chapter 1. statistics may be defined as "a body of methods for making wise decisions in the...

Chapter 1

Statistics may be defined as "a body of methods for making wise decisions in the face of uncertainty." ~W.A. Wallis

LEARNING GOAL

Understand the two meanings of the term statistics and the basic ideas behind any statistical study, including the relationships among the study’s population, sample, sample statistics, and population parameters.

Two Definitions of Statistics

•Statistics is the science of collecting, organizing, and interpreting data.

•Statistics are the data (numbers or other pieces of information) that describe or summarize something.

Definitions

The population in a statistical study is the complete set of people or things being studied.

Population parameters are specific characteristics of the population.

What are some specific characteristics of a population?

A political campaign worker wishes to conduct a poll to determine how her candidate is likely to fair in the upcoming state senate election.

A researcher wants to determine the average number of text messages sent per month by college students who have cell phones.

A college admissions counselor is interested in studying the relationship between high school GPA and composite score on the ACT.

For each of the following situations, describe the population being studied and identify some of the population parameters that would be of interest.

Definitions

A sample is a subset of the population from which data are actually obtained.

The raw data is the actual measurements or observations.

Sample statistics are characteristics of the sample found by consolidating or summarizing the raw data.

Remember:

Parameters are characteristics of the population.

Statistics are characteristics of a sample.

5000 Americans are polled and the average (mean) amount of time watching television is 4.6 hours per day.

As shown in the Crime Watch article of the local newspaper, 37% of the crimes in the county in the month of September were crimes against property.

•Sample statistics are used to estimate population parameters.•A confidence interval is a range of values likely to contain the population parameter.• The confidence interval is:

from (sample statistic – margin of error) to (sample statistic + margin of error)

•The margin of error is usually defined to give a 95% confidence interval, meaning that 95% of samples of the size used in the study would contain the actual population parameter (and 5% would not).

January 13, 2012In U.S., Slightly More Want Obama to Set Course Than GOPObama has maintained slight edge since early 2011by Jeffrey M. Jones

PRINCETON, NJ -- Americans say they want Barack Obama (46%) rather than the Republicans in Congress (42%) to have more influence over the direction the nation takes in the next year. U.S. preferences have been closely divided on this question since early 2011, after Republicans won a majority in the House of Representatives, but Obama has consistently had a slim advantage, suggesting a real lead for him. Prior to 2011, Americans favored congressional Republicans.

http://www.gallup.com/poll/152027/Slightly-Obama-Set-Course-GOP.aspx



ImplicationsGiven a choice, Americans are fairly evenly divided in their preferences for whether President Obama or the Republicans in Congress should have more influence over the course of the nation during the next year, with Obama having a slight edge.Obama's slightly better positioning is consistent with the president's usual advantage on this question, with Americans generally preferring the opposition party in Congress only when the president is highly unpopular. Obama is certainly not popular at the moment, but on a relative basis he is more popular than the Republicans in Congress.

Survey MethodsResults for this Gallup poll are based on telephone interviews conducted Jan. 5-8, 2012, with a random sample of 1,011 adults, aged 18 and older, living in all 50 U.S. states and the District of Columbia.For results based on the total sample of national adults, one can say with 95% confidence that the maximum margin of sampling error is ±4 percentage points.Interviews are conducted with respondents on landline telephones and cellular phones, with interviews conducted in Spanish for respondents who are primarily Spanish-speaking. Each sample includes a minimum quota of 400 cell phone respondents and 600 landline respondents per 1,000 national adults, with additional minimum quotas among landline respondents by region. Landline telephone numbers are chosen at random among listed telephone numbers. Cell phone numbers are selected using random-digit-dial methods. Landline respondents are chosen at random within each household on the basis of which member had the most recent birthday.Samples are weighted by gender, age, race, Hispanic ethnicity, education, region, adults in the household, and phone status (cell phone only/landline only/both, cell phone mostly, and having an unlisted landline number). Demographic weighting targets are based on the March 2011 Current Population Survey figures for the aged 18 and older non-institutionalized population living in U.S. telephone households. All reported margins of sampling error include the computed design effects for weighting and sample design.In addition to sampling error, question wording and practical difficulties in conducting surveys can introduce error or bias into the findings of public opinion polls.View methodology, full question results, and trend data.

