Math for Liberal Arts MAT 110: Chapter 5 Notes

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1 Math for Liberal Arts MAT 110: Chapter 5 Notes Statistical Reasoning David J. Gisch Fundamentals of Statistics Two Definitions of Statistics Statistics is the science of collecting, organizing, and interpreting data. Statistics are the data that describe or summarize something. Definitions The population in a statistical study is the complete set of people or things being studied. We study DMACC West students. All the students are the population. The sample is the subset of the population from which the raw data are actually obtained. Maybe we study a subset of West students randomly chosen from the Ellipse. Population parameters are specific characteristics of the population that a statistical study is designed to estimate. Maybe we ask each student the year of vehicle they drive. Sample statistics are numbers or observations that summarize the raw data. The actual years we record. 1

2 College Entrance Requirements Example: The admissions board at a college needs to decide who to accept into their institution. To do so they need to collect data to make their decision. (a) What is the population they are studying? (b) What would a sample be or should they even take a sample? (c) What are possible population parameters? (d) What are possible sample statistics? Basic Steps in a Statistical Study 1. State the goal of your study precisely. 2. Choose a representative sample from the population. 3. Collect data from the sample and summarize these data by finding sample statistics of interest. 4. Use the sample statistics to infer the population parameters. 5. Draw conclusions. Example: The admissions board only looks at high school GPA but finds that too many people have similar GPA s. So they decide to measure the amount of advanced level coursework taken and start over. Definitions In the admissions example you need to study the entire population but sometimes that is not possible. For example, what if you wanted to study the weight of Doritos bags leaving the factories. Common Sampling Techniques A representative sample is a sample in which the relevant characteristics of the sample members match those of the population. A statistical study suffers from bias if its design or conduct tends to favor certain results. 2

3 Sampling Example: Give an example of sampling bias. Sampling Techniques Example: Identify the sampling technique used. (a) You are conducting a survey of students in a dormitory. You choose your sample by knocking on the door of every fifth room. (b) To survey opinions on a proposed new water line, a research firm randomly draws the addresses of 200 homeowners from a public list of all homeowners. Types of Statistical Study In an observational study, researchers observe or measure characteristics of the sample members but do not attempt to influence or modify these characteristics. In an experiment, researchers apply a treatment to some or all of the sample members and then look to see whether the treatment has any effects Types of Statistical Study Example: Give an example of each. (a) Of an observational study. (b) Of an experiment. 3

4 Treatment and Control Groups The treatment group in an experiment is the group of sample members who receive the treatment being tested. The control group in an experiment is the group of sample members who do not receive the treatment being tested. It is important for the treatment and control groups to be selected randomly and to be alike in all respects except for treatment. Placebos and the Placebo Effect A placebo lacks the active ingredients of a treatment being tested in a study, but is identical in appearance to the treatment. Thus, study participants cannot distinguish the placebo from the real treatment. The placebo effect refers to the situation in which patients improve simply because they believe they are receiving a useful treatment. Blinding in Experiments An experiment is single-blind if the participants do not know whether they are members of the treatment group or members of the control group, but the experimenters do know. An experiment is double-blind if neither the participants nor the experimenters (people administering the treatment) know who belongs to the treatment group and who belongs to the control group. Definitions A case-control study is an observational study that resembles an experiment because the sample naturally divides into two (or more) groups. The participants who engage in the behavior under study form the cases. The participants who do not engage in the behavior are the controls. 4

5 Definitions The margin of error is used to describe a confidence interval that is likely to contain the true population parameter. A confidence interval is from (sample statistic margin of error) to (sample statistic + margin of error). Example: A study is conducted and concludes that 67% are frustrated with congress. The margin of error is 5%. Thus the actual results have what range? Breakdown Treatment Group Group that receives the drug. *Double-Blind Experiment* Statistical Study Control Group Control group who does not get the drug and we use for comparison purposes. Blind group who thinks they are getting the drug but are not. This is to check for the placebo effect. Observational Cases Those who we observe participating in a behavior. Control Those who do not participate in behavior. Example: You want to DMACC students opinion on the usefulness of MML. (a) This is what kind of study (experiment or observational)? (a) What would a sample be or should they even take a sample? Should You Believe A Statistical Study (b) What are possible population parameters? (c) What are possible sample statistics? 5

