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📚 Mostly Harmless Statistics
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1 Introduction to Data2 Organizing Data3 Descriptive Statistics4 Probability5 Discrete Probability Distributions6 Continuous Probability Distributions7 Confidence Intervals for One Population8 Hypothesis Tests for One Population9 Hypothesis Tests and Confidence Intervals for Two Populations10 Chi-Square Tests11 Analysis of Variance12 Correlation and Regression13 Nonparametric Tests

1 Introduction to Data

1.01: Introduction

Scientists seek to answer questions using rigorous methods and careful observations. These observations - collected from the likes of field notes, surveys, and experiments - form the backbone of a statistical investigation and are called data. Statistics is the study of how best to collect, analyze, and draw conclusions from data. It is helpful to put statistics in the context of a general process of investigation:

  1. Identify a question or problem.
  2. Collect relevant data on the topic.
  3. Analyze the data.
  4. Form a conclusion.

Statistics as a subject focuses on making stages 2-4 objective, rigorous, and efficient. That is, statistics has three primary components: How best can we collect data? How should the data be analyzed? What can we infer from the analysis?

The topics scientists investigate are as diverse as the questions they ask. However, many of these investigations can be addressed with a small number of data collection techniques, analytic tools, and fundamental concepts in statistical inference.

You are exposed to statistics regularly. If you are a sports fan, then you have the statistics for your favorite player. If you are interested in politics, then you look at the polls to see how people feel about certain issues or candidates. If you are an environmentalist, then you research arsenic levels in the water of a town or analyze the global temperatures. If you are in the business profession, then you may track the monthly sales of a store or use quality control processes to monitor the number of defective parts manufactured. If you are in the health profession, then you may look at how successful a procedure is or the percentage of people infected with a disease. There are many other examples from other areas.

“There are of course many problems connected with life, of which some of the most popular are: Why are people born? Why do they die? Why do they want to spend so much time wearing digital watches?” (Adams, 2002)

To understand how to collect and analyze data, you need to understand what the field of statistics is. Many of the words defined throughout this course have common definitions that are also used in non-statistical terminology. In statistics, some of these terms have slightly different definitions. It is important that you notice the difference and utilize the statistical definitions.

Statistics is the study of how to collect, organize, analyze, and interpret data collected from a group

There are two main branches of statistics. One is called descriptive statistics, which is where you collect, organize and describe data. The other branch is called inferential statistics, which is where you interpret data. First, you need to look at descriptive statistics since you will use the descriptive statistics when making inferences. In order to use inferential statistics, we will briefly touch on a completely new topic called Probability. Once we get some background in probability combined with your knowledge of descriptive statistics, we will move into Inferential Statistics.

1.02: Samples vs. Populations

The first thing to decide in a statistical study is whom you want to measure and what you want to measure. You always want to make sure that you can answer the question of whom you measured and what you measured. The “who” is known as the individual and the “what” is known as the variable.

Individual – a person, case or object that you are interested in finding out information about.

Variable (also known as a random variable) – the measurement or observation of the individual.

Population – is the total set of all the observations that are the subject of a study.

Notice, the population answers “who” you want to measure and the variable answers “what” you want to measure. Make sure that you always answer both of these questions or you have not given the audience reading your study the entire picture. As an example, if you just say that you are going to collect data from the senators in the United States Congress, you have not told your reader what you are going to collect. Do you want to know their income, their highest degree earned, their voting record, their age, their political party, their gender, their marital status, or how they feel about a particular issue? Without telling “what” you what to measure, your reader has no idea what your study is actually about.

Sometimes the population is very easy to collect. If you are interested in finding the average age of all of the current senators in the United States Congress, there are only 100 senators. This would not be hard to find. However, if instead you were interested in knowing the average age that a senator in the United States Congress first took office for all senators that ever served in the United States Congress, then this would be a bit more work. It is still doable, but it would take a bit of time to collect. However, what if you are interested in finding the average diameter at breast height of all Ponderosa Pine trees in the Coconino National Forest? This data would be impossible to collect. What do you do in these cases? Instead of collecting the entire population, you take a smaller group of the population, a snapshot of the population. This smaller group, called a sample, is a subset of the population, see Figure 1-1.

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Figure 1-1

Sample – a subset from the population.

