Sampling Distributions

Students on diets A sample survey interviews an SRS of $267$college women. Suppose (as is roughly true) that $70\%$ of college women have been on a diet within the past $12$ months. What is the probability that $75\%$ or more of the women in the sample have been on a diet? Follow the four-step process.

A sample survey interviews an SRS of $267$ college women. Suppose (as is roughly true) that $70\%$ of college women have been on a diet within the past $12$ months. What is the probability that $75\%$ or more of the women in the sample have been on a diet? Follow the four-step process. 

Harley-Davidson motorcycles make up $14\%$ of all the motorcycles registered in the United States. You plan to interview an SRS of $500$ motorcycle owners. How likely is your sample to contain $20\%$ or more who own Harleys? Follow the four-step process. 

Your mail-order company advertises that it ships $90\%$ of its orders within three working days. You select an SRS of $100$ of the $5000$ orders received in the past week for an audit. The audit reveals that $86$ of these orders were shipped on time.

(a) If the company really ships $90\%$ of its orders on time, what is the probability that the proportion in an SRS of $100$ orders is as small as the proportion in your sample or smaller? Follow the four-step process.

 (b) A critic says, “Aha! You claim $90\%$, but in your sample, the on-time percentage is lower than that. So the $90\%$ claim is wrong.” Explain in simple language why your probability calculation in (a) shows that the result of the sample does not refute the $90\%$ claim.

The Harvard College Alcohol Study finds that $67\%$ of college students support efforts to "crack down on underage drinking." The study took a random sample of almost $15,000$ students, so the population proportion who support a crackdown is close to $p=0.67$The administration of a local college surveys an SRS of $100$ students and finds that $62$ support a crackdown on underage drinking.

(a) Suppose that the proportion of all students attending this college who support a crackdown is $67\%$, the same as the national proportion. What is the probability that the proportion in an SRS of $100$students is as small as or smaller than the result of the administration’s sample? Follow the four-step process.

(b) A writer in the college’s student paper says that “support for a crackdown is lower at our school than nationally.” Write a short letter to the editor explaining why the survey does not support this conclusion 

The magazine Sports Illustrated asked a random sample of $750$ Division I college athletes, “Do you believe performance-enhancing drugs are a problem in college sports?” Suppose that $30\%$ of all Division I athletes think that these drugs are a problem. Let $\hat{p}$ be the sample proportion who say that these drugs are a problem.

The sampling distribution of $\hat{p}$ has mean

(a) $225$.

(b) $0.30$.

(c) $0.017$.

(d) $0$.

(e) none of these.

The magazine Sports Illustrated asked a random sample of $750$ Division I college athletes, “Do you believe performance-enhancing drugs are a problem in college sports?” Suppose that $30\%$ of all Division I athletes think that these drugs are a problem. Let $\hat{p}$be the sample proportion who say that these drugs are a problem.

The standard deviation of the sampling distribution is about 

(a) $0.0006$.

(b) $0.033$.

(c) $0.017$.

(d) $0$.

(e) None of these.

Decreasing the sample size from $750$ to $375$ would multiply the standard deviation by

(a) $2$.

(b) $\sqrt{2}$.

(c) $1/2$.

(d) $1/\sqrt{2}$.

(e) none of these.

The sampling distribution of $\hat{p}$ is approximately Normal because

(a) there are at least $7570$ Division I college athletes.

(b) $np=225$ and $n(1-p)=525$.

(c) a random sample was chosen.

(d) a large sample size like $n=750$ guarantees it.

(e) the sampling distribution of $\hat{p}$ always has this shape.

A sample survey reports that $29\%$Internet users download music files online, $21\%$ share music files from their computers, and $12\%$ both download and share music.6 Make a Venn diagram that displays this information. What per cent of Internet users neither download nor share music files? 

The California Department of Fish and Game publishes a list of the state’s endangered animals. The reptiles on the list are given across the bottom of the question. 

(a) Describe how you would use Table D at line 111 to choose an SRS of 3 of these reptiles to study. 

(b) Use your method from part (a) to select your sample. Identify the reptiles you chose.