6 Guidelines Example What are things to look for when you review statistical data? 1. Identify the goal, population, and type of study. 2. Consider the source. 3. Look for bias in the sample. 4. Look for problems in defining or measuring the variables of interest. 5. Watch out for confounding variables. 6. Consider the setting and wording in surveys. 7. Check that results are presented fairly. 8. Stand back and consider the conclusions. Example: The school board in a city announced that 35% of the school children were reading below grade level, and concluded that the methods of teaching reading needed to be changed. The announcement was based on a reading test on which 35% of the school children scored below the national average for their grade. Do these data support the boards decision? Example Example: Iowa Senate Majority Leader Michael Gronstal, D- Council Bluffs, hinted today he would likely oppose a citizenled legislative bill aimed at banning automated traffic cameras. Gronstal, who has the power to block any legislation from reaching the Senate floor, read reporters an sent to him by a Council Bluffs official which detailed dramatic reductions in traffic crashes in the western Iowa community since red light cameras have been installed there. Statistical Tables and Graphs Between 2002 and 2004, Council Bluffs averaged 3,900 traffic crashes per year. But during 2009 and 2010, the number plunged to an average of about 700 traffic crashes annually, Gronstal said, adding that much of the credit for the safety improvement was given to the traffic cameras. 6

7 Frequency Tables A basic frequency table has two columns: The first column lists the categories of data. The second column lists the frequency of each category, which is the number of times each category appears in the data set. Number of Super Bowl Victories NFL Team Frequency Packers 4 Vikings 0 Bears 1 Broncos 2 Steelers 6 Frequency Tables Additional columns may include relative frequency (frequency expressed as a fraction or percentage of the total) or cumulative frequency (total of frequencies for the given category and all previous categories). Number of Super Bowl Victories NFL Team Frequency Relative Freq. Cumulative Freq. Packers 4 4/13 =.31 4 Vikings 0 0/13 = 0% 4 Bears 1 1/13 = 8% 5 Broncos 2 2/13 = 15% 7 Steelers 6 6/13 = 46% 13 *We technically should include all the teams if we want to do relative frequencies. Frequency Table Year of Vehicle of MAT 110 Students Data Types and Binning Qualitative data describe qualities or categories. Year of Vehicle 1989 or older Frequency Relative Frequency Cumulative Frequency Quantitative data represent counts or measurements. When dealing with quantitative data categories, it is often useful to group (5-10 of them), or bin, the data into categories that cover a range of possible values or newer Example : What type of data was the previous example? 7

8 Summarizing Data Bar and Pie Charts Example: Consider the following 20 scores from a 100-point exam: Determine appropriate bins and make a frequency table including columns for relative and cumulative frequency. A bar chart shows each category with a bar whose length corresponds to its frequency or relative frequency. Pie charts are used primarily for relative frequencies, because the total pie must always represent the total relative frequency of 100%. The size of each wedge is proportional to the relative frequency of the category it represents. Bar and Pie Charts Example 5.C.3: The bar chart and pie chart below both show the data from table 5.1. Important Labels for Graphs Title/caption: 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. Vertical scale and title: Numbers along the vertical axis should clearly indicate the scale. The numbers should line up with the tick marks. Include a label that describes the variable. Horizontal scale and title: The categories should be clearly indicated along the horizontal axis. (Tick marks may not be necessary for qualitative data, but should be included for quantitative data.) Include a label that describes the variable. Legend: If multiple data sets are displayed on a single graph, include a legend or key to identify the individual data sets. 8

9 Definitions A histogram is a bar graph for quantitative data categories. The bars have a natural order and the bar widths have specific meaning. A line chart shows the data value for each category as a dot, and the dots are connected with lines. For each dot, the horizontal position is the center of the bin it represents and the vertical position is the data value for the bin. A time-series diagram is a histogram or line chart in which the horizontal axis represents time. Histogram and Line Chart Time-Series Diagram Review Example: If you invested $100 in bonds on July 7, how much would your investment be worth on August 25? Example: Finish the chart with a column for relative and cumulative frequency. Ages of Academy Award-Winning Actresses, Age Frequency Relative Frequency Cumulative Frequency 9

