Review for Test 2 Math 1231

1
Review for Test 2Math 1231

• This test covers
• 6-9
• Bring your laptop!
• You should have DataDesk, the Z-area and Inverse
  Normal tools available to you during the test.

2
Z-scores

• Know how to calculate a z-score and what it means
• A z-score represents how many standard deviations
  above or below the mean a data value is
• Provides a common scale for comparison
• (SAT vs ACT, for example)
• (using the Normal
  Model)
• (using statistics from
  data)

3
The Normal Model

• Know what the Normal Model is (unimodal,
  symmetric,
• bell-shaped) N(m,s)
• Know what the Standard Normal Curve is N(0,1)
• Know how to use the 68-95-99.7 Rule
• For approximately normal distributions only
• Estimation, not exact, and remember to use
  symmetry.
• 68 of data within 1 SD of mean, 95 within 2
  SDs, 99.7 within 3 SDs
• Adding a constant to all data values SHIFTS the
  distribution but does not change shape or spread.
• Multiplying each data value by a constant
  multiplies center and spread measures by that
  same constant.

4
Important More on Normal Model

• Given the mean and SD of a normal distribution,
  be able to determine
• The proportion above or below a data value
• The proportion between two data values
• The data value with a given proportion
  above/below it (the Inverse Normal procedure)

5
Z-area Tool

• You score 82 on French exam. The overall results
  of the exam were Normally distributed with a mean
  of 72, and SD of 8.
• What proportion of students did you do better
  than?
• Find z-score.
• Look up z-score on Table, or Zarea tool
• Zarea tool Set Lower bound to 100
• Set Upper Bound to 1.25 (your z-score)
• Area below z1.25 is 0.89435
• You scored better than 89.435 of students.
• Table Look on the Table Z
• Go down rows to z1.2
• Go across that row to column 0.05
• Area below is 0.8944

6
More Z-area
What percent of students scored better than
you? Using Zarea Set lower bound to 1.25
Set upper bound to 100
Area above z1.25 is 0.10565 10.565 did better
than you. Using Table below you above you
100 so above 100- below 100 -
89.44 10.56
7
More Z-area again
You scored 82 on the exam, and your friend scored
78. What percent of students scored between
you? Find friends z-score Using Z-area Set
lower bound to 0.75 Set
upper bound to 1.25 Area between you
0.120977 12.0977 scored between you. Using
Table Find below z0.75 0.7734 Subtract
below you - below friend 0.8944 - 0.7734
0.1210 12.10
8
Inverse Normal Procedure
How high must you score on the French test to
score in the top 20? Top 20 means 80 are
below you. Using Table 0.80 proportion
corresponds to a z-score of 0.84 Solve z
equation for y You must score a 78.82 on the
test. Using Inverse Normal Tool Set p to 0.80
(proportion below you) Use the z-score from
the tool in the z equation to solve for
y. You must score a 78.83 on the test.
9
Objectives Regression and Correlation

• Recognize explanatory and response variables
• Explanatory- explains or predicts the response
  var.
• Scatterplots Show relationships between 2
  Quantitative variables
• Explanatory var on x-axis
• Describing Scatterplot
• Direction (Positive/Negative)
• Form (Linear, Curved, Fan, etc)
• Strength
• Recognize outliers- deviations from pattern

10
Objectives Regression and Correlation

• Correlation If scatterplot is linear form,
  measures strength and direction.
• r positive positive association between
  variables
• r negative negative association between
  variables
• Association does not imply anything about
  causation.
• r values close to 1 or 1 indicate plot is close
  to linear.
• r values close to 0 indicate no linear
  relationship.
• Correlation is not resistant. It is effected by
  outliers.
• Correlation does not have units.
• Correlation only tells you association, not
  causation!

11
Objectives Regression and Correlation

• Regression line Describes how response variable
  changes as explanatory variable changes.
• Must have explanatory/response variable
  relationship between variables for regression
  line to be valid/useful.
• Scatterplot must have linear form.
• yhatb1xb0 here, yhat means predicted value
  of y
• Just plug a given x value into the equation to
  find the value of y predicted by the regression
  line.

12
Objectives Regression and Correlation

• Know how to plot a scatterplot, find correlation
  and regression line equation from DataDesk!
• Slope of regression tells you that the y variable
  changes by b1 y-units for every increase of 1
  x-units.
• Y-intercept of regression gives you the predicted
  initial value, or y-value when x is zero. This
  often has no realistic meaning.
• Residual observed y- predicted. Ie, data
  value-calculated y
• Residuals tell you if the regression line over-
  or under-predicts the actual data, and how far
  off you are.

13
More regression

• If the plot of residuals vs predicted values is
  horizontal without pattern, then the linear model
  was a good choice.
• Interpolation is reliable, extrapolation is not.
• If r is low, the regression line is useless.
• R2 tells you the of variation in the y-values
  that is accounted for by the model. This, again,
  does not mean that x causes y.
• To find R2, just square r!
• Regardless of how strongly correlated 2 variables
  are, only an experiment can show causation.

14
Outliers

• Remember that outliers can effect both the
  correlation and regression line equation.
• You dont need to know the following terms, but
  have a feel for what outliers can do to
  correlation and slope of the regression line.
• X-outliers have high leverage- can effect slope
  of regression line greatly.
• Pts that are both x-outliers and model outliers
  are influential, and can cause the regression
  model to be completely different from what it
  would be without the point (Bozo the clown
  example- shoe size vs IQ).
• If an x-outlier lines up with the model, R2 can
  be greatly increased.

15
DataDesk and Regression

• Select the explanatory var as X (use shift key)
  and response var as Y.
• Plot Scatterplot, or go straight to Calc menu and
  select Regression.
• Sample Regression output

b1, slope
b0, y-int
Equation y 673.529-88.2353x
16
Randomness

• Random event outcomes of individual events are
  uncertain, such as rolling a die. But a regular
  distribution of outcomes is seen in a large
  number of repetitions.
• Random events cannot be predicted with certainty
  in advance.
• Know how to use a random number table to simulate
  random events.

17
Study Suggestions

• If you havent already, DO THE HOMEWORK!
• Review your quizzes, particularly any questions
  you missed.
• Solutions to all quizzes are online
• Think about what the big idea is that each
  question is trying to ask.
• Work Practice Test from Website
• Dont forget the Review Problems in the end of
  the section (after Ch 11).
• Email or IM Dr. Matos with questions, or use
  office hours!
• Make sure youve gone through ActivStats!