Math notecards for Exam 3
Determine whether the following variable is qualitative or quantitative, and explain why. The eye color of students in a class
The variable is qualitative because colors are nonnumerical categories.
Consider an experiment in which the weights of 6-year-olds are measured. What is the variable of interest in this study?
The weights of 6-year-olds
What problems can confounding variables cause? Select all that apply.
They can cause the study to favor certain results unexpectedly, they can cause incorrect conclusions to be drawn from the study.
Choose the best answer to the following question. Explain your reasoning with one or more complete sentences. Suppose that four 6-sided dice are tossed. How many outcomes are possible?
6×6×6×6, because each of the four dice have six possible outcomes
You have a table listing ten tourist attractions and their annual numbers of visitors. Which type of display would be most appropriate for these data? Explain your reasoning.
A bar graph would be most appropriate because the goal is to compare the numbers of visitors that go to the attractions. A bar graph could do this because it uses a set of bars to represent the frequency of each category.
What is a frequency table? Explain what is meant by the categories and frequencies. What is meant by relative frequency? What is meant by cumulative frequency?
A frequency table has two columns. The first column lists all of the categories of data. The second column lists the frequency of each category, which is the number of data values in the category.
What is a placebo?
A placebo lacks the active ingredients of a treatment being tested in a study, but is identical in appearance to the treatment.
What is a self-selected survey?
A survey where people decide for themselves whether to be part of the survey
In order to gauge public opinion about how to handle Iran's growing nuclear program, a research group surveyed 990 American women by telephone and asked them to rate the threat Iran's nuclear program poses to the world on a scale of 1 to 10. Describe the population, sample, population parameters, and sample statistics. Identify the population in the given problem. Choose the correct answer below.
All American women
Give an example of qualitative data. Explain your reasoning.
An example is ratings of restaurants because they would be based on subjective opinions.
Events are overlapping if they _____ occur together. They are non-overlapping if they ______ occur together.
can, cannot
relative frequency probability
Based on observations or experiments, is the relative frequency of the event of interest
Combinations
Counting the number of ways something can happen
How did the gambling habits of Chevalier de Mère help launch the mathematical study of probability?
The Chevalier calculated his chance of winning incorrectly, but still won a lot of money. When he began losing money, he turned to a mathematician to figure out his error.
What two types of graphs are most common when the categories are quantitative data? Describe the construction of each.
For quantitative data categories, the two most common types of graphs are histograms and line charts. To make a histogram, bin the data, organize the bins into the natural order of the numerical categories, and then create the bars with no spaces in between each category. To make a line chart, use a dot to represent the frequency of each data category, and after the dots are placed, connect them with straight line segments.
the eight guidelines for evaluating statistical studies
Get a Big Picture View of the Study. Understand the goal of the study, population under the study, and whether the study was observational or an experiment, consider the Source. Look for sources of bias, look for Bias in the Sample. Decide if sample was a representative sample, look for Problems in Defining or Measuring the Variables of Interest. Ambiguity in variables make it difficult to interpret results, beware of Confounding Variables. Results may not be valid due to any confounding variables, consider the Setting and Wording in Surveys. Look for anything that may produce inaccurate/dishonest responses, check That Results Are Presented Fairly. Check if study supports conclusions presented, stand Back and Consider the Conclusions. Decide if the study achieved its goals. Do its conclusions make sense and are practical?
You have a list of the GPAs of 100 college graduates, precise to the nearest 0.001. You want to make a frequency table for these data. Explain your reasoning.
Group all the data into bins 0.2 of a grade point wide. It is very likely that small ranges of GPAs would be easier to analyze than the individual GPAs would be.
What is meant by cases and controls in a retrospective study?
The cases effectively represent a treatment group and the controls effectively represent the control group.
Much like sound bytes of news stories, statistical studies are often reduced to one- or two- sentence stat-bytes. For the following stat-byte, discuss what crucial information is missing and what more you would want to know before acting on the study. A cable network reports on a survey of America's top restaurants that found that "only nine restaurants achieved a rare 29 out of a possible 30 rating and none of those restaurants are in the Big Apple."
How the quality of restaurants was measured, who the respondents in the survey were, how the respondents were selected
How can making an experiment single-blind or double-blind help?
