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Presented by Del Ferster
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What’s in store for tonight?I have lots or “practice problems” that
cover the entire spectrum of statistics that are being considered this year.
The solutions to these problems will also be presented.
We’ll spend some time on the different types of sampling.
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What’s in store for tonight?We’ll consider the difference between
association and causation.I have a good M&M activity for us to do
(maybe you’re right, maybe it’s just an excuse to eat chocolate )
I have a nice Starburst activity that we’ll consider, too.
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Another look at some “test-like” problems that deal with a variety of statistics topics.
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You got a problem??This is a rather large set of problems, so I’m going to give you a while to work on them.
Hopefully, some of the ideas come back quickly.
When you’re set, we’ll look at the solutions.
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Solutions to Review Problems1. minimum x value is between 30 & 40, so A2. minimum y value is between 60 & 70, so D3. y-intercept (x=0) on final exam, overall
average is 59.351, so A4. C5. D6. A7. B8. C
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Solutions to Review Problems9. D10. D11. C12. B13. 8:00 Class 9:00 Class TOTAL
Earned an A 18 12 30
Did not earn an A
4 6 10
TOTAL 22 18 40
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Solutions to Review Problems14. 30/40=75%15 18
81.81%2212
66.7%1830
75%40
a
b
c
b c a so
C
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Solutions to Review Problems16. 18
60%304
40%1022
55%40
x
y
z
y z x so
C
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Solutions to Review Problems17A.
17B. 26/47=55.319%17C. 13/47=27.660%17D. Type O
Seniors: 9/26=34.615%Juniors: 6/21=28.571%, so Greater percentage
in Juniors.
Type A Type B Type AB Type O TOTAL
Junior 7 5 5 9 26Senior 1 6 8 6 21TOTAL 8 11 13 15 47
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Solutions to Review Problems18A. Approximately 86% (86.387%)18B. Approximately 95% (95.408%)18C. A student whose Quiz average is 0, has
a final course grade of 41.667.18D. For each change of 1 percent in quiz
average, final course average increases by 0.559 percent
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Solutions to Review Problems19.
19A. 537519B. 1004/5375=18.68%19C. 400/1780=22.47%
Student Smokes Student Does Not Smoke
TOTAL
Both Parents Smoke 400 1380 1780One Parent Smokes 416 1823 2239Neither Parent Smokes 188 1168 1356TOTAL 1004 4371 5375
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Solutions to Review Problems19D. 416/2239=18.58%19E. 188/1356=13.86%19F
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Solutions to Review Problems19G. 1380/4371=31.57%19H 188/1004=18.73%19I. 816/1004=81.27%20A. For each increase of 1 inch in wheel
diameter, coasting distance increases 5.332 inches.
20B. A wheel that has a diameter of 0 inches will have a coasting distance of 10.585 inches.
20C. Approximately 53 inches (53.241)
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Solutions to Review Problems20D. Approximately 17 inches (16.770)20E. No. The correlation is clearly positive; most
likely near POSITIVE 1
21A. Write each number on a slip of paper, put the paper slips in a hat (A Packers hat works BEST ), then select 10 slips from the hat.
21B. Group the numbers into 5 strata (1-20, 21-40, 41-60, 61-80, and 81-100), then randomly select 2 numbers from each stratum. (Impressive Latin knowledge, eh? )
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Solutions to Review Problems21B. Group the numbers into 5 strata
(1-20, 21- 40, 41-60, 61-80, and 81-100), then randomly select 2 numbers from each stratum. (Impressive Latin knowledge, eh? )
21C. Randomly select on of the groups (1-20, 21- 40, 41-60, 61-80 or 81-100), then randomly select 10 numbers from that group.
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Solutions to Review Problems21D. Pull a random number, then
include every 5th number after that number (note, it doesn’t have to be the 5th number, in reality, the “span number” should be random, too.
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Solutions to Review Problems22. Association implies some kind of
relationship exists between the two variables, but stops short of saying a change in x (the explanatory variable) causes a change in y (the response variable).
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Solutions to Review Problems22 (continued).
To conclude causation, an experiment (not an observational study) must be done, where subjects are randomly assigned to 2 groups—experimental and control. Other variables must be controlled or eliminated.
Association doesn’t require control, or random assignment of subjects to 2 groups. Observational studies can imply association.
