The Nature of Statistics

Megacities Risk. In an issue of Discover (Vol. 26, No. 5, p. 14). A. Casselman looked at the natural-hazards risk index of megacities to evaluate potential loss from catastrophes such as earthquakes, storms, and volcanic eruptions. Urban areas have more to lose from natural perils, technological risks, and environmental hazards than rural areas. The top 10 megacities in the world are Tokyo, San Francisco, Los Angeles, Osaka, Miami, New York, Hong Kong, Manila, London, and Paris. 

(a) Suppose that you decide to take a simple random sample of five of these 10 megacities. Use Table I in Appendix A to obtain five random numbers that you can use to specify your sample. 

(b) If you have access to a random-number generator, use it to solve part (a).

Element Hunters. In the article "Element Hunters" (National Geographic, Vol. 233, No. 5, pp. 112-120), R. Dunn reports about the search for new undiscovered elements. Since 1940, scientists have been synthesizing elements one by one. The first was neptunium (Np), element number 93. There are, as of this writing, a total of 26 new synthetic elements. The following table provides their element numbers and symbols.

(a) Suppose that you decide to take a simple random sample of eight of these new elements. Use Table I in Appendix A to obtain eight random numbers that you can use to specify your sample.

(b) If you have access to a random-number generator, use it to solve part (a).

Undercoverage. Oftentimes, an accurate and complete list of the population is unavailable. In such cases, one or more groups will be omitted from the sampling process because they are not listed as part of the population. This type of bias is called undercoverage. 

(a) Explain why a sample survey of households will generally suffer from undercoverage. 

(b) Provide another example where bias due to undercoverage is likely to occur. 

Nonresponse. When responses are not obtained from some of the individuals in the sample because either those individuals cannot be reached or refuse to participate, we have nonresponse bias.

(a) Discuss some of the dangers of nonresponse. 

(b) Many sample surveys that are reported in the media have rates as low as 10%. Explain the consequences of such low response rates in trying to generalize the results to the entire population.

Response bias. When the behavior of the interviewer or respondent results in inaccurate responses, we have response bias. 

(a) Explain why a survey question "Do you smoke marijuana" might result in response bias? 

(b) Provide some additional survey situations that might be conducive to response bias. 

(c) Provide some additional factors that might lead to response bias.

Surveys that combine one or more of simple random sampling, systematic random sampling, cluster sampling, and stratified sampling employ what is called                 sampling.

Ideally, in stratified sampling, the members of each stratum should be                relative to the characteristic under consideration.

 Ideally, in cluster sampling, each cluster should                     the entire population.

In each of Exercises 1.55-1.58, fill in the blank (s).

Systematic random sampling is easier to execute than simple random sampling and usually provides comparable results. The exception is the presence of some kind of                                       in the listing of the members of the population. 

The members of a population have been numbered 1-372. A sample of size 5 is to be taken from the population, using systematic random sampling. 

(a) Apply Procedure 1.1 on page 17 to determine the sample ( i.e .. the numbers corresponding to the members of the population that are included in the sample ). 

(b) Suppose that, in Step 2 of Procedure 1.1, the random number chosen is 10 ( i.e., k = 10 ). Determine the sample. 

The members of a population have been numbered 1-500. A sample of size 9 is to be taken from the population, using systematic random sampling.

(a) Apply Procedure 1.1 on page 17 to determine the sample (i.e., the numbers corresponding to the members of the population that are included in the sample).

(b) Suppose that, in Step 2 of Procedure 1.1, the random number chosen is 48 (i.e., k = 48). Determine the sample.

The members of a population have been numbered 1-50. A sample of size 20 is to be taken from the population, using cluster sampling. The clusters are of equal size 10, where cluster #1 consists of the members of the population numbered 1-10, cluster #2 consists of the members of the population numbered 11-20, and so forth.

(a) Apply Procedure 1.2 on page 18 to determine the sample (i.e., the numbers corresponding to the members of the population that are included in the sample).

(b). Suppose that, in Step 2 of Procedure 1.2, clusters #1 and #3 are selected. Determine the sample.

The members of a population have been numbered 1-100. A sample of size 30 is to be taken from the population, using cluster sampling. The clusters are of equal size 10, where cluster #1 consists of the members of the population numbered 1-10, cluster #2 consists of the members of the population numbered 11-20, and so forth.