If an experiment is blinded, then any effect arising from psychological factors should affect all groups equally.
Why is it so important that a statistical study use a representative sample?
If the sample fairly represents the population as a whole, then it is reasonable to make inferences from the sample to the population.
What two types of graphs are most common when the categories are qualitative data? Describe the construction of each.
In a bar graph, the categories are clearly indicated along the horizontal axis. Over each category is a rectangle whose height indicates the frequency or relative frequency of the category. Numbers along the vertical axis clearly indicate the scale, In a pie chart, each category corresponds to a wedge of a circle. The size of each wedge is proportional to the relative frequency of the category it represents.
Describe and contrast observational studies and experiments. Choose the correct answer below.
In an observational study, researchers do not attempt to influence the characteristics of the sample members, while in an experiment, researchers apply a treatment and then observe the effects of the treatment.
What is a histogram?
It is a bar graph for quantitative data.
In a class of 100 students, 68 students received a grade of B. What was the relative frequency of a B grade? Explain your reasoning.
It was .68 because the relative frequency of a category is the number of data values that fall in that category divided by the total number of data values.
Consider a study designed to learn about the social networks of all college freshmen, in which researchers randomly interviewed students living in on-campus dormitories. How will the study suffer due to the way this sample was chosen?
It will suffer from selection bias. Selecting students living in on-campus dormitories may not accurately reflect the population.
What are confounding variables, and what problems can they cause?
Items or quantities that are not intended to be part of the study
A teacher has 26 students, and 5 of them will be chosen to participate in a play that has 5 distinct characters. Which of the following questions requires calculating permutations to solve? Explain your reasoning.
Once the 5 children have been chosen, how many different ways can their roles be arranged? To answer this question, the total number of possible arrangements, or permutations, needs to be found. Since the roles are distinct, the order of the children matters.
Distinguish between an outcome and an event in probability. Give an example in which the same event can occur through two or more outcomes.
Outcomes are the most basic possible results of observations or experiments. An event consists of one or more outcomes that share a property of interest.
Imagine that a survey of randomly selected people finds that people who used sunscreen were more likely to have been sunburned in the past year. Which explanation for this result seems most likely?
People who use sunscreen are more likely to spend time in the sun.
theoretical probability
Probability based on comparing the number of possible favorable outcomes to the number of total possible outcomes
What is the distinction between qualitative data and quantitative data? Give a few examples of each.
Qualitative data describe categories, while quantitative data represent counts or measures. Brand names of shoes in a consumer survey and eye colors are examples of qualitative data. Heights of students and quiz scores are examples of quantitative data.
Identify the sample for the given problem. Choose the correct answer below
The 990 American women surveyed by telephone
Identify any potential sources of bias in the following study. An exit poll designed to predict the winner of a local election uses voluntary surveys with every Republican who votes between 7:00 and 9:30 a.m
Section bias only
four common sampling methods
Simple random sampling; Systematic sampling; Convenience sampling; Stratified sampling
Give an example in which the same event can occur through two or more outcomes. Choose the correct answer below.
Suppose you roll a fair, six-sided die. The possible outcomes are rolling the number 1, 2, 3, 4, 5, or 6. The event of rolling an even number will occur with the three outcomes 2, 4, and 6.
Describe the importance of labeling on a graph, and briefly discuss the kinds of labels that should be included on graphs.
The categories should be clearly indicated along the horizontal axis, and there should be a label that describes the variable that the categories represent. Without either of these, it would not be possible to know what data the graph is supposed to show, If multiple data sets are displayed on a single graph, there should be a legend or key to identify the individual data sets. Otherwise, comparisons among the data sets would not be possible and confusion among the data sets could occur, Numbers along the vertical axis should clearly indicate the scale, and there should be a label that describes the variable shown on the vertical axis. Without either of these, there would be no way of interpreting the data shown in the graph, The graph should have a title or caption (or both) that explains what is being shown and, if applicable, lists the source of the data. Without this label, it might not be clear what the graph is supposed to illustrate.
Consider an experiment designed to test whether cash incentives improve school attendance. The researcher chooses two groups of 100 high school students. She offers one group $10 for every week of perfect attendance. She tells the other group that they are part of an experiment but does not give them any incentive. How would you describe the experiment above?