http://www.gallup.com/file/poll/152039/More_Influence_Congress_Approval_120113.pdf

Gallup Daily: Obama Job ApprovalEach result is based on a three-day rolling average

January 13, 2011

•On January 16, 2012, President Obama’s approval rating was 46%.•Gallup tracks daily the percentage of Americans who approve or disapprove of the job Barack Obama is doing as president. Results are based on telephone interviews with approximately 1,500 national adults; Margin of error is ±3 percentage points.•Find the 95% confidence interval.•Is it possible that the majority of Americans approve of the job President Obama is doing?

An internet supplier of refilled ink cartridges for ink jet printers sold cartridges to 30,000 customers over the past two months. A random sample of 1000 of those customers revealed that 18% were not happy with their purchase. The margin of error was 2%. •Identify the population.•Identify the sample.•Is 18% a parameter or a statistic?•What range is likely to contain the population parameter?

Definitions

Descriptive Statistics—describes raw data in the form of graphics and sample statistics

Inferential Statistics—infers or estimates population parameters from sample data

We will use descriptive statistics from sample data to make inferences about population parameters.

Basic Steps in a Statistical Study1. State the goal of your study precisely. Identify

the population and what you want to determine.

2. Choose a sample from the population using an appropriate sampling technique.

3. Collect raw data from the sample. Summarize by finding sample statistics of interest.

4. Use the sample statistics to make inference about the population parameter.

5. Draw conclusions. Determine what you have learned and if the goal was achieved.

A researcher wanted to study the relationship between coffee drinking and heart problems. She interviewed all patients at several area hospitals and found that patients in the hospital for treatment of heart problems drank an average of 2.7 cups of coffee per day (margin of error is 1.4 cups) while those patients being treated for other problems averaged only 1.4 cups of coffee per day (margin of error is 0.8 cups). What conclusion can you draw about coffee as a cause of heart problems from this information?

a. Coffee is bad for you.b. Coffee is good for you.c. Heart patients drink more coffee than other patients.d. None, because only patients in hospitals were

interviewed.

End of 1.1

1.2 Sampling

Not everything that can be counted counts, and not everything that counts can be counted.

Albert EinsteinLearning Goal—Understand the importance

of choosing a representative sampling and become familiar with several common methods of sampling.

Definitions Census—collection of data from every member of a

population. Sample—collecting data from part of a population. Representative sample—a sample in which the relevant

characteristics of the sample members are generally the same as the characteristics of the population.

What are advantages and disadvantages of collecting data using a census?

What are advantages and disadvantages of collecting data using a representative sample?

How will you know if the sample represents the population?

Countdown to Census Day: April 1, 2010

•The census is a count of everyone living in the United States every 10 years. •The census is mandated by the U.S. Constitution. •2010 is a census year. •Your participation in the census is required by law. •It takes less than 10 minutes to complete. •Federal law protects the personal information you share during the census. •Census data are used to distribute Congressional seats to states, to make decisions about what community services to provide, and to distribute $300 billion in federal funds to local, state and tribal governments each year.

What is the census?

http://2010.census.gov/2010census/

Bias A statistical study suffers from bias if its design or

conduct tends to favor certain results.When can bias occur? When members of the sample differ in some specific

way from members of the population. Example—Use members of SEMO’s football team to predict the average weight of all college males.

When a researcher has a personal stake in the outcome. The researcher may intentionally or unintentionally distort the true meaning of the data.

Even if the study is done well, it may be reported in a biased fashion.