Consider the following three research questions:

  1. What is the average mercury content in albacore tuna in the Pacific Ocean?
  2. Over the last 5 years, what is the average time to complete a degree for Portland State University undergraduate students?
  3. Does a new drug reduce the number of deaths in patients with severe heart disease?

Each research question refers to a target population. In the first question, the target population is all albacore tuna in the Pacific Ocean, and each fish represents a case.

A sample represents a subset of the cases and is often a small fraction of the population. For instance, 60 albacore tuna in the population might be selected and the mercury level is measured in each fish. The sample average of the 60 fish may then be used to provide an estimate of the population average of all the fish and answer the research question.

We use the lower-case n to represent the number of cases in the sample and the upper-case N to represent the number of cases in the population.

n = sample size.

N = population size.

How the sample is collected can determine the accuracy of the results of your study. There are many ways to collect samples. No sampling method is perfect, but some methods are better than other methods. Sampling techniques will be discussed in more detail later.

For now, realize that every time you take a sample you will find different data values. The sample is a snapshot of the population, and there is more information than is in this small picture. The idea is to try to collect a sample that gives you an accurate picture, but you will never know for sure if your picture is the correct picture. Unlike previous mathematics classes, where there was always one right answer, in statistics there can be many answers, and you do not know which are right.

The sample average in this case is the statistic, and the population average is the parameter. We use sample statistics to make inferences, educated guesses made by observation, about the population parameter.

Once you have your data, either from a population or from a sample, you need to know how you want to summarize the data.

As an example, suppose you are interested in finding the proportion of people who like a candidate, the average height a plant grows to using a new fertilizer, or the variability of the test scores. Understanding how you want to summarize the data helps to determine the type of data you want to collect. Since the population is what we are interested in, then you want to calculate a number from the population. This is known as a parameter.

Parameter – An unknown quantity from the population. Usually denoted with a Greek letter (for example μ “mu”). This number is a fixed, unknown number that we want to estimate.

As mentioned already, it is hard to collect the entire population. Even though this is the number you are interested in, you cannot really calculate it. Instead, you use the number calculated from the sample, called a statistic, to estimate the parameter.

Statistic – a number calculated from the sample. Usually denoted with a ^ (called a hat, for example \(\hat{p}\) “p-hat”) or a – (called a bar, for example \(\bar{x}\) “x-bar”) above the letter.

Since most samples are not exactly the same, the statistic values are going to be different from sample to sample. Statistics estimate the value of the parameter, but again, you do not know for sure if your statistic is correctly estimating the parameter.

1.03: Collecting Data and Sampling Techniques

Qualitative vs. Quantitative

Variables can be either quantitative or qualitative. Quantitative variables are numeric values that count or measure an individual. Qualitative variables are words or categories used to describe a quality of an individual. Qualitative variables are also called categorical variables and can sometimes have numeric responses that represent a category or word.

Qualitative or categorical variable – answer is a word or name that describes a quality of the individual.

Quantitative or numerical variable – answer is a number (quantity), something that can be counted or measured from the individual.

Each type of variable has different graphs, parameters and statistics that you find. Quantitative variables usually have a number line associated with graphical displays. Qualitative variables usually have a category name associated with graphical displays.

Examples of quantitative variables are number of people per household, age, height, weight, time (usually things we can count or measure). Examples of qualitative variables are eye color, gender, sports team, yes/no (usually things that we can name).

When setting up survey questions it is important to know what statistical questions you would like the data to answer. For example, a company is trying to target the best age group to market a new game. They put out a survey with the ordinal age groupings: baby, toddler, adolescent, teenager, adult, and elderly. We could narrow down a range of ages for, say, teenagers to 13-19, although many 19-year-olds may record their response as an adult. The company wants to run an ad for the new game on television and they realize that 13-year-olds do not watch the same shows nor in the same time slots as 19-year-olds. To narrow down the age range the survey question could have just asked the person’s age. Then the company could look at a graph or average to decide more specifically that 17-year-olds would be the best target audience.

Types of Measurement Scales

There are four types of data measurement scales: nominal, ordinal, interval and ratio.

Nominal data is categorical data that has no order or rank, for example the color of your car, ethnicity, race, or gender.