The experiment is not blind because the researcher and students know which groups the students are in.
What do we mean by variables of interest in a study?
The items or quantities that the study seeks to measure
Identify the population parameter in the given study. Choose the correct answer below.
The mean threat rating of all American women on Iran's nuclear program
Identify the sample statistic. Choose the correct answer below.
The mean threat rating on Iran's nuclear program of the 990 American women surveyed by telephone
Describe the placebo effect and how it can make experiments difficult to interpret. Choose the correct answer below.
The placebo effect refers to a situation in which a patient improves simply because they believe they are receiving a useful treatment. It can sometimes be difficult or impossible to distinguish between effects that arise from the actual treatment and those that arise from psychological factors.
A researcher wants to determine the percentage given below. Describe how the researcher should apply the five basic steps in a statistical study. (Assume that all the people in the poll answered truthfully.) Determine how to apply the first basic step in a statistical study in this situation. Choose the correct answer below.
The population is all elementary school teachers. The researcher wants to estimate the percentage in this population that eat while teaching.
You conduct a poll in which you randomly select 970 registered voters from Texas and ask if they approve of the job their governor is doing. What is the population for this study?
The population is all registered voters in the state of Texas because this is the entire group that is being selected from randomly.
Outcomes
The possible results of an experiment
What is the purpose of a placebo?
The purpose of a placebo is to prevent participants from knowing whether they belong to the treatment group or the control group.
What is the purpose of binning? Give an example in which binning is useful.
The purpose of binning is to analyze the frequency of quantitative data grouped into categories that cover a range of possible values. A useful example is grouping quiz scores with a maximum score of 40 points with 10-point bins. The first bin contains scores 0-9, the second bin contains scores 10-19, and so on.
Discuss the differences between the following questions, each of which could be the basis for a statistical study. bullet What percentage of Internet dates lead to marriage? bullet What percentage of marriages begin with Internet dates?
The questions have different populations.
Determine how to apply the second basic step in a statistical study in this situation. Choose the correct answer below.
The researcher should gather data about eating while teaching from the largest sample of elementary school teachers from which the researcher can gather data.
Determine how to apply the fifth basic step in a statistical study in this situation. Choose the correct answer below.
The researcher should use the methods of statistics to determine the quality of the estimate of the population parameter and draw conclusions based on this estimate accordingly.
Determine how to apply the fourth basic step in a statistical study in this situation. Choose the correct answer below.
The researcher should use the sample statistic as an estimate for the population value of the percentage of elementary school teachers that eat while teaching.
You read about an issue that was the subject of an observational study when clearly it should have been studied with a double-blind experiment. How can the results from the observational study be classified?
The results are essentially meaningless because there is no way to adjust the results to reflect what type of study should have been performed.
It can be said that a sample is representative of the population. What is meant by this?
The results found for the sample are similar to those we would expect to find for the entire population.
Determine how to apply the third basic step in a statistical study in this situation. Choose the correct answer below.
The sample statistic of interest is the percentage of elementary school teachers in the sample that eat while teaching.
Decide whether the statement below makes sense or does not make sense. Explain your choice. A pollster plans to improve survey results by only conducting polls in which the margin of error is zero.
The statement does not make sense. A margin of error of zero implies that there is no uncertainty in a survey result. This could happen only if the entire population was surveyed, rather than just a sample.
Decide whether the following statements makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. I made a frequency table with two columns, one labeled "State" and one labeled "State Capitol."
The statement does not make sense. In a frequency table, one of the columns lists the frequency of each category, which is the number of data values in the category. The table described in the given statement does not have this column.
I wanted to test the effects of vitamin C on colds, so I gave the treatment group vitamin C and gave the control group vitamin D.
The statement does not make sense. The control group should only receive a placebo, not another treatment.
The TV survey got more than 1 million phone-in responses, so it is clearly more valid than the survey by the professional pollsters, based on interview with only a few hundred people.
The statement does not make sense. The eight guidelines for evaluating a statistical study need to be reviewed before one study can be called more valid than another.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. I don't believe the results of the experiment because the results were based on interviews and the study was not double-blind.