(a) Apply Procedure 1.2 on page 18 to determine the sample (i.e., the numbers corresponding to the members of the population that are included in the sample).

(b) Suppose that, in Step 2 of Procedure 1.2, clusters #2, #6, and #9 are selected. Determine the sample.

The members of a population have been numbered 1-500. A sample of size 10 is to be taken from the population, using stratified random sampling with proportional allocation. The strata are of sizes 200, 150, and 150, where stratum #1 consists of the members of the population numbered 1-200, stratum #2 consists of the members of the population numbered 201-350, and so forth.

(a) Determine the sample sizes that will be taken from the strata.

(b) Apply Procedure 1.3 on page 21 to determine the sample (i.e., the numbers corresponding to the members of the population that are included in the sample).

The members of a population have been numbered 1-1000. A sample of size 20 is to be taken from the population, using stratified random sampling with proportional allocation. The strata are of sizes $300,200,400$, and 100, where stratum #1 consists of the members of the population numbered 1-300, stratum $\#2$ consists of the members of the population numbered 301-500, and so forth.

a. Determine the sample sizes that will be taken from the strata.

b. Apply Procedure $1.3$ on page 21 to determine the sample (i.e., the numbers corresponding to the members of the population that are included in the sample).

Ghost of Speciation Past. In the article, "Ghost of Speciation Past" (Nature, Vol. 435, pp. 29-31), T. Kocher looked at the origins of a diverse flock of cichlid fishes in the lakes of southeast Africa. Suppose that you wanted to select a sample from the hundreds of species of cichlid fishes that live in the lakes of southeast Africa. If you took a simple random sample from the species of each lake and combined all the simple random samples into one sample, which type of sampling design would you have used? Explain your answer.

Number of Farms. The National Agricultural Statistics Service (NASS) conducts studies of the number of acres devoted to farms in each county of the United States. Suppose that we divide the United States into the four census regions (Northeast, North Central, South, and West), take a simple random sample of counties from each of the four regions, and combine all four simple random samples into one sample. What type of sampling design have we used? Explain your answer. 

John F. Kennedy. In one of his books, Ted Sorenson, Special Counsel to President John F. Kennedy, presents an intimate biography of the extraordinary man. According to Sorenson, Kennedy "read every fiftieth letter of the thirty thousand comings weekly to the White House." What type of sampling design was Kennedy using in this case? Explain your answer. [SOURCE: From Kennedy: The Classical Biography by Ted Sorenson. Published by Harper Perennial, © 2009.]

Immunization of Schoolchildren. In the article, "Reasons for Non - uptake of Measles, Mumps, and Rubella Catch Up Immunisation in a Measles Epidemic and Side Effects of the Vaccine" (British Medical Journal, Vol. 310, pp. 1629-1632), R. Roberts et al . discussed a follow-up survey to examine why almost 10,000 children, ages 11-15, whose records showed no previous immunization were not immunized. In the survey, 10 of the 46 schools participating in the immunization campaign were randomly chosen and then the parents of all the nonimmunized children at the 10 selected schools were sent a questionnaire. What type of sampling design was used by these authors in their survey? Explain your answer.

Litigation Surveys. In the article, "Non - probability Sampling Designs for Litigation Surveys (Trademark Reporter, Vol. 81, pp. 169-179), J. Jacoby and H. Handlin discussed the controversy about whether nonprobability samples are acceptable as evidence in litigation. The authors randomly selected 26 journals from a list of 1285 scholarly journals in the social and behavioral sciences. They examined all articles published during one year in each of the 26 journals selected with regard to sampling methods. What type of sampling design was used by these two authors in their investigation? Explain your answer.

Sampling Dorm Residents. Students in the dormitories of a university in the state of New York live in clusters of four double rooms, called suites. There are 48 suites, with eight students per suite. 

(a) Describe a cluster sampling procedure for obtaining a sample of 24 dormitory residents.

(b) Students typically choose friends from their classes as suitemates. With that in mind, do you think cluster sampling is a good procedure for obtaining a representative sample of dormitory residents? Explain your answer.

