Sampling Distributions

The number of undergraduates at Johns Hopkins University is approximately $2000$, while the number at Ohio State University is approximately $40,000$. At both schools, a simple random sample of about $3\%$ of the undergraduates is taken. Each sample is used to estimate the proportion p of all students at that university who own an iPod. Suppose that, in fact, $p=0.80$ at both schools. Which of the following is the best conclusion?

(a) The estimate from Johns Hopkins has less sampling variability than that from Ohio State. 

(b) The estimate from Johns Hopkins has more sampling variability than that from Ohio State. 

(c) The two estimates have about the same amount of sampling variability. 

(d) It is impossible to make any statement about the sampling variability of the two estimates since the students surveyed were different. 

(e) None of the above

 A researcher initially plans to take an SRS of size n from a population that has a mean of 80 and a standard deviation of $20$. If he were to double his sample size (to $2n$), the standard deviation of the sampling distribution of the sample mean would be multiplied by

(a) $2$.

(b)$\frac{1}{\sqrt{2}}$ . 

(c) $2$.

(d) $\frac{1}{2}$.

(e) $\frac{1}{\sqrt{2n}}$

 The student newspaper at a large university asks an SRS of $250$ undergraduates, “Do you favor eliminating the carnival from the term-end celebration?” All in all, $150$ of the $250$ are in favor. Suppose that (unknown to you) $55\%$ of all undergraduates favor eliminating the carnival. If you took a very large number of SRSs of size $n=250$ from this population, the sampling distribution of the sample proportion $\stackrel{\wedge }{P}$ would be

(a) exactly Normal with mean $0.55$ and standard deviation of $0.03$ 

(b) approximately Normal with mean $0.55$ and standard deviation $0.03$.

(c) exactly Normal with mean $0.60$ and standard deviation $0.03$.

(d) approximately Normal with mean $0.60$ and standard deviation $0.03$.

(e) heavily skewed with mean $0.55$ and standard deviation $0.03$.

 Which of the following statements about the sampling distribution of the sample mean is incorrect? (a) The standard deviation of the sampling distribution will decrease as the sample size increases. 

(b) The standard deviation of the sampling distribution is a measure of the variability of the sample mean among repeated samples. 

(c) The sample mean is an unbiased estimator of the true population mean. 

(d) The sampling distribution shows how the sample mean will vary in repeated samples. 

(e) The sampling distribution shows how the sample was distributed around the sample mean 

 A machine is designed to fill $16$-ounce bottles of shampoo. When the machine is working properly, the mean amount poured into the bottles is $16.05$ ounces with a standard deviation of $0.1$ ounces. Assume that the machine is working properly. If four bottles are randomly selected each hour and the number of ounces in each bottle is measured, then $95\%$ of the observations should occur in which interval?

(a) $16.05$ to $16.15$ ounces

(b) $0.30$ to $0.30$ ounces

(c) $15.95$ to $16.15$ ounces 

(d) $15.90$ to $16.20$ ounces  

(e) None of the above  

Suppose that you are a student aide in the library and agree to be paid according to the “random pay” system. Each week, the librarian flips a coin. If the coin comes up heads, your pay for the week is $\backslash (80$. If it comes up tails, your pay for the week is $\backslash )40$. You work for the library for $100$ weeks. Suppose we choose an SRS of 2 weeks and calculate your average earnings x. The shape of the sampling distribution of x will be 

(a) Normal. 

(b) approximately Normal 

(c) right-skewed 

(d) left-skewed.  

(e) symmetric but not Normal.

The five-number summary for a data set is given by $Min=5$, $Q_1=18$, $M=20$, $Q_3=40$, $Max=75$. If you wanted to construct a modified boxplot for the data set (that is, one that would show outliers, if any existed), what would be the maximum possible length of the right-side “whisker”? (a) $33$ (b)$35$ (c) $45$  (d)  (e) $55$

The probability distribution for the number of heads in four tosses of a coin is given by 

The probability of getting at least one tail in four tosses of a coin is

 (a) $0.2500$. (b)$0.3125$ (c) $0.6875$ (d) $0.9375$ (d) None of these

In a certain large population of adults, the distribution of IQ scores is strongly left-skewed with a mean of $122$ and a standard deviation of $5$. Suppose  adults are randomly selected from this population for a market research study. The distribution of the sample mean of IQ scores is

(a) left-skewed with a mean of $122$ and a standard deviation of $0.35$.