The statement makes sense because all studies should be double-blinded. A study that is not double-blind would risk the participant and/or the interviewer influencing the results by knowing which group the participant was in.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. By choosing my sample carefully, I can make a good estimate of the average height of Americans by measuring the heights of only 500 people.
The statement makes sense because carefully choosing a sample is likely to result in a sample that closely resembles the population.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. I followed all the guidelines for sample selection carefully, but my sample still did not reflect the characteristics of the population.
The statement makes sense because even if a sample is chosen in the best possible way, it may not be representative of the population by random chance.
Decide whether the statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. The survey of religious beliefs suffered from selection bias because the questionnaires were handed out only at Catholic churches.
The statement makes sense because no religions other than Catholicism will be represented in the responses.
Decide whether the following statements makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. The relative frequency of B grades in our class was 0.3.
The statement makes sense. Suppose there were 20 people in the class. If 6 received Bs, then the relative frequency of B grades was 6 Over 20 which equals 0.3.
Consider an experiment designed to test whether cash incentives improve school attendance. The researcher chooses two groups of 100 high school students. She offers one group $10 for every week of perfect attendance. She tells the other group that they are part of an experiment but does not give them any incentive. What do the students who do not receive an incentive represent?
The students who do not receive an incentive represent the control group.
Briefly describe the differences among theoretical, relative frequency, and subjective techniques for finding probabilities. Give an example of each.
The theoretical technique is based on the assumption that all outcomes are equally likely, while the relative frequency technique is based on observations or experiments, and the subjective technique is an estimate based on experience or intuition.
What is meant by the treatment group and the control group in an experiment?
The treatment group in an experiment is the group of sample members who receive the treatment being tested, while the control group does not receive the treatment being tested.
Poll X predicts that a presidential candidate will receive 46% of the vote, while Poll Y predicts that he will receive 56% of the vote. Both polls have a margin of error of 6 percentage points. What can you conclude?
The two polls are consistent with each other because the confidence intervals overlap.
In which of the following situations would we be interested in an either/or probability? Select all that apply.
We roll a die, hoping for a 2 or a 5, We want to know whether a person selected at random is a Democrat or a man.
In which of the following situations are the events non-overlapping? Select all that apply.
We roll a die, hoping for a 2 or a 5.
In which of the following situations are the events overlapping? Select all that apply.
We want to know whether a person selected at random is a Democrat or a man
relative frequency method
a method of assigning probabilities that is appropriate when data are available to estimate the proportion of the time the experimental outcome will occur if the experiment is repeated a large number of times
sample statistic
a number that describes a characteristic of a sample
population parameter
a number that describes a characteristic of population
Coincidences
accidental events that happen at the same time but seem to have a connection
permutations
an arragement or listing in which an order or placement is important
bias
any problem in the design or conduct of a statistical study that tends to favor certain results
event
anything that happens or is regarded as happening
Population
complete set of people or things being studied
population
complete set of people or things being studied`
independent
free
probability distribution
list of possible outcomes with associated probabilities
Participation bias occurs any time
participation in a study is voluntary.
A margin of error is used to describe the range of values likely to contain a _____ and is _______ a ______ to establish a confidence interval. The confidence interval is used to estimate a ______
population parameter, added to and subtracted from, sample statistic, population parameter.
dependent
relying on someone or something else for support
Several forms or bias
researcher with a personal stake in the outcome distorts the true meaning of data, An experiment that is not blinded, and A non-representative sample
Selection bias occurs whenever
researchers select their sample in a biased way.
sample
set of people or things from which the data are obtained
raw data
the individual measurements collected.
subjective probability
the individual's personal estimate of the chance of loss
Two meanings of statistics
the science of collecting, organizing, and interpreting data and statistics are the data (numbers or other pieces of information) that describe or summarize something
overlapping
way of showing depth that deals with placing one object in front of the other
Decide whether the statement below makes sense (or is clearly true) or does not make sense (or is clearly false). Explain. In my statistical study, I used a sample that was larger than the population.
No, the statement does not make sense. A sample is a subset of the population and cannot be larger than the population.
What does it mean when we write P(A)? What is the possible range of values for P(A), and why?
P(A) means the probability that event A will occur, The range of possible values for P(A) is from 0 to 1 (inclusive), with 0 meaning there is no chance that event A will occur and 1 meaning it is certain that event A will occur.