Desert tortoise  Green sea turtle   Loggerhead sea turtle    Giant garter snake Olive Ridley sea turtle     Leatherback sea turtle       Barefoot banded gecko San Francisco garter snake      Island night lizard      Alameda whip snake Coachella Valley fringe-toed lizard       Flat-tailed horned lizard         Southern rubber boa         blunt-nosed leopard lizard 

The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean $266$days and standard deviation $16$days.

1. Find the probability that a randomly chosen pregnant woman has a pregnancy that lasts for more than $270$days. Show your work.

Suppose we choose an$\mathrm{SRS}$ of 6 pregnant women. $\overline{x}=$ the mean pregnancy length for the sample.

The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean $266$days and standard deviation $16$ days.

 What is the mean of the sampling distribution of $\overline{x}$? Explain.

The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean $266$days and standard deviation $16$days.

 Compute the standard deviation of the sampling distribution of$\overline{x}$. Check that the $10\%$ condition is met.

The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean$266$days and standard deviation $16$days.

 Find the probability that the mean pregnancy length for the women in the sample exceeds $270$ davs, Show your work.

Songs on an iPod David’s iPod has about $10000$ songs. The distribution of the playtimes for these songs is heavily skewed to the right with a mean of $225$ seconds and a standard deviation of $60$ seconds. Suppose we choose an SRS of 10 songs from this population and calculate the mean playtime x of these songs. What are the mean and the standard deviation of the sampling distribution of x? Explain. 

A grinding machine in an auto parts plant prepares axles with a target diameter $\mu =40.125$ millimeters (mm). The machine has some variability, so the standard deviation of the diameters is $\sigma =0.002$ mm. The machine operator inspects a random sample of $4$ axles each hour for quality control purposes and records the sample mean diameter $\overline{x}$. Assuming that the process is working properly, what are the mean and standard deviation of the sampling distribution of $\overline{x}$? Explain 

Songs on an iPod David's iPod has about $10000$ songs. The distribution of the play times for these songs is heavily skewed to the right with a mean of $225$ seconds and a standard deviation of $60$ seconds. How many songs would you need to sample if you wanted the standard deviation of the sampling distribution of $\overline{x}$to be$30$seconds? justiify your answer.

Making auto parts $A$ grinding machine in an auto parts plant prepares axles with a target diameter $\mu =40.125$ millimeters (mm). The machine has some variability, so the standard deviation of the diameters is $\sigma =0.002mm$. The machine operator inspects a random sample of $4$ axles cach hour for quality control purposes and records the sample mean diameter $\overline{x}$,how many axles would you need to sample if you wanted the standard deviation of the sampling distribution $\overline{x}$to be$0.0005mm$?justify your answer.

Larger sample Suppose that the blood cholesterol level of all men aged $20-34$ follows the Normal distribution with mean $\mu =188$milligrams per deciliter (mg/dl) and standard deviation $\sigma =41 mg/dl$

(a) Choose an SRS of $100$men from this population What is the sampling distribution of $\overline{x}$ ?

(b) Find the probability that $\overline{x}$ estimates $\mu$ within $\pm 3 mg/dl$. (This is the probability that$\overline{x}$ takes a value between $185 and 191 mg/dl$ .) Show your work.

(c) Choose an SRS of $1000$men from this population. Now what is the probability that $\overline{x}$ falls within $\pm 3 mg/dl$of$\mu$? show your wrok.in what sense is the large sample "better".

Stop the carl A car company has found that the lifetime of its disc brake pads varies from car to car according to a Normal distrilustion with mean $\mu =55000$miles and standard deviation$\sigma =4500$ miles. The company installs a new brand of brake pads on an SRS of $8$ cars.

(a) If the new brand has the same lifetime distribution as the previous bye of brake pad, what is the sampling distribution of the mean lifetime $\overline{x}$ ?

(b) The average life of the pads an these $8$ cars turns out to be $\overline{x}=51800$ miles. Find the probability that the sample mean lifetime is $51800$ miles or less if the lifetime distribution is unchanged, What conclusion would you draw?

Bottling cola A hattling compamy uses a fillimg maichine to fill plastic botles with cola. The bottles are supposed to contain $300$ milliliters (ml) . In fact, the contents vary according to a Normal distribution with mean  $\mu =298$ml and standard deviation $\sigma =3 ml$

(a) What is the probability that in individual bottle contains less than $295 ml$ ? Show you work.

(b) What is the probability that the mean contents of six randomly selected bottles is less than  $295 ml$? Show your work.