(c) The university housing office has separate lists of dormitory residents by class level. The number of dormitory residents in each class level is as follows. Use the table to design a procedure for obtaining a stratified sample ( with proportional allocation ) of 24 dormitory residents.

Keno. In the game of keno, 20 balls are selected at random from 80 balls numbered 1-80. In Exercise 1.48 on page 16, you used simple random sampling to simulate one game of keno. 

(a) Use systematic random sampling to obtain a sample of 20 of the 80 balls. 

(b) Which method is easier: simple random sampling or systematic random sampling? 

(c) Does it seem reasonable to use systematic random sampling to simulate one game of keno? Explain your answer.

The International 500. In Exercise 1.49 on page 16, you used simple random sampling to obtain a sample of 10 firms from Fortune Magazine's list of "The International 500."

(a) Use systematic random sampling to accomplish that same task. 

(b) Which method is easier: simple random sampling or systematic random sampling? 

(c) Does it seem reasonable to use systematic random sampling to obtain a representative sample? Explain your answer.

University Parking Facilities. During one year, a university wanted to gauge the sentiment of the people using the university's parking facilities. Each of the 8493 people that used the parking facilities had a sticker with a unique number between 1 and 8493. The university committee on parking decided to sample 30 users of the parking facilities and obtain their views on those facilities. The committee selected a number at random between 1 and 283 and got the number 10. The people interviewed were the ones whose stickers had numbers 10, 293, 576, ..., 8217. What type of sampling design was used by the university committee on parking? Explain your answer.

Best High Schools. In an issue of Newsweek (Vol. CXLV, No. 20, pp. 48-57), B. Kantrowitz listed "The 100 best high schools in America" according to a ranking devised by J. Mathews. Another characteristic measured from the high school is the percent free lunch, which is the percentage of student body that is eligible for free and reduced-price lunches, an indicator of socioeconomic status. A percentage of 40% or more generally signifies a high concentration of children in poverty. The top 100 schools, grouped according to their percent free lunch, is as follows.

(a) Use the table to design a procedure for obtaining a stratified sample (with proportional allocation) of 25 high schools from the list of the top 100 ranked high schools. 

(b) If stratified random sampling with proportional allocation is used to select the sample of 25 high schools, how many would be selected from the stratum with a percent-free-lunch value of 30-under 40?

In simple random sampling, all samples of a given size are equally likely. Is that true in systematic random sampling? Explain your answer.

In simple random sampling, it is also true that each member of the population is equally likely to be selected, the chance for each member being equal to the sample size divided by the population size. Show that this fact is also true for stratified random sampling with proportional allocation.

The Terri Schiavo Case.  In the early part of 2005, the Terri Schiavo case received national attention as her husband sought to have life support removed, and her parents sought to maintain that life support. The courts allowed the life support to be removed, and her death ensued. A Harris Poll of 1010 U.S. adults was taken by telephone on April 21, 2005, to determine how common it is for life support systems to be removed. Those questioned in the sample were asked: (1) Has one of your parents, a close friend, or a family member died in the last 10 years? (2) Before (this death/these deaths) happened, was this person/were any of these people, kept alive by any support system? (3) Did this person die while on a life support system, or had it been withdrawn? Respondents were also asked questions about age, sex, race, education, region, and household income to ensure that results represented a cross-section of U.S. adults. 

(a) What kind of sampling design was used in this survey? Explain your answer

(b) If 78% of the respondents answered the first question in the affirmative, what was the approximate sample size for the second question? 

(c) If 28% of those responding to the second question answered "yes," what was the approximate sample size for the third question?

U.S. House of Representatives.  There are 435 representatives in the 113th session of the U.S. House of Representatives. On the website www.house.gov, you can find an alphabetized list of the 435 congresspersons. In 2013, the first representative listed is Robert Aderholt, a Republican from Alabama, and the last representative listed is Todd Young, a Republican from Indiana. Suppose that the alphabetized list is indexed 1 through 435. 

(a) Use systematic random sampling to obtain a sample of 15 of the 435 representatives. 

(b) Suppose that, in Step 2 of Procedure 1.1, the random number chosen is 12 (i.e., k = 12). Determine the sample.

In simple random sampling, it is also true that each member of the population is equally likely to be selected, the chance for each member being equal to the sample size divided by the population size. 