Ordinal data is categorical data that has a natural order to it, for example, year in school (freshman, sophomore, junior, senior), a letter grade (A, B, C, D, F), the size of a soft drink (small, medium, large) or Likert scales. A Likert scale is a numeric scale that indicates the extent to which they agree or disagree with a series of statements. Interval data is numeric where there is a known difference between values, but zero does not mean “nothing.”

Interval data is ordinal, but you can now subtract one value from another and that subtraction makes sense. You can do arithmetic on this data. For example, Fahrenheit temperature, 0° is cold but it does not mean that no temperature exists. Time, dates and IQ scores are other examples.

Ratio data is numeric data that has a true zero, meaning when the variable is zero nothing is there. Most measurement data are ratio data. Some examples are height, weight, age, distance, or time running a race.

Here are some ways to help you decide if the data are nominal, ordinal, interval, or ratio. First, if the variable is words instead of numbers then it is either nominal or ordinal data. Now ask yourself if you can put the data in a particular order. If you can order the names then this is ordinal data. Otherwise, it is nominal data. If the variable is numbers (not including words coded as numbers like Yes = 1 and No = 0), then it is either interval or ratio data. For ratio data, a value of zero means there is no measurement. This is known as the absolute zero. If there is an absolute zero in the data, then it means it is ratio. If there is no absolute zero, then the data are interval. An example of an absolute zero is if you have $0 in your bank account, then you are without money. The amount of money in your bank account is ratio data. Word of caution, sometimes ordinal data is displayed using numbers, such as 5 being strongly agree, and 1 being strongly disagree. These numbers are not really numbers. Instead, they are used to assign numerical values to ordinal data. In reality, you should not perform any computations on this data, though many people do. If there are numbers, make sure the numbers are inherent numbers, and not numbers that were randomly assigned.

Likert scales are frequently misused as quantitative data. A Likert scale is a numeric scale that indicates the extent to which they agree or disagree with a series of statements. For example, if we look at the following 5-point Likert Scale:

  1. Strongly disagree
  2. Disagree
  3. Neither agree nor disagree
  4. Agree
  5. Strongly agree

Nominal and ordinal data are qualitative, while interval and ratio data are quantitative.

Likert scales are ordinal in that one can easily see that the larger number corresponds to a higher level of agreeableness. Some people argue that since there is a one-unit difference between the numeric values Likert scales should be interval data. However, the number 1 is just a placeholder for someone that strongly disagrees. There is no way to quantify a one-unit difference between two different subjects that answered 1 or 2 on the scale. For example, one person’s response for strongly disagree could stem from the exact same reasoning behind another person’s response of disagree. People view subjects at different intensities that is not quantifiable.

Discrete vs. Continuous

Quantitative variables are discrete or continuous. This difference will be important later on when we are working with probability. Discrete variables have gaps between points that are countable, usually integers like the number of cars in a parking garage or how many people per household. A continuous variable can take on any value and is measurable, like height, time running a race, distance between two buildings. Usually, just asking yourself if you can count the variable then it is discrete and if you can measure the variable then it is continuous. If you can actually count the number of outcomes (even if you are counting to infinity), then the variable is discrete.

Discrete variables can only take on particular values like integers.

Discrete variables have outcomes you can count.

Continuous variables can take on any value.

Continuous variables have outcomes you can measure.

For example, think of someone’s age. They may report in a survey an integer value like 28 years-old. The person is not exactly 28 years-old though. From the time of their birth to the point in time that the survey respondent recorded, their age is a measurable number in some unit of time. A person’s true age has a decimal place that can keep going as far as the best clock can measure time. It is more convenient to round our age to an integer rather than 28 years 5 months, 8 days, 14 hours, 12 minutes, 27 seconds, 5 milliseconds or as a decimal 28.440206335775. Therefore, age is continuous.

However, a continuous variable like age could be broken into discrete bins, for example, instead of the question asking for a numeric response for a person’s age they could have had discrete age ranges where the survey respondent just checks a box.

  1. Under 18
  2. 18-24
  3. 25-35
  4. 36-45
  5. 46-62
  6. Over 62

When a survey question takes a continuous variable and chunks it into discrete categories, especially categories with different widths, you limit what type of statistics you can do on that data.

Figure 1-2 is a breakdown of the different variable and data types.

