams is weighed repeatedly.The scale readings are Normally distributed with unknown mean (this mean is 10 grams if the scale has no bias). The standard deviation of the scale readings
Sample size needed for apartment rental rates.How large a sample of one-bedroom apartments in Exercise 6.20 would be needed to estimate the meanμ within ±$50 with 90% confidence?
CHAL LENGE Adjusting required sample size for drop out. Refer to the previous exercise. In similar previous studies, about 20% of the subjects drop out before the study is completed. Adjust your sample size requirement to have enough subjects at the end of the study to meet the margin of error
Required sample size for specifed margin of error. A new bone study is being planned that will measure the biomarker TRAP described in Exercise 6.17. Using the value of σ given there, 6.5 U/l, find the sample size required to provide an estimate of the mean TRAP with a margin of error of 2.0 U/l
APPLET Percent coverage of 95% confidence interval. The Confidence Interval applet lets you simulate large numbers of confidence intervals quickly. Select 95% confidence and then sample 50 intervals. Record the number of intervals that cover the true value (this appears in the “Hit” box in the
Fuel efficiency in metric units. In the previous exercise you found an estimate with a margin of error for the average miles per gallon. Convert your estimate and margin of error to the metric units kilometers per liter (kpl). To change mpg to kpl, use the facts that 1 mile = 1.609 kilometers and 1
Fuel efficiency. Computers in some vehicles calculate various quantities related to performance.One of these is the fuel efficiency, or gas mileage, usually expressed as miles per gallon (mpg). For one vehicle equipped in this way, the mpg were recorded each time the gas tank was filled, and the
Calories consumed by women in the U.S. The mean number of calories consumed by women in the United States who are 19 to 30 years of age isμ = 1791 calories per day. The standard deviation is 31 calories.9 You will study a sample of 200 women in this age range, and one of the variables you will
Average minutes per week on the Internet. Refer to the previous exercise.(a) Give the mean and standard deviation in minutes.(b) Calculate the 95% confidence interval in minutes from your answer to part (a).(c) Explain how you could have directly calculated this interval from the 95% interval that
Average hours per week on the Internet. The Student Monitor surveys 1200 undergraduates from 100 colleges semiannually to understand trends among college students.8 Recently, the Student Monitor reported that the average amount of time spent per week on the Internet was 15.1 hours.Assume that the
CHALLENGE Inference based on skewed data. The mean OC for the 31 subjects in Exercise 6.18 was 33.4 ng/ml. In our calculations, we assumed that the standard deviation was known to be 19.6 ng/ml. Use the 68–95–99.7 rule from Chapter 1(page 59) to find the approximate bounds on the values of OC
More on apartment rental rates. Refer to the previous exercise. Will the 95% confidence interval include approximately 95% of the rents of all unfurnished one-bedroom apartments in this area?Explain why or why not.
Apartment rental rates. You want to rent an unfurnished one-bedroom apartment in Boston next year. The mean monthly rent for a random sample of 10 apartments advertised in the local newspaper is $1400. Assume that the standard deviation is $220. Find a 95% confidence interval for the mean monthly
Populations sampled and margins of error.Consider the following two scenarios. (A) Take a simple random sample of 100 sophomore students at your college or university. (B) Take a simple random sample of 100 sophomore students in your major at your college or university. For each of these samples
Mean OC in young women. Refer to the previous exercise. A biomarker for bone formation measured in the same study was osteocalcin (OC), measured in the blood. The units are nanograms per milliliter(ng/ml). For the 31 subjects in the study the mean was 33.4 ng/ml. Assume that the standard deviation
Mean serum TRAP in young women. For many important processes that occur in the body, direct measurement of characteristics of the process is not possible. In many cases, however, we can measure a biomarker, a biochemical substance that is relatively easy to measure and is associated with the
Inference based on integer values. Refer to Exercise 6.13. The data for this study are integer values between 1 and 10. Explain why the confidence interval based on the Normal distribution will be a good approximation.
Importance of quality professors. Refer to Exercise 6.13. In this same sample, the factor“quality of professors and ability to interact with them” resulted in a mean score of 8.7. Assuming a standard deviation of 3.5, find the 95% confidence interval.
More on the importance of recreational sports.Refer to Exercise 6.13. Repeat the calculations for a 99% confidence interval. How do the results compare with those in Exercise 6.13?