(b) exactly Normal with mean $122$ and standard deviation $5$.

(c) exactly Normal with mean $122$ and standard deviation $0.35$.

(d) approximately Normal with mean$122$and standard deviation $5$.

 (e) approximately Normal with a mean$122$ and standard deviation $0.35$.

A $10$-question multiple-choice exam offers $5$ choices for each question. Jason just guesses the answers, so he has a probability $1/5$ of getting any one answer correct. You want to perform a simulation to determine the number of correct answers that Jason gets. One correct way to use a table of random digits to do this is the following:

(a) One digit from the random digit table simulates one answer, with 5$=$ right and all other digits $=$ wrong. Ten digits from the table simulate $10$ answers. (b) One digit from the random digit table simulates one answer, with 0 or 1 $=$ right and all other digits $=$wrong. Ten digits from the table simulate $10$ answers.

(c) One digit from the random digit table simulates one answer, with odd $=$ right and even $=$ wrong. Ten digits from the table simulate $10$ answers.

(d) Two digits from the random digit table simulate one answer, with $00$ to $20$ $=$ right and $21$to $99=$ wrong. Ten pairs of digits from the table simulate $10$ answers.

(e) Two digits from the random digit table simulate one answer, with $00$ to $05=$ right and $06$ to $99$ $=$ wrong. Ten pairs of digits from the table simulate $10$ answers. 

The amount that households pay service providers for access to the Internet varies quite a bit, but the mean monthly fee is $\backslash (28$ and the standard deviation is $\backslash )10$. The distribution is not Normal: many households pay about $\backslash (10$ for limited dial-up access or about $\backslash )30$ for unlimited dial-up access, but some pay much more for faster connections. A sample survey asks an SRS of 500 households with Internet access how much they pay. Let $\overline{x}$ be the mean amount paid.

(a) Explain why you can't determine the probability that the amount a randomly selected household pays for access to the Internet exceeds $\backslash (29$.

(b) What are the mean and standard deviation of the sampling distribution of $\overline{x}$ ?

(c) What is the shape of the sampling distribution of $\overline{x}$ ? Justify your answer.

(d) Find the probability that the average fee paid by the sample of households exceeds $\backslash )29$. Show your work.

According to government data, $22\%$American children under the age of six live in households with incomes less than the official poverty level. A study of learning in early childhood chooses an SRS of $300$ children. Find the probability that more than $20\%$of the sample is from poverty households. Be sure to check that you can use the Normal approximation.

Suppose we roll a fair die four times. The probability that a $6$ occurs on exactly one of the rolls is 

(a) $4{\left(\frac{1}{6}\right)}^3{\left(\frac{5}{6}\right)}^1$

(b) $4{\left(\frac{1}{6}\right)}^1{\left(\frac{5}{6}\right)}^3$

(c) $6{\left(\frac{1}{6}\right)}^1{\left(\frac{5}{6}\right)}^3$

(d) ${\left(\frac{1}{6}\right)}^3{\left(\frac{5}{6}\right)}^1$

(e) ${\left(\frac{1}{6}\right)}^1{\left(\frac{5}{6}\right)}^3$

You want to take an SRS $50$ of the $816$ students who live in a dormitory on a college campus. You label the students $001$ to $816$ in alphabetical order. In the table of random digits, you read the entries

$95592$  $94007$  $69769$  $33547$  $72450$  $16632$  $81194$  $14873$ 

The first three students in your sample have labels 

(a) $955,929,400$ 

(b) $400,769,769$ 

(c) $559,294,007$

(d) $929,400,769$

(e) $400,769,335$

The number of unbroken charcoal briquets in a twenty-pound bag filled at the factory follows a Normal distribution with a mean of$450$briquets and a standard deviation of $20$ briquets. The company expects that a certain number of the bags will be underfilled, so the company will replace for free the 5% of bags that have too few briquets. What is the minimum number of unbroken briquets the bag would have to contain for the company to avoid having to replace the bag for free?