Sampling Methods

Simple Random Samples Systematic Sampling Convenience Samples Cluster Samples Stratified Samples

Simple Random Samples (SRS) A random sample is one in which every

member of the population has an equal chance of being selected to be part of the sample.

With simple random sampling every possible sample of a particular size has an equal chance of being selected.

Use a random number generator to find samples of size 5 from this class. Go to http://www.randomizer.org/

Systematic sampling

randomly select a member of the sampling frame for the sample

using a set procedure or rule, select the rest of the individuals for the sample for example, randomly select an individual from the

sampling frame, and then select every 25th member of the sampling frame to be in the sample

Each individual has an equal likelihood of being chosen, but not every group has an equal likelihood of being chosen.

Convenience Samples

• The sample is chosen for convenience rather than by a more sophisticated procedure.

• Self-selected sample—when people choose whether or not to be part of the sample.

Cluster Samples

Cluster sampling involves the selection of all members in randomly selected groups, or clusters.

Divide the population into clusters. Randomly choose clusters. All subjects in the chosen clusters make up the sample. For example—The clusters could be the ER in each

hospital in a city. Randomly choose 5 hospitals and survey all the ER triage nurses in the 5 hospitals about certain practices.

Stratified SamplingStratified sampling involves randomly selecting members

from each stratum. first divide the population into groups of similar

individuals, called strata second, choose a separate simple random sample in

each stratum third, combine these simple random samples to form the

full sample For example—choose a simple random sample of 50

freshmen, 50 sophomores, 50 juniors and 50 seniors. Together, these students make up the full sample of size 200.

Summary of Sampling Methods

Keep in mind the following three key ideas:

• A study can be successful only if the sample is representative of the population.

• A biased sample is unlikely to be a representative sample.

• Even a well-chosen sample may still turn out to be unrepresentative just because of bad luck in the actual drawing of the sample.


Simple Random Sampling (SRS): Every sample of the same size has an equal chance of being selected. Computers are often used to generate random numbers.

Summary of Sampling Methods


Systematic Sampling: Select every kth member.


Convenience Sampling: Use results that are readily available.


Cluster Sampling: Divide the population into clusters, randomly select some of those clusters, then choose all members of the selected clusters.


Stratified Sampling: Partition the population into at least two strata, then draw a sample from each.

Identify each sample as SRS, systematic sample, convenience sample, stratified sample, or cluster sample. In each case, state whether you think the sampling method is likely to yield a representative sample or a biased sample, and explain why.

A quality improvement technician samples every 500th bag of potato chips coming off the assembly line to test the chips for fat content. Identify the type of sampling.

A political science student randomly selects 100 names from the voter list of registered Democrats and 100 from the list of registered Republicans. Identify the type of sampling.

The Gallup Organization plans to conduct a poll of New York City residents with the “212” area code. Computers are used to randomly generate telephone numbers that are automatically called.

Southeast conducts a study of student drinking by randomly selecting 10 different classes and surveys all of the students in each of those classes.

A student surveys all students on his floor about residence life issues.

End of 1.2

1.3 Types of Statistical Studies

You can observe a lot by just watching.

Yogi Berra

LEARNING GOAL

Understand the differences between observational studies and experiments; recognize key issues in experiments, including the selection of treatment and control groups, the placebo effect, and blinding.


Definition

The subjects of a study are the people, animals (or other living things), or objects chosen for the sample; if the subjects are people, they may also be called the participants in the study.

Two Basic Types of Statistical Studies In an observational study, researchers

observe or measure characteristics of the subjects but do not attempt to influence or modify these characteristics

In an experiment, researchers apply some treatment and observe its effects on the subjects of the experiment.

Identify the study as an observational study or an experiment.

Researchers at a medical school want to compare two methods of treating blocked arteries. Fifty patients in similar condition are randomly assigned to two groups. One group is treated surgically and the other group is treated with drugs.