Importance of recreational sports. The National Intramural-Recreational Sports Association(NIRSA) performed a study to look at the value of recreational sports on college campuses.6 A total of 2673 students were asked to indicate how important(on a 10-point scale) each of 21 factors was in terms of
Changing the confidence level. A study with 36 observations had a mean of 70. Assume that the standard deviation is 12. Make a diagram similar to Figure 6.6 (page 364) that illustrates the effect of the confidence level on the width of the interval.Use 80%, 90%, 95%, and 99%. Summarize what the
Changing the sample size. Suppose that the sample mean is 50 and the standard deviation is assumed to be 5. Make a diagram similar to Figure 6.5 (page 362) that illustrates the effect of sample size on the width of a 95% interval. Use the following sample sizes: 10, 20, 40, and 80. Summarize what
Margin of error and the confidence interval.A study based on a sample of size 25 reported a mean of 93 with a margin of error of 11 for 95%confidence.(a) Give the 95% confidence interval.(b) If you wanted 99% confidence for the same study, would your margin of error be greater than, equal to, or
Internet users. A survey of users of the Internet found that males outnumbered females by nearly 2 to 1. This was a surprise, because earlier surveys had put the ratio of men to women closer to 9 to 1. Later in the article we find this information:Detailed surveys were sent to more than 13,000
Changes in sample size. Suppose that in the setting of the previous exercise you have the resources to contact 1000 recent graduates. If all respond, will your margin of error be larger or smaller than $900?What if only 45% respond? Verify your answers by performing the calculations.
Starting salaries. You are planning a survey of starting salaries for recent marketing majors. In 2005, the average starting salary was reported to be $37,832.3 Assuming the standard deviation for this study is $10,500, what sample size do you need to have a margin of error equal to $900 with 95%
Changing the confidence level. In the setting of the previous exercise, would the margin of error for 99% confidence be larger or smaller?Verify your answer by performing the calculations.
College freshmen anxiety level. An SRS of 100 incoming freshmen was taken to look at their college anxiety level. The mean score of the sample was 83.5 (out of 100). Assuming a standard deviation of 4, give the 95% confidence interval for μ, the average anxiety level among all freshmen.
80% confidence intervals. The idea of an 80% confidence interval is that the interval captures the true parameter value in 80% of all samples. That’s not high enough confidence for practical use, but 80%hits and 20% misses make it easy to see how a confidence interval behaves in repeated samples
Constructing a 95% confidence interval. In the setting of the previous two exercises, about 95% of all samples will capture the true mean in the interval x plus or minus $ . Fill in the blank.
Applying the 68–95–99.7 rule. In the setting of the previous exercise, the 68–95–99.7 rule says that the probability is about 0.95 that x is within $ of the population mean μ. Fill in the blank.
How much do you spend on lunch? The average amount you spend on a lunch during the week is not known. Based on past experience, you are willing to assume that the standard deviation is about $2. If you take a random sample of 36 lunches, what is the value of the standard deviation for x?
CHALLENGE A random walk. A particle moves along the line in a random walk. That is, the particle starts at the origin (position 0) and moves either right or left in independent steps of length 1.If the particle moves to the right with probability 0.6, its movement at the ith step is a random
Income of working couples. A study of working couples measures the income X of the husband and the income Y of the wife in a large number of couples in which both partners are employed.Suppose that you knew the means μX and μY and the variances σ2 X and σ2 Y of both variables in the
CHALLENGE Summer employment of college students.Suppose (as is roughly true) that 88% of college men and 82% of college women were employed last summer. A sample survey interviews SRSs of 400 college men and 400 college women.The two samples are of course independent.(a) What is the approximate
CHALLENGE Learning a foreign language. Does delaying oral practice hinder learning a foreign language? Researchers randomly assigned 25 beginning students of Russian to begin speaking practice immediately and another 25 to delay speaking for 4 weeks. At the end of the semester both groups took a
Colors of cashmere sweaters. The unique colors of the cashmere sweaters your firm makes result from heating undyed yarn in a kettle with a dye liquor. The pH (acidity) of the liquor is critical for regulating dye uptake and hence the final color.There are 5 kettles, all of which receive dye liquor
Plastic caps for motor oil containers. A machine fastens plastic screw-on caps onto containers of motor oil. If the machine applies more torque than the cap can withstand, the cap will break.Both the torque applied and the strength of the caps vary. The capping-machine torque has the Normal
A survey of college women. A sample survey interviews an SRS of 280 college women. Suppose(as is roughly true) that 70% of all college women have been on a diet within the last 12 months. What is the probability that 75% or more of the women in the sample have been on a diet?