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A Quick look at basic terms, ways to represent results, and 2-way tables.
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Qualitative variables classify the data into categories.
The categories may or may not have a natural ordering to them.
Qualitative variables are also called categorical variables.
EXAMPLESEye colorFavorite NFL teamGenderDo you smoke?
Qualitative Variables/Categorical Variables
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Distribution of a categorical variableThe distribution of a categorical variable provides the possible values that a variable can take on and how often these possible values occur.
The distribution of a categorical variable shows the pattern of variation of the variable.
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According to the Bureau of Justice, the following data represent the number of inmates by ethnicity in 2007.
Example #1
White 338,400Black 301,900
Hispanic 125,600
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Graphing Qualitative DataOften, rather than simply presenting
numerical values, we choose to graph our data.
When generating a graph of 1 categorical variable, we might consider the following types of graph.Pie ChartBar Graph
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Pie ChartA pie chart displays the distribution
of the qualitative variable by dividing the circle into wedges corresponding to the categories of the variable such that the angle of each wedge is proportional to the percentage of items in that category.
Pie Charts are easy to do in EXCEL.
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A Pie Chart for the Prison DataWhite 338,400
Black 301,900
Hispanic 125,600
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A Pie Chart for the Prison Data (Using Percents)
White 338,400
Black 301,900
Hispanic 125,600
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Bar GraphA bar graph displays the distribution
of a qualitative variable by listing the categories of the variable along one axis and drawing a bar over each category with a height equal to the percentage of items in that category.
The bars should all be of equal width. We could also do one using percents.
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Bar Graph for the Prison Data
White 338,400
Black 301,900
Hispanic 125,600
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Categorical Variables place individuals into one of several groups or categories.
The values of a categorical variable are labels for the different categories.
The distribution of a categorical variable lists the count or percent of individuals who fall into each category.
Comparing 2 Categorical Variables
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When a dataset involves two categorical variables, we begin by examining the counts or percents in various categories for one of the variables.
Comparing 2 Categorical Variables
Two-way Table – describes two categorical variables, organizing counts according to a row variable and a column variable.
Two-way Table – describes two categorical variables, organizing counts according to a row variable and a column variable.
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Two-Way TablesTwo-way tables come about when
we are interested in the relationship between two categorical variables.One of the variables is the row variable.
The other is the column variable.The combination of a row variable and a column variable is a cell.
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Dr. F is hosting 38 of his friends to a cookout. Now, Dr. F. has limited cooking skills, so everyone is having a burger. However, he has bought sufficient tomatoes so that anyone who wants tomato on his or her burger will be happy.
The following slide details the results of his burger and tomato survey.For the record….a good burger needs only
2 things…CHEESE….and KETCHUP!
Example #2
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Burger/Tomato Two-Way TableLet’s look at the components of a 2 way table
GENDER * TOMATOES Crosstabulation
Count
11 8 19
6 13 19
17 21 38
F
M
GENDER
Total
N Y
TOMATOES
Total
Row variable
Column variable
Column Totals
Row Totals
Overall Total
Cells
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A quick look at basic terms, and an introduction to linear regression and correlation.
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Quantitative variables have numerical values that are measurements (length, weight, and so on) or counts (of how many).
Examples:How many are in your family?How many cars do you own?
Quantitative Variables
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We further distinguish quantitative variables based on whether or not the values fall on a continuum.A discrete variable is one for which
you can count the number of possible values. How many siblings a person has
A continuous variable can take on any value within a given interval.A person’s weight
More on Quantitative Variables
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SCATTER PLOTS
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ScatterplotA graphical display of two
quantitative variablesWe plot the explanatory
(independent) variable on the x-axis and the response (dependent) variable on the y-axis
Each dot represents a single observation and its ordered pair (x,y)
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Describing Scatterplots
When we consider scatterplots, we focus on 4 things:DirectionFormScatterUnusual elements
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DirectionPositive: as values of the
explanatory variable increase, values in the response variable tend to increase
As x gets larger, y gets larger
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DirectionNegative: as values of the
explanatory variable increase, values in the response variable tend to decrease
As x gets larger, y gets smaller
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DirectionNull: no discernible patter of
change in the response variable
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Form (Shape)Linear: The shape has the
appearance of a linear relationship.