Researchers at a medical school want to compare the rates of birth defects in babies born to mothers whose diets are classified as low fat with those born to mothers whose diets are high in fat. For one year, they collect data on all babies and their mothers in area hospitals, using information obtained from the mothers to determine whether they fall in low fat or high fat diet category.

Definitions A variable is any item or quantity that can vary

or take on different values. The variables of interest in a statistical study

are the items or quantities that the study seeks to measure.

The explanatory variable may explain or cause the effect.

The response variable responds to changes in the explanatory variable.

State the type of study, observational or experiment, that would best answer the question. Identify the variables of interest. State the explanatory variable and response variable.

Do people who take a vitamin supplement have lower rates of colon cancer than those who do not take the supplement?

Does texting while driving cause accidents?

Observational Studies

In most observational studies the data is collected all at once (or as close to that as possible. Other types of observational studies:•Retrospective (or case-control) study—uses data from the past, such as official records or past interviews•Prospective (or longitudinal) study—set up to collect data in the future from groups that share common factors.

What type of study should be used?

You want to know how meth abuse during pregnancy effects the learning ability of a child?

You want to know whether ACT scores as a senior in high school predicts college GPA.

Experiments Most of the time, a treatment and a control group is

used to fairly reach conclusions in an experiment. Treatment group—group of subjects who receive

the treatment Control group—group of subjects who do not

receive the treatment Important—Both groups should be randomly

selected. Why?

An educator wants to compare the effectiveness of using songs to teach number and counting skills with that of a a standard curriculum. He tests the number and counting skills of each student in a class of kindergartners, then divides them into two groups. One group uses songs to learn, while the other studies a standard curriculum. At the end of the year, he retests all the students and compares the increase in the skills of the two groups.

Is this an experiment? Why or why not? Identify the treatment. Identify the treatment and control groups? How

should the students be put into these groups?

Confounding

A study suffers from confounding if the effects of different variables are mixed so we cannot determine the specific effects of the variables of interest.

The variables that lead to the confusion are called confounding variables.

Confounding An article in a women’s magazine says that women who nurse their babies feel

warmer and more receptive toward the infants than mothers who bottle-fee. The author concludes that nursing has desirable effects on the mother’s attitude toward the child. But women choose whether to nurse or bottle-feed. What are some confounding variables that may make you skeptical of this conclusion.

Most colleges determine scholarships using a student’s high school GPA and ACT score. The assumption is that a student with a high GPA and ACT score will be successful in college. What are some confounding variables that may effect this premise?

Strategies for Selecting Treatment and Control Groups1. Select groups at random. Each subject should have an equal chance of being assigned to either the treatment or control

2. Use sufficiently large groups. Make sure that the treatment and control groups are both sufficiently large that they are unlikely to differ in a significant way (aside from the fact that one group gets the treatment and the other does not).

Placebos A placebo lacks the active ingredients of a

treatment being tested in a study, but looks or feels like the treatment so that participants cannot distinguish whether they are receiving the placebo or the real treatment.

The placebo effect—patients improve simply because they believe they are receiving a useful treatment.

Are placebos ethical?

A nutritionist wants to conduct a study to determine the efficacy of an herb as an aid in weight loss. She randomly assigns half of a group of overweight persons to a treatment group who are given herbs with instructions for its use and a planned diet for six weeks.

•The other half of the group is to be the control group. What will they be given and told to do?

•Identify confounding variables.

A mathematics teacher wanted to determine whether assigning homework had a beneficial effect on student learning in first year algebra. His class met a 8:00 in the morning and he obtained the cooperation of another teacher of the same class that met at 2:00 in the afternoon. He gave his class no homework while the other teacher continued to assign homework as he usually did. Both teachers gave the same tests so that they could compare the results.

•What is the treatment?

•What is the explanatory variable?

•What is the response variable?

•Name some confounding variables?

•Is this good experimental design? Why or why not?

DefinitionAn experimenter effect occurs when a researcher or experimenter somehow influences subjects through such factors as facial expression, tone of voice, or attitude.