Losses of British aircraft inWorldWar II. Serving in a bomber crew in World War II was dangerous.The British estimated that the probability of an aircraft loss due to enemy action was 1/20 for each mission. A tour of duty for British airmen in Bomber Command was 30 missions. What is the probability
The weight of a dozen eggs. The weight of the eggs produced by a certain breed of hen is Normally distributed with mean 65 grams (g) and standard deviation 5 g. If cartons of such eggs can be considered to be SRSs of size 12 from the population of all eggs, what is the probability that the weight
Genetics of peas. According to genetic theory, the blossom color in the second generation of a certain cross of sweet peas should be red or white in a 3:1 ratio. That is, each plant has probability 3/4 of having red blossoms, and the blossom colors of separate plants are independent.(a) What is the
Benford’s law. It is a striking fact that the first digits of numbers in legitimate records often follow a distribution known as Benford’s law. Here it is:First digit 1 2 3 4 5 6 7 8 9 Proportion 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046 Fake records usually have fewer first digits
Common last names. The Census Bureau says that the 10 most common names in the United States are (in order) Smith, Johnson, Williams, Jones, Brown, Davis, Miller, Wilson, Moore, and Taylor. These names account for 5.6% of all U.S.residents. Out of curiosity, you look at the authors of the textbooks
Carpooling. Although cities encourage carpooling to reduce traffic congestion, most vehicles carry only one person. For example, 70% of vehicles on the roads in the Minneapolis–St. Paul metropolitan area are occupied by just the driver.(a) If you choose 12 vehicles at random, what is the
CHALLENGE SAT scores. Example 4.37 (page 284) notes that the total SAT scores of high school seniors in a recent year had mean μ = 1026 and standard deviation σ = 209. The distribution of SAT scores is roughly Normal.(a) Julie scored 1110. If scores have a Normal distribution, what percentile of
Auto accidents. The probability that a randomly chosen driver will be involved in an accident in the next year is about 0.2. This is based on the proportion of millions of drivers who have accidents. “Accident” includes things like crumpling a fender in your own driveway, not just highway
CHALLENGE The effect of sample size on the standard deviation. Assume that the standard deviation in a very large population is 100.(a) Calculate the standard deviation for the sample mean for samples of size 1, 4, 25, 100, 250, 500, 1000, and 5000.(b) Graph your results with the sample size on the
Concrete blocks and mortar. You are building a wall from precast concrete blocks. Standard “8 inch”blocks are 758 inches high to allow for a 38 inch layer of mortar under each row of blocks. In practice, the height of a block-plus-mortar row varies according to a Normal distribution with mean 8
CHALLENGE Investments in two funds. Linda invests her money in a portfolio that consists of 70%Fidelity 500 Index Fund and 30% Fidelity Diversified International Fund. Suppose that in the long run the annual real return X on the 500 Index Fund has mean 9% and standard deviation 19%, the annual real
CHALLENG E Treatment and control groups. The two previous exercises illustrate a common setting for statistical inference. This exercise gives the general form of the sampling distribution needed in this setting. We have a sample of n observations from a treatment group and an independent sample of
Advertisements and brand image. Many companies place advertisements to improve the image of their brand rather than to promote specific products. In a randomized comparative experiment, business students read ads that cited either the Wall Street Journal or the National Enquirer for important facts
Treatment of cotton fabrics. “Durable press”cotton fabrics are treated to improve their recovery from wrinkles after washing. Unfortunately, the treatment also reduces the strength of the fabric. The breaking strength of untreated fabric is Normally distributed with mean 57 pounds and standard
Holes in engine blocks. A hole in an engine block is 2.5 centimeters (cm) in diameter. Shafts manufactured to go through this hole must have 0.024 cm clearance for unforced fit. That is, shaft diameter cannot exceed 2.476 cm. The shafts vary in diameter according to the Normal distribution with
Returns on common stocks. The distribution of annual returns on common stocks is roughly symmetric, but extreme observations are more frequent than in a Normal distribution. Because the distribution is not strongly non-Normal, the mean return over even amoderate number of years is close to Normal.