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Figure 1-2

Types of Sampling

If you want to know something about a population, it is often impossible or impractical to examine the entire population. It might be too expensive in terms of time or money to survey the population. It might be impractical: you cannot test all batteries for their length of lifetime because there would not be any batteries left to sell.

When you choose a sample, you want it to be as similar to the population as possible. If you want to test a new painkiller for adults, you would want the sample to include people of different weights, age, etc. so that the sample would represent all the demographics of the population that would potentially take the painkiller. The more similar the sample is to the population, the better our statistical estimates will be in predicting the population parameters.

There are many ways to collect a sample. No sampling technique is perfect, and there is no guarantee that you will collect a representative sample. That is unfortunately the limitation of sampling. However, several techniques can result in samples that give you a semi-accurate picture of the population. Just remember to be aware that the sample may not be representative of the whole population. As an example, you can take a random sample of a group of people that are equally distributed across all income groups, yet by chance, everyone you choose is only in the high-income group. If this happens, it may be a good idea to collect a new sample if you have the time and money.

When setting up a study there are different ways to sample the population of interest. The five main sampling techniques are:

  1. Simple Random Sample
  2. Systematic Sample
  3. Stratified Sample
  4. Cluster Sample
  5. Convenience Sample

A simple random sample (SRS) means selecting a sample size of n objects from the population so that every sample of the same size n has equal probability of being selected as every other possible sample of the same size from that population. For example, we have a database of all PSU student data and we use a random number generator to randomly select students to receive a questionnaire on the type of transportation they use to get to school. See Figure 1-3. Simple random sampling was used to randomly select the 18 cases.

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Retrieved from OpenIntroStatistics.

Figure 1-3

A stratified sample is where the population is split into groups called strata, then a random sample is taken from each stratum. For instance, we divide Portland by ZIP code and then randomly select n registered voters out of each ZIP code. See Figure 1-4. Cases were grouped into strata, then simple random sampling was employed within each stratum.

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Retrieved from OpenIntroStatistics.

Figure 1-4

A cluster sample is where the population is split up into groups called clusters, then one or more clusters are randomly selected and all individuals in the chosen clusters are sampled. Similar to the previous example, we split Portland up by ZIP code, randomly pick 5 ZIP codes and then sample every registered voter in those 5 ZIP codes. See Figure 1-5. Data were binned into nine clusters, three of these clusters were sampled, and all observations within these three clusters were included in the sample.

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Retrieved from OpenIntroStatistics.

Figure 1-5

A systematic sample is where we list the entire population, then randomly pick a starting point at the nth object, and then take every nth value until the sample size is reached. For example, we alphabetize every PSU student, randomly choose the number 7. We would sample the 7th, 14th, 21st, 28th, 35th, etc. student.

A convenience sample is picking a sample that is conveniently at hand. For example, asking other students in your statistics course or using social media to take your survey. Most convenience samples will give biased views and are not encouraged.

There are many more types of sampling, snowball, multistage, voluntary, purposive, and quota sampling to name some of the ways to sample from a population. We can also combine the different sampling methods. For example, we could stratify by rural, suburban and urban school districts, then take 3rd grade classrooms as clusters.

Guidelines for planning a statistical study

  1. Identify the individuals that you are interested in studying. Realize that you can only make conclusions for these individuals. As an example, if you use a fertilizer on a certain genus of plant, you cannot say how the fertilizer will work on any other types of plants. However, if you diversify too much, then you may not be able to tell if there really is an improvement since you have too many factors to consider.
  2. Specify the variable. You want to make sure the variable is something that you can measure, and make sure that you control for all other factors too. For example, if you are trying to determine if a fertilizer works by measuring the height of the plants on a particular day, you need to make sure you can control how much fertilizer you put on the plants (which is what we call a treatment), and make sure that all the plants receive the same amount of sunlight, water, and temperature.
  3. Specify the population. This is important in order for you to know for whom and what conclusions you can make.
  4. Specify the method for taking measurements or making observations.
  5. Determine if you are taking a census or sample. If taking a sample, decide on the sampling method.
  6. Collect the data.
  7. Use appropriate descriptive statistics methods and make decisions using appropriate inferential statistics methods.
  8. Note any concerns you might have about your data collection methods and list any recommendations for future.