(a) $404$

(b) $411$

 (c) $418$

(d) $425$

(e) $448$

You work for an advertising agency that is preparing a new television commercial to appeal to women. You have been asked to design an experiment to compare the effectiveness of three versions of the commercial. Each subject will be shown one of the three versions and then asked about her attitude toward the product. You think there may be large differences between women who are employed and those who are not. Because of these differences, you should use 

(a) a block design, but not a matched pairs design

(b) a completely randomized design. 

(c) a matched pairs design. 

(d) a simple random sample. 

(e) a stratified random sample.  

Suppose that you have torn a tendon and are facing surgery to repair it. The orthopedic surgeon explains the risks to you. Infection occurs in$3\%$ such operations, the repair fails in $14\%$, and both infection and failure occur together $1\%$ at the time. What is the probability that the operation is successful for someone who has an operation that is free from infection?

(a) $0.0767$

(b) $0.8342$

(c) $0.8400$

(d) $0.8660$

(e) $0.9900$

Social scientists are interested in the association between the high school graduation rate (HSGR) and the percentage of U.S. families living in poverty (POV). Data were collected from all $50$ states and the District of Columbia, and a regression analysis was conducted. The resulting least-squares regression line is given by $\overline{POV}=59.2-0.620$(HSGR) with $r^2=0.802$. Based on the information, which of the following is the best interpretation for the slope of the least-squares regression line?

(a) For each $1\%$ increase in the graduation rate, the per cent of families living in poverty is predicted to decrease by approximately $0.896$.

(b) For each $1\%$ increase in the graduation rate, the per cent of families living in poverty is predicted to decrease by approximately $0.802$.

(c) For each $1\%$ increase in the graduation rate, the per cent of families living in poverty is predicted to decrease by approximately $0.620$.

(d) For each$1\%$ increase in the percentage of families living in poverty, the graduation rate is predicted to increase by approximately $0.802$.

(e) For each $1\%$ increase in the per cent of families living in poverty, the graduation rate is predicted to decrease by approximately $0.620$. 

T7.11. Below are histograms of the values taken by three sample statistics in several hundred samples from the same population. The true value of the population parameter is marked with an arrow on each histogram 

Which statistic would provide the best estimate of the parameter? Justify your answer 

Here is a dot-plot of the adult literacy rates in $177$ countries in$2008$, according to the United Nations. For example, the lowest literacy rate was $23.6\%$, in the African country of Burkina Faso. Use the dot-plot below to answer the Question

The overall shape of this distribution is 

(a) clearly skewed to the right. 

(b) clearly skewed to the left.

(c) roughly symmetric. 

(d) uniform. 

(e) There is no clear shape.

Here is a dot plot of the adult literacy rates in $177$ countries in $2008$, according to the United Nations. For example, the lowest literacy rate was $23.6\%$, in the African country of Burkina Faso. Use the dot-plot below to answer the Question

The mean of this distribution (don’t try to find it) will be 

(a) very close to the mode. 

(b) greater than the median. 

(c) less than the median. 

(d) You can’t say, because the median is random. 

(e) You can’t say, because the mean is random. 

Here is a dot plot of the adult literacy rates in $177$ countries in $2008$, according to the United Nations. For example, the lowest literacy rate was $23.6\%$, in the African country of Burkina Faso. Use the dot plot below to answer Questions

Based on the shape of this distribution, what numerical measures would best describe it? 

(a) The five-number summary 

(b) The mean and standard deviation 

(c) The mean and the quartiles 

(d) The median and the standard deviation 

(e) It is not possible to determine which numerical values to use. 

The correlation between the age and height of children under the age of $12$ is found to be$r=0.60$. Suppose we use the age$x$ of a child to predict the height y of the child. What can we conclude?
(a) The height is generally $60\%$ of a child’s weight.
(b) About $60\%$ of the time, age will accurately predict height.
(c) The fraction of the variation in heights explained by the least-squares regression line of $y$on $x$ is $0.36$.
(d) The least-squares regression line of $y$ on $x$ has a slope of $0.6$.
(e) Thirty-six percent of the time, the least-squares regression line accurately predicts height.