Risks and insurance. The idea of insurance is that we all face risks that are unlikely but carry high cost. Think of a fire destroying your home.So we form a group to share the risk: we all pay a small amount, and the insurance policy pays a large amount to those few of us whose homes burn down.An
Weights of airline passengers. In response to the increasing weight of airline passengers, the Federal Aviation Administration told airlines to assume that passengers average 190 pounds in the summer, including clothing and carry-on baggage.But passengers vary: the FAA gave a mean but not a
Flaws in carpets. The number of flaws per square yard in a type of carpet material varies with mean 1.5 flaws per square yard and standard deviation 1.3 flaws per square yard. This population distribution cannot be Normal, because a count takes only whole-number values. An inspector studies 200
Defining a high glucose reading. In Exercise 5.51, Sheila’s measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with μ = 125 mg/dl andσ = 10 mg/dl. What is the level L such that there is probability only 0.05 that the mean glucose level
A lottery payoff. A $1 bet in a state lottery’s Pick 3 game pays $500 if the three-digit number you choose exactly matches the winning number, which is drawn at random. Here is the distribution of the payoff X:Payoff X $0 $500 Probability 0.999 0.001 Each day’s drawing is independent of other
Diabetes during pregnancy. Sheila’s doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational
Grades in an English course. North Carolina State University posts the grade distributions for its courses online.8 Students in one section of English 210 in the spring 2006 semester received 31% A’s, 40% B’s, 20% C’s, 4% D’s, and 5% F’s.(a) Using the common scale A = 4, B = 3, C = 2, D =
Gypsy moths threaten oak and aspen trees. The gypsy moth is a serious threat to oak and aspen trees. A state agriculture department places traps throughout the state to detect the moths. When traps are checked periodically, the mean number of moths trapped is only 0.5, but some traps have several
ACT scores of high school seniors. The scores of high school seniors on the ACT college entrance examination in 2003 had mean μ = 20.8 and standard deviation σ = 4.8. The distribution of scores is only roughly Normal.(a) What is the approximate probability that a single student randomly chosen
CHALLENGE Cholesterol levels of teenagers. A study of the health of teenagers plans to measure the blood cholesterol level of an SRS of 13- to 16-year olds. The researchers will report the mean x from their sample as an estimate of the mean cholesterol level μ in this population.(a) Explain to
Lightning strikes. The number of lightning strikes on a square kilometer of open ground in a year has mean 6 and standard deviation 2.4. (These values are typical of much of the United States.)Counts of strikes on separate areas are independent.The National Lightning Detection Network uses
Axle diameters. Averages are less variable than individual observations. Suppose that the axle diameters in Exercise 5.43 vary according to a Normal distribution. In that case, the mean x of an SRS of axles also has a Normal distribution.(a) Make a sketch of the Normal curve for a single axle. Add
Play times for songs on an iPod. Averages of several measurements are less variable than individual measurements. Suppose the true mean duration of the play time for the songs in the iPod of Exercise 5.42 is 350 seconds.(a) Sketch on the same graph the two Normal curves, for sampling a single song
A grinding machine for auto axles. An automatic grinding machine in an auto parts plant prepares axles with a target diameter μ = 40.135 millimeters(mm). The machine has some variability, so the standard deviation of the diameters isσ = 0.003 mm. A sample of 4 axles is inspected each hour for
Songs on an iPod. An iPod has about 10,000 songs.The distribution of the play time for these songs is highly skewed. Assume that the standard deviation for the population is 300 seconds.(a) What is the standard deviation of the average time when you take an SRS of 10 songs from this population?(b)
What is wrong? Explain what is wrong in each of the following scenarios.(a) If the standard deviation of a population is 10, then the variance of the mean for an SRS of 20 observations from this population will be 10/√20.(b) When taking SRS’s from a population, larger sample sizes will have
Find a probability. Refer to the example above. Find the probability that the mean time spent working on 70 units is less than 1.1 hours.
The effect of increasing the sample size. In the setting of Exercise 5.38, suppose we increase to the sample size to 400. Use the 95 part of the 68–95–99.7 rule to describe the variability of this sample mean. Compare your results with those you found in Exercise 5.38.
Use the 68–95–99.7 rule. You take an SRS of size 100 from a population with mean 200 and standard deviation 10. According to the central limit theorem, what is the approximate sampling distribution of the sample mean? Use the 95 part of the 68–95–99.7 rule to describe the variability of
The effect of increasing the sample size. In the setting of the previous exercise, repeat the calculations for a sample size of 100. Explain the effect of the increase on the sample mean and standard deviation.
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