Observational vs. Experimental

The section is an introduction to experimental design. This is a brief introduction on how to design an experiment or a survey so that they are statistically sound. Experimental design is a very involved process, so this is just a small overview.

There are two types of studies:

  1. An observational study is when the investigator collects data by observing, measuring, counting, watching or asking questions. The investigator does not change anything.
  2. An experiment is when the investigator changes a variable or imposes a treatment to determine its effect.

For instance, if you were to poll students to see if they favor increasing tuition, this would be an observational study since you are asking a question and getting data. Give a patient a medication that lowers their blood pressure. This is an experiment since you are giving the treatment and then getting the data.

Many observational studies involve surveys. A survey uses questions to collect the data and needs to be written so that there is no bias.

Bias is the tendency of a statistic to incorrectly estimate a parameter. There are many ways bias can seep into statistics. Sometimes we don’t ask the correct question, give enough options for answers, survey the wrong people, misinterpret data, sampling or measurement errors, or unrepresentative samples.

In an experiment, there are different options to assign treatments.

  1. Completely Randomized Experiment: In this experiment, the individuals are randomly placed into two or more groups. One group gets either no treatment or a placebo (a fake treatment); this group is called the control group. The groups getting the treatment are called the treatment groups. The idea of the placebo is that a person thinks they are receiving a treatment, but in reality, they are receiving a sugar pill or fake treatment. Doing this helps to account for the placebo effect, which is where a person’s mind makes their body respond to a treatment because they think they are taking the treatment when they are not really taking the treatment. Note, not every experiment needs a placebo, such as when using animals or plants. In addition, you cannot always use a placebo or no treatment. For example, if you are testing a new Ebola vaccination you cannot give a person with the disease a placebo or no treatment because of ethical reasons.
  2. Matched Pairs Design: This is a subset of the randomized block design where the treatments are given to two groups that can be matched up with each other in some way. One example would be to measure the effectiveness of a muscle relaxer cream on the right arm and the left arm of individuals, and then for each individual you can match up their right arm measurement with their left arm. Another example of this would be before and after experiments, such as weight of a person before and weight after a diet.
  3. Randomized Block Design: A block is a group of subjects that are considered similar or the same subject measured multiple times, but the blocks differ from each other. Then randomly assign treatments to subjects inside each block. For instance, a company has several new stitching methods for a soccer ball and would like to pick the ball that travels the fastest. We would expect variation in different soccer player’s abilities which we do not want affect our results. We randomly choose players to kick each of the new types of balls where the order of the ball design is also randomized. Figure 1-6 shows blocking using a variable depicting patient risk. Patients are first divided into low-risk and high-risk blocks, then each block is evenly separated into the treatment groups using randomization. This strategy ensures an equal representation of patients in each treatment group from both the low-risk and high-risk categories.
  4. Factorial Design: This design has two or more independent categorical variables called factors. Each factor has two or more different treatment levels. The factorial design allows the researcher to test the effect of the different factors simultaneously on the dependent variable. For example, an educator believes that both the time of day (morning, afternoon, evening) and the way an exam is delivered (multiple-choice paper, short answer paper, multiplechoice electronic, short answer electronic) affects a student’s grade on their exam.

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Retrieved from OpenIntroStatistics.

Figure 1-6

No matter which experiment type you conduct, you should also consider the following:

Replication: repetition of an experiment on more than one subject so you can make sure that the sample is large enough to distinguish true effects from random effects. It is also the ability for someone else to duplicate the results of the experiment.

Blind study is where the individual does not know which treatment they are getting or if they are getting the treatment or a placebo.

Double-blind study is where neither the individual nor the researcher knows who is getting the treatment and who is getting the placebo. This is important so that there can be no bias in the results created by either the individual or the researcher.

One last consideration is the time-period that you are collecting the data. There are different time-periods that you can consider.

Cross-sectional study: observational data collected at a single point in time.

Retrospective study: observational data collected from the past using records, interviews, and other similar artifacts.

Prospective (or longitudinal or cohort) study: Subjects are measured from a starting point over time for the occurrence of the condition of interest.

1.04: Chapter 1 Exercises

Chapter 1 Exercises

1. The dotplot shows the height of some 5-year-old children measured in inches. Use the distribution of heights to find the approximate answer to the question, “How many inches tall are 5-year-olds?”