ACT scores The composite scores of individual students on the ACT college entrance examination in $2009$ followed a Normal distribution with mean $21.1$ and standard deviation$5.1$ 

(a) What is the probability that a single student randomly chosen from all those taking the test scores  $23$or higher? Show your work.

(b) Now take an SRS of $50$ students Who took the fest. What is the probability that the mean score  $\overline{x}$of these students is $23$or higher? Show your work

What does the CLT say? Asked what the central limit theorem says, a student replies, "As you take larger and larger samples from a population, the histogram of the sample values looks mote and more Normal." ls the student right? Explain your answer.

The CLT applet Go to the textbook Web site (www whfreeman com/tp4e) and click on "Statistical Applets." Launch the Central limit Theorem applet. You should see a screen like the one shown here. Click and drag the slider to change the sample size, and watch how the density curve for the sampling distribution changes with it. Write a few sentences describing what in happening.

Songs on an iPod David's iPod has about $10000$ songs. The distribution of the play times for these songs is heavily skewed to the right with a mean of $225$ seconds and a standard deviation of $60$ seconds. 

(a) Explain why you cannot safely calculate the probability that the mean play time $\overline{x}$ is more than $4$ minutes $\left(240\right)$ seconds for an SRS of $10$ songs.

(b) Suppose we take an SRS of  $36$songs instead. Explain how the central limit theorem allows us to find the probability that the mean playtime is more than $240$ seconds. Then calculate this probability. Show your work.

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.) The National lighting Detection Network (NLDN) uses automatic sensors to watch for lightning in a random sample of 10 one-square-kilometer plots of land.

(a) What are the mean and standard deviation of $\overline{x}$ the sample mean number of strikes per square kilometers?

(b) Explain why you can't safely calculate the probability that $\overline{x}<5$ based on a sample of size $10$ .

(c) Suppose the NLDN takes a random sample of $n=50$ square kilometers instead. Explain how the central limit theorem allows us to find the probability that the mean number of lightning strikes per square kilometer is less than $5$ . Then calculate this probability. Show your work.

 Airline passengers get heavier In response to the increasing weight of airline passengers, the Federal Aviation Administration (FAA) in $2003$ told airlines to assume that passengers average $190$ pounds in the summer, including clothes and carry-on baggage. But passengers vary, and the FAA did not specify a standard deviation. A reasonable standard deviation is $35$ pounds. Weights are not Normally distributed, especially when the population includes both men and women, but they are not very non-Normal. A commuter plane carries $30$ passengers.

(a)  Explain why you cannot calculate the probability that a randomly selected passenger weighs more than $200$ pounds.

(b)  Find the probability that the total weight of the passengers on a full flight exceeds $6000$ pounds. Show your work. (Hint: To apply the central limit theorem, restate the problem in terms of the mean weight.)

How many people are in a car? A study of rush-hour traffic in San Francisco counts the number of people in each car entering a freeway at a suburban interchange. Suppose that this count has a mean of $1.5$. and a standard deviation of $0.75$ in the population of all cars that enter this interchange during rush hours.

(a)  Could the exact distribution of the count be Normal? Why or why not? 

(b) Traffic engineers estimate that the capacity of the interchange is $700$ cars per hour. Find the probability that $700$ cars will carry more than $1075$ people. Show your work. (Hint: Restate this event in terms of the mean number of people x per car.) 

More on insurance An insurance company knows that in the entire population of homeowners, the mean annual loss from fire is $\mu =250$ and the standard deviation of the loss is $\sigma =300$. The distribution of losses is strongly right-skewed: many policies have $0$ loss, but a few have large losses. If the company sells $10,000$ policies, can it safely base its rates on the assumption that its average loss will be no greater than $275$? Follow the four-step process

 Bad carpet The number of flaws per square yard is a type of carpet material that varies with mean $1.6$ flaws per square yard and standard deviation $1.2$ flaws per square yard. The population distribution cannot be Normal, because a count takes only whole-number values. An inspector studies $200$ square yards of the material records the number of flaws found in each square yard, and calculates $x$, the mean number of flaws per square yard inspected. Find the probability that the mean number of flaws exceeds $2$ per square yard. Follow the four-step process. 

The mean of the average scores you get should be close to

(a) $515$ .