(a) Under what circumstances is that fact also true for systematic random sampling? Explain your answer. 

(b) Provide an example in which that fact is not true for systematic random sampling.

Peace Corps Volunteers. The Peace Corps is an independent U.S. government agency that provides trained volunteers for countries requesting assistance. According to Peacecorps.org, as of September 2012, volunteers currently serve in about 76 different host countries. The average age of a volunteer is about 28 years old. The following table reports the percentage of total volunteers serving by geographic region.

(a) Use the table to design a procedure for obtaining a stratified sample (with proportional allocation) of 50 Peace Corps volunteers.

(b) If stratified random sampling with proportional allocation is used to select the sample of 50 Peace Corps volunteers, how many would be selected from the Caribbean?

In a designed experiment, there is one factor with four levels. How many treatments are there? 

In this section, we discussed two types of statistical designs. Identify and explain the meaning of each one.

Define each of the following terms in the context of experimental design. 

(a)  Response variable 

(b)  Factor 

(c)   Levels 

(d)  Treatments 

State and explain the significance of the three basic principles of experimental design.

In a designed experiment, 

(a) what are the experimental units? 

(b)  if the experimental units are humans, what term is often used in place of experimental units?

In a designed experiment, there are two factors, say, Factor A and Factor B. Factor A has three levels, say, $a_1, a_2 \mathrm{and} a_3$; Factor B has four levels, say, $b_1, b_2, b_3 \mathrm{and} b_4$.

(a)  Construct a schematic for the treatments similar to Table 1.8 on page 27. 

(b)  Use part ( a ) to determine the number of treatments. 

(c)  Could you have determined the answer for part ( b ) without constructing the schematic in part ( a )? Explain your answer.

In a designed experiment, there is one factor with five levels. How many treatments are there?

In a designed experiment, there are two factors, say, Factor A and Factor B. Factor A has four levels, say, $a_1, a_2, a_3 and a_4$; Factor B has two levels, say, $b_1 \mathrm{and} b_2$.

(a)  Construct a schematic for the treatments similar to Table 1.8 on page 27. 

(b)  Use part ( a ) to determine the number of treatments. 

(c) Could you have determined the answer for part ( b ) without constructing the schematic in part ( a )? Explain your answer. 

In a designed experiment, there are two factors. One factor has m levels and the other factor has n levels. Determine the number of treatments. 

Hint: Refer to Exercise 1.87 or Exercise 1.88.

Adverse Effects of Prozac.  Prozac ( fluoxetine hydrochloride ), a product of Eli Lilly and Company, is used for the treatment of depression, obsessive-compulsive disorder ( OCD ), and bulimia nervosa. An issue of the magazine Arthritis Today contained an advertisement reporting on the " ... treatment-emergent adverse events that occurred in 2 % or more patients treated with Prozac and with incidence greater than placebo in the treatment of depression, OCD, or bulimia. " In the study, 2444 patients took Prozac and 1331 patients were given placebo. Identify the 

(a) treatment group. 

(b)  control group. 

(c)  treatments.

In Exercises 1.92-1.97, we present descriptions of designed experiments. In each case, identify the 

(a)  experimental units

(b)  response variable

(c)  factor (s)

(d)  levels of each factor:- 

(e)  treatments.

Increasing Unit Sales. Supermarkets are interested in strategies to increase temporarily the unit sales of a product. In one study, researchers compared the effect of display type and price on unit sales for a particular product. The following display types and pricing schemes were employed.

• Display types: normal display space interior to an aisle, normal display space at the end of an aisle, and enlarged display space.
• Pricing schemes: regular price, reduced price, and cost.

In Exercises 1.92-1.97, we present descriptions of designed experiments. In each case, identify the 

(a)  experimental units

(b)  response variable

(c)  factor (s)

(d)  levels of each factor:- 

(e)  treatments.

Highway Signs. A driver's ability to detect highway signs is an important consideration in highway safety. In his dissertation, High way Construction Safety and the Aging Driver, S. Younes investigated the distance at which drivers can first detect highway caution signs. This distance is called the detection distance. Younes analyzed the effect that sign size and sign material have on detection distance. Drivers were randomly assigned to one combination of sign size ( small, medium, and large ), and sign material ( 1, 2, and 3 ). Each driver covered the same stretch of highway at a constant speed during the same time of day, and the detection distance, in feet, was determined for the driver's assigned caution sign.