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“‘Alright,’ he said, ‘but where do we start? How should I know? They say the Ultimate Answer or whatever is Forty‐two, how am I supposed to know what the question is? It could be anything. I mean, what's six times seven?’ Zaphod looked at him hard for a moment. Then his eyes blazed with excitement. ‘Forty-two!’ he cried.” (Adams, 2002)

2. The dotplot shows the height of some 3-year-old children measured in cm. You are asked, “How many cm tall are 3-year-olds?”

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  1. Is this a statistical question?
  2. Use the distribution of heights to approximate the answer for the question, “How many cm tall are 3-year-olds?”

3. What are statistics?

  1. A question with a variety of answers.
  2. A way to measure the entire population.
  3. The science of collecting, organizing, analyzing and interpreting data.
  4. A question from a survey.

4. What is a statistical question?

  1. A question where you expect to get a variety of answers and you are interested in the distribution and tendency of those answers.
  2. A question using reported statistics.
  3. A question on a survey.
  4. A question on a census.

5. Which of the following are statistical questions? Select all that apply.

  1. How old are you?
  2. What is the weight of a mouse?
  3. How tall are all 3-year-olds?
  4. How tall are you?
  5. What is the average blood pressure of adult men?

6. In 2010, the Pew Research Center questioned 1,500 adults in the United States to estimate the proportion of the population favoring marijuana use for medical purposes. It was found that 73% are in favor of using marijuana for medical purposes. Identify the individual, variable, population, sample, parameter and statistic.

  1. Percentage who favor marijuana for medical purposes calculated from sample.
  2. Set of 1,500 responses of United States adults who are questioned.
  3. All adults in the United States.
  4. Percentage who favor marijuana for medical purposes calculated from population.
  5. The response to the question “should marijuana be used for medical purposes?”
  6. An adult in the United States.

7. Suppose you want to estimate the percentage of videos on YouTube that are cat videos. It is impossible for you to watch all videos on YouTube so you use a random video picker to select 1,000 videos for you. You find that 2% of these videos are cat videos. Determine which of the following is an observation, a variable, a sample statistic, or a population parameter.

  1. Percentage of all videos on YouTube that are cat videos.
  2. A video in your sample.
  3. 2%
  4. Whether a video is a cat video.

8. A doctor wants to see if a new treatment for cancer extends the life expectancy of a patient versus the old treatment. She gives one group of 25 cancer patients the new treatment and another group of 25 the old treatment. She then measures the life expectancy of each of the patients. Identify the individual, variable, population, sample, parameter and statistic.

  1. Cancer patient given the new treatment and cancer patient given the old treatment.
  2. The two groups of 25 cancer patients given the old and new treatments.
  3. Average life expectancy of 25 cancer patients given the old treatment and average life expectancy of 25 cancer patients given the new treatment.
  4. Average life expectancy of all cancer patients given the old and new treatment.
  5. All cancer patients.
  6. Life expectancy of the cancer patients.

9. The 2010 General Social Survey asked the question, “After an average workday, about how many hours do you have to relax or pursue activities that you enjoy?” to a random sample of 1,155 Americans. The average relaxing time was found to be 1.65 hours. Determine which of the following is an individual, a variable, a sample statistic, or a population parameter.

  1. Average number of hours all Americans spend relaxing after an average workday.
  2. 1.65
  3. An American in the sample.
  4. Number of hours spent relaxing after an average workday. 10. In a study, the sample is chosen by dividing the population by gender, and choosing 30 people of each gender. Which sampling method is used?

11. In a study, the sample is chosen by separating all cars by size, and selecting 10 of each size grouping. What is the sampling method?

12. In a study, the sample is chosen by writing everyone’s name on a playing card, shuffling the deck, then choosing the top 20 cards. What is the sampling method?

13. In a study, the sample is chosen by asking people on the street. What is the sampling method?

14. In a study, the sample is chosen by selecting a room of the house, and appraising all items in that room. What is the sampling method?

15. In a study, the sample is chosen by surveying every 3rd driver coming through a tollbooth. What is the sampling method?