(b) $515/100=5.15$.

(c) $515/\sqrt{100}=51.5$.

(d) $0$.

(e) none of these.

The standard deviation of the average scores you get should be close to

(a) $114$.

(b) $114/100=1.14$.

(c) $114/\sqrt{100}=11.4$.

(d) $1$ .

(e) none of these.

A newborn baby has extremely low birth weight (ELBW) if it weighs less than $1000$ grams. A study of the health of such children in later years examined a random sample of $219$ children. Their mean weight at birth was $\overline{x}=810$ grams. This sample mean is an unbiased estimator of the mean weight$\mu$ in the population of all ELBW babies, which means that

(a) in all possible samples of size $219$ from this population, the mean of the values of $\overline{x}$ will equal $810$ .

(b) in all possible samples of size 219 from this population, the mean of the values of $\overline{x}$ will equal $\mu$.

(c) as we take larger and larger samples from this population, $\overline{x}$ will get closer and closer to $\mu$.

(d) in all possible samples of size $219$ from this population, the values of $\overline{x}$ will have a distribution that is close to Normal.

(e) the person measuring the children's weights does so without any systematic error.

The number of hours a light bulb burns before failing varies from bulb to bulb. The distribution of burnout times is strongly skewed to the right. The central limit theorem says that

(a) as we look at more and more bulbs, their average burnout time gets ever closer to the mean $\mu$ for all bulbs of this type.

(b) the average burnout time of a large number of bulbs has a distribution of the same shape (strongly skewed) as the population distribution.

(c) the average burnout time of a large number of bulbs has a distribution with a similar shape but not as extreme (skewed, but not as strongly) as the population distribution.

(d) the average burnout time of a large number of bulbs has a distribution that is close to Normal.

(e) the average burnout time of a large number of bulbs has a distribution that is exactly Normal.

In the language of government statistics, you are “in the labor force” if you are available for work and either working or actively seeking work. The unemployment rate is the proportion of the labor force (not of the entire population) who are unemployed. Here are data from the Current Population Survey for the civilian population aged $25$ years and over in a recent year. The table entries are counted by thousands of people.

Unemployment ($1.1$) Find the unemployment rate for people with each level of education. How does the unemployment rate change with education? 

In the language of government statistics, you are “in the labor force” if you are available for work and either working or actively seeking work. The unemployment rate is the proportion of the labor force (not of the entire population) who are unemployed. Here are data from the Current Population Survey for the civilian population aged $25$ years and over in a recent year. The table entries are counted by thousands of people.

Unemployment (5.1) What is the probability that a randomly chosen person 25 years of age or older is in the labor force? Show your work.

In the language of government statistics, you are “in the labor force” if you are available for work and either working or actively seeking work. The unemployment rate is the proportion of the labor force (not of the entire population) who are unemployed. Here are data from the Current Population Survey for the civilian population aged $25$ years and over in a recent year. The table entries are counts in thousands of people. 

If you know that a randomly chosen person $25$ years of age or older is a college graduate, what is the probability that he or she is in the labor force? Show your work.

Are the events “in the labor force” and “college graduate” independent? Justify your answer .

Sale of eggs that are contaminated with salmonella can cause food poisoning in consumers. A large egg producer takes an SRS of $200$ eggs from all the eggs shipped in one day. The laboratory reports that$9$of these eggs had salmonella contamination. Unknown to the producer, $0.1\%$ (one-tenth of 1%) of all eggs shipped had salmonella. Identify the population, the parameter, the sample, and the statistic.

(a) Sketch a possible graph of the distribution of sample data for an SRS of size $5$  with a range of $1000$ grams.

(b) Explain why the dotplot of sample ranges above is not the actual sampling distribution of the sample range. 

Researchers in Norway analyzed data on the birth weights of $400,000$ newborns over a six-year period. The distribution of birth weights is approximately Normal with a mean of $3668$ grams and a standard deviation of $511$ grams.$9$ In this population, the range (maximum – minimum) of birth weights is $3417$ grams. We used Fathom software to take $500$ SRSs of size $n=5$ and calculate the range (maximum – minimum) for each sample. The dotplot below shows the results. 

(a) Is the sample range an unbiased estimator of the population range? Give evidence from the graph above to support your answer.

(b) Explain how we could decrease the variability of the sampling distribution of the sample range. 