An agronomist wants to test three different types of fertilizer (A, B, and C) on the yield of a new variety of wheat. The yield will be measured in bushels per acre. Six one-acre plots of land were randomly assigned to each of the three fertilizers. The treatment, experimental unit, and
response variable are, respectively,
(a) a specific fertilizer, bushels per acre, a plot of land.
(b) a plot of land, bushels per acre, a specific fertilizer.
(c) random assignment, a plot of land, bushels per acre.
(d) a specific fertilizer, a plot of land, bushels per acre.
(e) a specific fertilizer, the agronomist, bushels per acre

According to the U.S. census, the proportion of adults in a certain county who owned their own home was $0.71$. An SRS of $100$ adults in a certain section of the county found that $65$ owned their home. Which one
of the following represents the approximate probability of obtaining a sample of $100$ adults in which fewer than $65$own their home, assuming that this section of the county has the same overall proportion of adults who own their home as does the entire county?

(a) $\left(\begin{array}{c}100\\ 65\end{array}\right)(0.71{)}^{65}(0.29{)}^{35}$ 

(b) $\left(\begin{array}{c}100\\ 65\end{array}\right)(0.29{)}^{65}(0.71{)}^{35}$
(c) $P\left(z<\frac{0.65-0.71}{\sqrt{\frac{(0.71)(0.29)}{100}}}\right)$ 

(d) $P\left(z<\frac{0.65-0.71}{\sqrt{\frac{(0.65)(0.35)}{100}}}\right)$
(e) $P\left(z<\frac{0.65-0.71}{\left.\frac{(0.71)(0.29)}{\sqrt{100}}\right)}\right.$

Which one of the following would be a correct interpretation if you have a z-score of $+2.0$ on an exam?

(a) It means that you missed two questions on the exam.
(b) It means that you got twice as many questions correct as the average student.
(c) It means that your grade was two points higher than the mean grade on this exam.
(d) It means that your grade was in the upper $2\%$ of all grades on this exam.
(e) It means that your grade is two standard deviations above the mean for this exam.

Records from a random sample of dairy farms yielded the information below on the number of male and female calves born at various times of the day.

What is the probability that a randomly selected calf was
born in the night or was a female?
(a)$\frac{369}{513}$ 

(b)$\frac{485}{513}$

(c) $\frac{116}{513}$

(d)$\frac{116}{252}$

(e) $\frac{116}{233}$

When people order books from a popular online source, they are shipped in standard-sized boxes. Suppose that the mean weight of the boxes is$1.5$ pounds with a standard deviation of $0.3$ pounds, the mean weight of the packing material is $0.5$ pounds with a standard deviation of 0.1 pounds, and the mean weight of the books shipped is $12$ pounds with a
standard deviation of $3$ pounds. Assuming that the weights are independent, what is the standard deviation of the total weight of the boxes that are shipped from this source?
(a) $1.84$  
(b) $2.60$ 

(c) $3.02$

(d) $3.40$

(e) $9.10$

A grocery chain runs a prize game by giving each customer a ticket that may win a prize when the box is scratched off. Printed on the ticket is a dollar value $(\backslash (500, \backslash )100, \backslash (10)$ or the statement, “This ticket is not a winner.” Monetary prizes can be redeemed for groceries at the store. Here are the distribution of the prize values and the associated probabilities for each prize:

Which of the following are the mean and standard deviation, respectively, of the winnings?
$\backslash )15.00, \backslash (2900.00$
(b) $\backslash )15.00, \backslash (53.85$
(c) $\backslash )15.00, \backslash (26.9$

(d) $\backslash )156.25, \backslash (53.85$

(e) $\backslash )156.25, \$26.93$

 A large company is interested in improving the efficiency of its customer service and decides to examine the length of the business phone calls made to clients by its sales staff. A cumulative relative frequency graph is shown below from data collected over the past year. According to the graph, the shortest $80\%$of calls will take how long to complete?