16. Researchers collected data to examine the relationship between air pollutants and preterm births in Southern California. During the study air pollution levels were measured by air quality monitoring stations. Specifically, levels of carbon monoxide (CO) were recorded in parts per million, nitrogen dioxide (NO2) and ozone (03) in parts per hundred million, and coarse particulate matter (PM10) in μg/m3 . Length of gestation data were collected on 143,196 births between the years 1989 and 1993, and air pollution exposure during gestation was calculated for each birth. The analysis suggested that increased ambient PM10 and, to a lesser degree, CO concentrations may be associated with the occurrence of preterm births. [B. Ritz et al. “Effect of air pollution on preterm birth among children born in Southern California between 1989 and 1993.” In: Epidemiology 11.5 (2000), pp. 502– 511.] In this study, identify the variables. Select all that apply.

  1. Ozone
  2. Carbon Monoxide
  3. PM10
  4. Preterm Births in California
  5. Length of Gestation
  6. 143,196 Births
  7. 1989-1993
  8. Nitrogen Dioxide

17. State whether each study is observational or experimental.

  1. You want to determine if cinnamon reduces a person’s insulin sensitivity. You give patients who are insulin sensitive a certain amount of cinnamon and then measure their glucose levels.
  2. A researcher wants to evaluate whether countries with lower fertility rates have a higher life expectancy. They collect the fertility rates and the life expectancies of countries around the world.
  3. A researcher wants to determine if diet and exercise together helps people lose weight over just exercising. The researcher solicits volunteers to be part of the study, and then randomly assigns the volunteers to be in the diet and exercise group or the exercise only group.
  4. You collect the weights of tagged fish in a tank. You then put an extra protein fish food in water for the fish and then measure their weight a month later.

18. The Buteyko method is a shallow breathing technique developed by Konstantin Buteyko, a Russian doctor, in 1952. Anecdotal evidence suggests that the Buteyko method can reduce asthma symptoms and improve quality of life. In a scientific study to determine the effectiveness of this method, researchers recruited 600 asthma patients aged 18-69 who relied on medication for asthma treatment. These patients were split into two research groups: one practiced the Buteyko method and the other did not. Patients were scored on quality of life, activity, asthma symptoms, and medication reduction on a scale from 0 to 10. On average, the participants in the Buteyko group experienced a significant reduction in asthma symptoms and an improvement in quality of life. [McGowan. “Health Education: Does the Buteyko Institute Method make a difference?” In: Thorax 58 (2003).] Which of the following is the main research question?

  1. The Buteyko method causes shallow breathing.
  2. The Buteyko method can reduce asthma symptoms and an improvement in quality of life.
  3. Effectiveness of the Buteyko method.
  4. The patients score on quality of life, activity, asthma symptoms and medication reduction.

19. Researchers studying the relationship between honesty, age and self-control conducted an experiment on 160 children between the ages of 5 and 15. Participants reported their age, sex, and whether they were an only child or not. The researchers asked each child to toss a fair coin in private and to record the outcome (white or black) on a paper sheet, and said they would only reward children who report white. Half the students were explicitly told not to cheat and the others were not given any explicit instructions. In the no instruction group, the probability of cheating was found to be uniform across groups based on child’s characteristics. In the group that was explicitly told to not cheat, girls were less likely to cheat, and while rate of cheating did not vary by age for boys, it decreased with age for girls. [Alessandro Bucciol and Marco Piovesan. “Luck or cheating? A field experiment on honesty with children.” In: Journal of Economic Psychology 32.1 (2011), pp. 73–78.] In this study, identify the variables. Select all that apply.

  1. Age
  2. Sex
  3. Paper Sheet
  4. Cheated or Not
  5. Reward for White Side of Coin
  6. White or Black Side of Coin
  7. Only Child or Not

20. Select the measurement scale Nominal, Ordinal, Interval or Ratio for each scenario.

  1. A person’s age.
  2. A person’s race.
  3. Age groupings (baby, toddler, adolescent, teenager, adult, elderly).
  4. Clothing brand.
  5. A person’s IQ score.
  6. Temperature in degrees Celsius.
  7. The amount of mercury in a tuna fish.

21. Select the measurement scale Nominal, Ordinal, Interval or Ratio for each scenario.

  1. Temperature in degrees Kelvin.
  2. Eye color.
  3. Year in school (freshman, sophomore, junior, senior).
  4. The weight of a hummingbird.
  5. The height of a building.
  6. The amount of iron in a person’s blood.
  7. A person’s gender.
  8. A person’s race.