10 Review Example: Use the data to answer the following (a) What type of data is this? Quantitative or Qualitative (b) Should we group the data into bins? Yes or No Graphics in the Media (c) What type of chart should we use? Bar Pie Histogram Line Chart Graphics Beyond the Basics A multiple bar graph has two or more sets of bars that allow comparison between two or more data sets. All the data sets must involve the same categories so that they can be displayed on the same graph. A stack plot shows different data sets in a vertical stack. Stack plots usually use either stacked bars or stacked line graphs. A Multiple Bar Graph The following multiple bar graph shows how education affects personal employment. 10

11 A Stack Plot Reading a Stack Plot The following stack plot shows trends in death rates from four diseases. What was the death rate for tuberculosis in 1930? About = 120 deaths per 100,000 A Few Cautions about Graphics Perceptual distortions: Many graphics are drawn in a way that distorts our perception of them. Watch the scales: It can be visually deceptive if you do not study the scales carefully. Percentage change graphs: Graphs that show percentage change can be misleading unless you interpret them with great care. Perceptual Distortion The lengths of the dollars to the right represent the data, but your eyes tend to focus on the area. Pictographs: Pictographs are graphs embellished with additional artwork. The artwork may make the graph more appealing, but can also mislead. 11

12 Different Vertical Scales Different Horizontal Scales Both graphs show the same data, but they look very different because their vertical scales have different ranges. It appears that the world population has been rising linearly. However, the time intervals on the horizontal axis are not uniform in size. Is it a Good Chart? Is it a Good Chart? 12

13 Is it a Good Chart? Is it a Good Chart? U.S. Light Pollution Definitions A correlation exists between two variables when higher values of one variable consistently go with higher values of another or when higher values of one variable consistently go with lower values of another. Correlation and Causality A scatter diagram is a graph in which each point represents the values of two variables. 13

14 Relationships Between Two Data Variables No correlation: There is no apparent relationship between the two variables. Positive correlation: Both variables tend to increase (or decrease) together. Negative correlation: One variable increases while the other decreases. Strength of a correlation: The more closely two variables follow the general trend, the stronger the correlation. In a perfect correlation, all data points lie on a straight line. Examples Ignore the r-values. Positive Correlation Example : The scatter diagram shows that higher diamond weight generally goes with higher price. (a) Is it a strong or weak correlation? Negative Correlation Example: The scatter diagram shows that higher life expectancy generally goes with lower infant mortality. (a) Is it a strong or weak correlation? 14

15 MAT Chapter 5 Negative Correlation Correlation Example: The scatter diagram shows that the higher years of education of a person generally goes with lower prejudice. Example: State the type (positive/negative) and strength (strong/weak) of correlation, or state none if there is no correlation. (a) Is it a strong or weak correlation? (a) The amount of education a person has and religious convictions. (b) Price of bread and the year. (c) GPA and the amount of credits taken online. (d) The amount of trash you have each weak and the amount of coffee you drink. Correlation In groups come up with examples of two things that have a correlation. Possible Explanations for a Correlation 1. The correlation may be a coincidence. 2. Both variables might be directly influenced by some common underlying cause. 3. One variable may be a cause of the other. Most of the time. 15

16 Explain a Correlation Example: Consider the negative correlation between infant mortality and life expectancy. Which of the three explanations for correlation applies? The negative correlation is probably due to a common underlying cause the quality of health care. In countries where health care is better in general, infant mortality is lower and life expectancy is higher. Guidelines for Establishing Causality To investigate whether a suspected cause actually causes an effect, follow these guidelines. 1. Look for situations where the effect is correlated with the suspected cause. 2. Check that the effect is present or absent among groups that differ only in the presence or absence of the suspected cause. 3. Look for evidence that larger amounts of the suspected cause produce larger effects. 4. Account for other potential causes. 5. Test the suspected cause with an experiment. 6. Try to determine how the suspected cause produces the effect. Explain a Correlation Example: A study was conducted and showed a strong positive correlation to ocean temperature and shark attacks. Therefore our conclusion is that sharks are crazed by high temperature water and turn into blood-thirsty human eaters. Explain a Correlation Example: Explain each of the following correlations. (a) Children with bigger feet spell better. (b) Nations that add fluoride to their water have a higher cancer rate than those who don t. (c) The education and religion example. (d) Price of bread and the year. 16