In Exercises 1.92-1.97, we present descriptions of designed experiments. In each case, identify the 

(a)  experimental units

(b)  response variable

(c)  factor (s)

(d)  levels of each factor:- 

(e)  treatments.

Dexamethasone and IQ.  In the paper "Outcomes at School Age After Postnatal Dexamethasone Therapy for Lung Disease of Prematurity" (New England Journal of Medicine, Vol . 350, No. 13, pp . 1304-1313), T. Yeh et al . studied the outcomes at school age in children who had participated in a double-blind, placebo-controlled trial of early postnatal dexamethasone therapy for the prevention of chronic lung disease of prematurity. One result reported in the study was that, at school age, the control group of 74 children had an average IQ score of 84.4, whereas the dexamethasone group of 72 children had an average IQ score of 78.

In Exercises 1.92-1.97, we present descriptions of designed experiments. In each case, identify the 

(a)  experimental units

(b)  response variable

(c)  factor (s)

(d)  levels of each factor:- 

(e)  treatments.

Sexual Signals. In a study by A. Elliot et al . , titled " Women's Use of Red Clothing as a Sexual Signal in Intersexual Interaction " (Journal of Experimental Social Psychology, Vol. 49, Issue 3, pp. 599-602), women were studied to determine the effect of apparel color choice based on perceived attractiveness and gender of a new acquaintance. The following experiment is based on the researchers' investigation. Women are randomly assigned to be told that they would be conversing with an attractive male, an unattractive male, an attractive female, or an unattractive female. The women must wear a standardized shirt before the conversation but can choose between red, green, and blue.

In Exercises 1.92-1.97, we present descriptions of designed experiments. In each case, identify the 

(a)  experimental units

(b)  response variable

(c)  factor (s)

(d)  levels of each factor:- 

(e)  treatments.

The Lion's Mane. In a study by P. M. West titled "The Lion's Mane" ( American Scientist, Vol. 93, No. 3, pp. 226-236), the effects of the mane of a male lion as a signal of quality to mates and rivals was explored. Four life-sized dummies of male lions provided a tool for testing female response to the unfamiliar lions whose manes varied by length (long or short) and color (blonde or dark). The female lions were observed to see whether they approached each of the four life-sized dummies.

Lifetimes of Flashlight Batteries. Two different options are under consideration for comparing the lifetimes of four brands of flashlight battery, using 20 flashlights. 

(a) One option is to randomly divide 20 flashlights into four groups of 5 flashlights each and then randomly assign each group to use a different brand of battery. Would this statistical design be a completely randomized design or a randomized block design? Explain your answer. 

(b) Another option is to use 20 flashlights - five different brands of 4 flashlights-each and randomly assign the 4 flashlights of each sign be a completely randomized design or a randomized block design? Explain your answer.

The Salk Vaccine. In Exercise 1.17 on page 8, we discussed the Salk vaccine experiment. The experiment utilized a technique called double-blinding because neither the children nor the doctors involved knew which children had been given the vaccine and which had been given placebo. Explain the advantages of using double-blinding in the Salk vaccine experiment.

Dental Hygiene: Which Toothbrush? In an experiment reported by J. Singer and D. Andrade in the article " Regression Models for the Analysis of Pretest / Posttest Data " ( Biometrics, Vol . 53, pp . 729-735 ), the effect of using either a conventional or experimental ( hugger ) toothbrush was investigated. Twelve female and 12 male preschoolers were selected. Within each gender group, six were randomly assigned to the conventional toothbrush and the remaining six to the experimental toothbrush. After each subject brushed with the assigned toothbrush, a dental plaque index was measured. The higher the dental plaque index, the greater was the amount of plaque on an individual's teeth. 

(a) Is the statistical design described here a completely randomized design or a randomized block design? Explain your answer. 

(b) If the statistical design is a randomized block design, what are the blocks?

In sampling from a population, state which type of sampling design corresponds to each of the following experimental designs: 

(a)  Completely randomized design 

(b)  Randomized block design