.

Blinding in Experiments Single-blind experiment--participants do not

know whether they are members of the treatment or control group, but the experimenters do know

Double-blind experiment--neither the participants or the experimenters know who belongs to the treatment or control group

Why is single-blind important? Double-blind? How is a double-blind experiment feasible?

Design an experiment to test the effectiveness of Sleepeze, a new drug that allegedly cures insomnia. There are 90 subjects suffering from insomnia that have agreed to participate.

•Specify how the groups will be chosen and the treatment given to each group.

•Should the experiment be single-blind, double-blind or neither?


Meta-Analysis

DefinitionIn a meta-analysis, researchers review many past studies. The meta-analysis considers these studies as a combined group, with the aim of finding trends that were not evident in the individual studies.

End of 1.3

1.4 Should You Believe a Statistical Study?

If I have ever made any valuable

discoveries, it has been owing more to patient attention, than to any other talent.

Isaac Newton

LEARNING GOAL

Be able to evaluate statistical studies that you hear about in the media, so that you can decide whether the results are meaningful.


Eight Guidelines for Critically Evaluating a Statistical Study

1. Identify the goal of the study, the population considered, and the type of study.

2. Consider the source, particularly with regard to whether the researcher may be biased.

3. Examine the sampling method to decide whether it is likely to produce a representative sample.

4. Look for problems in defining or measuring the variables of interest, which can make it difficult to interpret any reported results.


5. Watch out for confounding variables that can invalidate the conclusions of a study.

6. Consider the setting and wording in surveys or polls, looking for anything that might tend to produce inaccurate or dishonest responses.

7. Check that results are fairly represented in graphics and concluding statements, because both researchers and media often create misleading graphics or jump to conclusions that the results do not support.

8. Stand back and consider the conclusions. Did the study achieve its goals? Do the conclusions make sense? Do the results have any practical significance?

Eight Guidelines for Critically Evaluating a Statistical Study


Guideline 1: Identify the Goal, Population and Type of Study

Based on what you hear or read about a study, try to answer these basic questions:

• What was the study designed to determine?

• What was the population under study? Was the population clearly and appropriately defined?

• Was the study an observational study, an experiment, or a meta-analysis?


Guideline 2: Consider the SourceStatistical studies are supposed to be objective, but the people who carry them out and fund them may be biased.

It is therefore important to consider the source of a study and evaluate the potential for biases that might invalidate the study’s conclusions.

DefinitionPeer review is a process in which several experts in a field evaluate a research report before the report is published.


Guideline 3: Examine the Sampling MethodA statistical study cannot be valid unless the sample is representative of the population under study.

DefinitionsSelection bias (or a selection effect) occurs whenever researchers select their sample in a biased way.

Participation bias occurs any time participation in a study is voluntary.

A self-selected survey (or voluntary response survey) is one in which people decide for themselves whether to be included in the survey.


Guideline 4: Look for Problems in Defining or Measuring the Variable of Interest

Results of a statistical study may be difficult to interpret if the variables under study are difficult to define or measure.


Guideline 5: Watch Out for Confounding Variables

Often, variables that are not intended to be part of the study can make it difficult to interpret results properly.


Guideline 6: Consider the Setting and Wording in Surveys

Even when a survey is conducted with proper sampling and with clearly defined terms and questions, you should watch for problems in the setting or wording that might produce inaccurate or dishonest responses.

Dishonest responses are particularly likely when the survey concerns sensitive subjects.


Guideline 7: Check That Results Are Fairly Represented in Graphics or Concluding Statements

Even when a statistical survey is done well, it may be misrepresented in graphics or concluding statements.


Guideline 8: Stand Back and Consider the ConclusionsAsk yourself questions such as these:

• Did the study achieve its goals?

• Do the conclusions make sense?

• Can you rule out alternative explanations for the results?

• If the conclusions make sense, do they have any practical significance?

End of Chapter 1

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