22. State which type of variable each is, qualitative or quantitative?

  1. A person’s age.
  2. A person’s gender.
  3. The amount of mercury in a tuna fish.
  4. The weight of an elephant.
  5. Temperature in degrees Fahrenheit.

23. State which type of variable each is, qualitative or quantitative?

  1. The height of a giraffe.
  2. A person’s race.
  3. Hair color.
  4. A person’s ethnicity.
  5. Year in school (freshman, sophomore, junior, senior).

24. State whether the variable is discrete or continuous.

  1. A person’s weight.
  2. The height of a building.
  3. A person’s age.
  4. The number of floors of a skyscraper.
  5. The number of clothing items available for purchase.

25. State whether the variable is discrete or continuous.

  1. Temperature in degrees Celsius.
  2. The number of cars for sale at a car dealership.
  3. The time it takes to run a marathon.
  4. The amount of mercury in a tuna fish.
  5. The weight of a hummingbird.

26. State whether each study is cross-sectional, retrospective or prospective.

  1. To see if there is a link between smoking and bladder cancer, patients with bladder cancer are asked if they currently smoke or if they smoked in the past.
  2. The Nurses Health Survey was a survey where nurses were asked to record their eating habits over a period of time, and their general health was recorded.
  3. A new study is underway to track the eating and exercise patterns of people at different time-periods in the future, and see who is afflicted with cancer later in life.
  4. The prices of generic items are compared to the prices of the equivalent named brand items.

27. Which type of sampling method is used for each scenario, Simple Random Sampling (SRS), Systematic, Stratified, Cluster or Convenience?

  1. The quality control officer at a manufacturing plant needs to determine what percentage of items in a batch are defective. The officer chooses every 15th batch off the line and counts the number of defective items in each chosen batch.
  2. The local grocery store lets you survey customers during lunch hour on their preference for a new bottle design for laundry detergent.
  3. Put all names in a hat and draw a certain number of names out.
  4. The researcher randomly selects 5 hospitals in the United States then measures the cholesterol level of all the heart attack patients in each of those hospitals.

28. Which type of sampling method is used for each scenario, Simple Random Sampling (SRS), Systematic, Stratified, Cluster or Convenience?

  1. If you want to calculate the average price of textbooks, you could divide the individuals into groups by major and then conduct simple random samples inside each group.
  2. Obtain a list of patients who had surgery at a hospital. Divide the patients according to type of surgery. Draw simple random samples from each group.
  3. You want to measure whether a tree in the forest is infected with bark beetles. Instead of having to walk all over the forest, you divide the forest up into sectors, and then randomly pick the sectors that you will travel to. Then record whether a tree is infected or not for every tree in that sector.
  4. You select every 3rd customer that comes that orders from your website.
Answer to Odd Numbered Exercises

1) 42

3) c

5) b, c, e

7)

  1. A question where you expect to get a variety of answers and you are interested in the distribution and tendency of those answers.
  2. A question using reported statistics.
  3. A question on a survey.
  4. A question on a census.
a) Population Parameter b) Observation c) Sample Statistic d) Variable

9)

  1. A question where you expect to get a variety of answers and you are interested in the distribution and tendency of those answers.
  2. A question using reported statistics.
  3. A question on a survey.
  4. A question on a census.
a) Population Parameter b) Sample Statistic c) Individual d) Variable

11) Stratified

13) Convenience

15) Systematic

17)

  1. Experimental
  2. Observational
  3. Experimental
  4. Experimental

19) a, b, d, f, g

21)

  1. Ratio
  2. Nominal
  3. Ordinal
  4. Ratio
  5. Ratio
  6. Ratio
  7. Nomina

23)

  1. Quantitative
  2. Qualitative
  3. Qualitative
  4. Qualitative
  5. Qualitative

25)

  1. Continuous
  2. Discrete
  3. Continuous
  4. Continuous
  5. Continuous

27)

  1. Systematic
  2. Convenience
  3. Random
  4. Cluster

From Mostly Harmless Statistics by Rachel L. Webb, adapted from LibreTexts. Licensed CC BY-SA 4.0. XYZ Homework OER web edition, adapted with 7 verified corrections (see errata).