(a) Less than $10$ minutes.
(b) At least $10$ minutes.
(c) Exactly $10$ minutes.
(d) At least $5.5$ minutes.
(e) Less than $5.5$minutes.

AP2.22. A health worker is interested in determining if omega-$3$ fish oil can help reduce cholesterol in adults. She obtains permission to examine the health records of $200$ people in a large medical clinic and classifies them
according to whether or not they take omega-$3$ fish oil. She also obtains their latest cholesterol readings and finds that the mean cholesterol reading for those who are taking omega-$3$ fish oil is $18$points lower than the mean for the group not taking omega-$3$ fish oil.
(a) Is this an observational study or an experiment? Explain.
(b) Do these results provide convincing evidence that taking omega-$3$ fish oil lowers cholesterol?
(c) Explain the concept of confounding in the context of this study and give one example of a possible confounding variable.

AP2.23. There are four major blood types in humans: $\mathrm{O},\mathrm{A},\mathrm{B}$, and $\mathrm{AB}$. In a study conducted using blood specimens from the Blood Bank of Hawaii, individuals were classified according to blood type and ethnic group. The ethnic groups were Hawaiian, Hawaiian-White, HawaiianChinese, and White. Suppose that a blood bank specimen is selected at random.

(a) Find the probability that the specimen contains type $\mathrm{O}$ blood or comes from the Hawaiian-Chinese ethnic group. Show your work.

(b) What is the probability that the specimen contains type AB blood, given that it comes from the Hawaiian ethnic group? Show your work.

(c) Are the events "type B blood" and "Hawaiian ethnic group” independent? Give appropriate statistical evidence to support your answer.

Now suppose that two blood bank specimens are selected at random.

(d) Find the probability that at least one of the specimens contains type A blood from the White ethnic group.

AP2.25. Five cards, each with a different symbol, are shuffled and you choose one. If it is the diamond, you win $\backslash (5.00$. The cards are reshuffled after each draw. You must pay $\backslash )1.00$ for each selection. You continue to play until you select the diamond. Is this a fair game (that is, on average
will you win the same amount as you lose)?

 (a) Describe how you will carry out a simulation of this game using the random digit table below. Be sure to indicate what the digits will represent. 

(b) Perform $10$ repetitions of your simulation. Copy the random digit table onto your paper. Mark on or above the table so that someone can follow your work.

12975   13258   13048   45144   72321   81940   00360 

02428  96767  35964   23822  96012  94591   65194 

50842  53372  72829   50232  97892  63408 77919 

44575  24870   04178   81565  42628  17797   49376 

61762   16953    88604  12724   62964  88145  83083  

69453  46109   59505  69680 00900  19687  12633

57857   95806   09931  02150  43163

(c) Based on your simulation, what is the average number of cards you would need to draw in order to obtain a diamond? Justify your answer.
(d) Is this a fair game (that is, on average will you win the same amount as you lose)? Explain your reasoning.

AP2.24. Every $17$ years, swarms of cicadas emerge from the ground in the eastern United States, live for about six weeks, and then die. (There are several different “broods,” so we experience cicada eruptions more often than every $17$years.) There are so many cicadas that their dead bodies can serve as fertilizer and increase plant growth. In a study, a researcher added $10$ cicadas under $39$ randomly selected plants in a natural plot of American bell flowers on the forest floor, leaving other plants undisturbed. One of the response variables measured was the size of seeds produced by the plants. Here are the box plots and summary statistics of seed mass (in milligrams) for $39$cicada plants and $33$ undisturbed (control) plants:

Variable:          n   Minimum Q1   Median  Q3  Maximum 

Cicada plants: 39    0.17     0.22  0.25    0.28  0.35
Control plants: 33   0.14      0.19   0.25   0.26   0.29
(a) Is this an observational study or an experiment? Explain.
(b) Based on the graphical displays, which distribution has
the larger mean? Justify your answer.
(c) Do the data support the idea that dead cicadas can
serve as fertilizer? Give graphical and numerical evidence
to support your conclusion.