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Review for Test 2 Math 1231

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... the Z-area and Inverse Normal tools available to you during the test. ... be without the point (Bozo the clown example- shoe size vs IQ) ... are online ... – PowerPoint PPT presentation

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Title: 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!
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