Thousands of travelers pass through the airport in Guadalajara, Mexico, each day. Before leaving the airport, each passenger must pass through the Customs inspection area. Customs agents want to be sure that passengers do not bring illegal items into the country. But they do not have time to search every traveler’s luggage. Instead, they require each person to press a button. Either a red or a green bulb lights up. If the red light shows, the passenger will be searched by Customs agents. A green light means “go ahead.” Customs agents claim that the proportion of all travelers who will be stopped (red light) is$0.30$, because the light has probability $0.30$ of showing red on any push of the button. To test this claim, a concerned citizen watches a random sample of $100$ travelers push the button. Only $20$ get a red light.

(a) Assume that the Customs agents’ claim is true. Find the probability that the proportion of travelers who get a red light is as small as or smaller than the result in this sample. Show your work. 

(b) Based on your results in (a), do you believe the Customs agents’ claim? Explain. 

Do you jog? The Gallup Poll once asked a random sample of$1540$ adults, "Do you happen to jog?" Suppose that in fact $15\%$ of all adults jog.

(a) What is the mean of the sampling distribution of $\hat{p}$ ? Justify your answer.

(b) Find the standard deviation of the sampling distribution of $\hat{p}$. Check that the $10\%$ condition is met.

(c) Is the sampling distribution of $\hat{p}$ approximately Normal? Justify your answer.

(d) Find the probability that between $13\%$ and $17\%$ of a random sample of $1540$ adults are joggers. Show your work.

A study of voting chose $663$ registered voters at random shortly after an election. Of these,  $72\%$said they had voted in the election. Election records show that only$56\%$ of registered voters voted in the election. Which of the following statements is true about the boldface numbers?

 (a)$72\%$ is a sample; $56\%$ is a population.

(b) $72\%$ and$56\%$ are both statistics.

(c)$72\%$ is a statistic and $56\%$ is a parameter.

(d) $72\%$ is a parameter and $56\%$ is a statistic.

(e) $72\%$ and$56\%$ are both parameters.

The Gallup Poll has decided to increase the size of its random sample of voters from about $1500$ people to about $4000$ people right before an election. The poll is designed to estimate the proportion of voters who favor a new law banning smoking in public buildings. The effect of this increase is to

(a) reduce the bias of the estimate. 

(b) increase the bias of the estimate. 

(c) reduce the variability of the estimate. 

(d) increase the variability of the estimate. 

(e) have no effect since the population size is the same 

Suppose we select an SRS of size $n=100$ from a large population having proportion p of successes. Let $\stackrel{\wedge }{p}$ be the proportion of successes in the sample. For which value of p would it be safe to use the Normal approximation to the sampling distribution of $\stackrel{\wedge }{p}$?

(a) $0.01$ 

(b) $\frac{1}{11}$

(c) $0.85$ 

(d) $0.975$ 

(e) $0.999$

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 moths. The distribution of moth counts is discrete and strongly skewed, with standard deviation$0.7$.

 (a) What are the mean and standard deviation of the average number of moths $x$ in$50$ traps?

 (b) Use the central limit theorem to help you find the probability that the average number of moths in $50$ traps is greater than$0.6$.

IQ tests The Wechsler Adult Intelligence Scale (WAIS) is a common "IQ test" for adults. The distribution of WAIS scores for persons over $16$ years of age is approximately Normal with mean 100 and standard deviation $15$.

(a) What is the probability that a randomly chosen individual has a WAIS score of $105$ or higher? Show your work.

(b) Find the mean and standard deviation of the sampling distribution of the average WAIS score $\overline{x}$ for an SRS of $60$ people.

(c) What is the probability that the average WAIS score of an SRS of $60$ people is $105$ or higher? Show your work.

(d) Would your answers to any of parts (a), (b), or (c) be affected if the distribution of WAIS scores in the adult population were distinctly non-Normal? Explain.

The central limit theorem is important in statistics because it allows us to use the Normal distribution to make inferences concerning the population mean 

(a) if the sample size is reasonably large (for any population). 

(b) if the population is normally distributed and the sample size is reasonably large. 

(c) if the population is normally distributed (for any sample size). 

(d) if the population is normally distributed and the population variance is known (for any sample size). 

(e) if the population size is reasonably large (whether the population distribution is known or not,