There doesn’t have to be a perfect fit.
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FormCurvedWe can use logarithms to
transform into linear forms.
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FormNoneNo discernible form
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Strength (Scatter)Strong association: very little
scatter
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StrengthModerate strength:
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StrengthWeak strength: lots of scatter
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Unusual FeaturesOutliers—They just don’t fit the
trend
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Determining the LINE that best fits our data.
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Regression LineA regression line is a straight line
that describes how a response variable y changes as an explanatory variable x changes.
A regression line summarizes the relationship between two variables, but only in a specific setting: when one of the variables helps explain or predict the other.
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Regression LineWe often use a regression line to predict the value of y for a given value of x.
Regression, unlike correlation, requires that we have an explanatory variable and a response variable
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Regression LineFitting a line to data means drawing a
line that comes as close as possible to the points.
Extrapolation-the use of a regression line for prediction far outside the range of values of the explanatory variable x that you used to obtain the line.Such predictions are often not
accurate.
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Linear RegressionRegression analysis finds the equation
of the line that best describes the relationship between the two variables.
In other words, what line best fits the data that is represented on our scatterplot.
While there are formulas to calculate this line, most of the time we’d use a graphing calculator or app for our ipad.
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Interpreting our lineThe slope, b, is the amount by which y changes when x increases by one unit.
The intercept, a, is the value of y when
.0x
y a bx
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A way to measure the strength of a LINEAR trend.
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CORRELATION, denoted by r measures the direction and strength of the linear relationship between two quantitative variables.
General PropertiesIt must be between -1 and 1, or (-1≤ r ≤ 1).If r is negative, the relationship is
negative.If r = –1, there is a perfect negative
linear relationship (extreme case).If r is positive, the relationship is
positive.
Some facts about CORRELATION
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General PropertiesIf r = 1, there is a perfect positive linear
relationship (extreme case).If r is 0, there is no linear relationship.r measures the strength of the linear
relationship.If explanatory and response are switched,
r remains the same.r has no units of measurement associated
with itScale changes do not affect r
Some facts about CORRELATION
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Association does not imply causationCorrelation does not imply causationSlope is not correlationA scale change does not change the correlation.
Correlation doesn’t measure the strength of a non-linear relationship.
Summary of Correlation
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A look at the different ways in which we can acquire a sample.
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Data CollectionIn research, statisticians use data in many
different ways. Data can be used to describe situations. Data can be collected in a variety of ways, BUT if the sample data is not collected in an appropriate way, the data may be so completely useless that no amount of statistical torturing can salvage them.
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Basic Methods of SamplingRandom Sampling
Selected by using chance or random numbers
Each individual subject (human or otherwise) has an equal chance of being selected
Examples: Drawing names from a
hat Random Numbers
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Basic Methods of SamplingSystematic Sampling
Select a random starting point and then select every kth subject in the population
Simple to use so it is used often
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Basic Methods of Sampling
Convenience SamplingUse subjects that are easily accessible Examples:
Using family members or students in a classroom Mall shoppers
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Basic Methods of SamplingStratified Sampling
Divide the population into at least two different groups with common characteristic(s), then draw SOME subjects from each group (group is called strata or stratum)
Results in a more representative sample
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Basic Methods of SamplingCluster Sampling
Divide the population into groups (called clusters), randomly select some of the groups, and then collect data from ALL members of the selected groups
Used extensively by government and private research organizations
Examples: Exit Polls
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A look at the differences.
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Types of ExperimentsObservational Studies
The researcher merely observes what is happening or what has happened in the past and tries to draw conclusions based on these observations
No interaction with subjects, usuallyNo modifications on subjects Occur in natural settings, usuallyCan be expensive and time consumingExample:
Surveys---telephone, mailed questionnaire, personal interview
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Types of ExperimentsExperimental Studies
The researcher manipulates one of the variables and tries to determine how the manipulation influences other variables
Interaction with subject occurs, usuallyModifications on subject occursMay occur in unnatural settings (labs or
classrooms)Example:
Clinical trials of new medications ,treatments, etc.
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Wrapping it all upAgain, I thank you for your attention, participation, and effort—I truly do know how long the day is for you!
I hope that you and your family enjoy a wonderful Thanksgiving and Christmas time.Take some time to relax, and be with the ones that matter to you!