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Wednesday, May 13, 2015
Report at 11:30 to Prairieview
AP Exam
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We started by describing univariate (that means one variable) data graphically and numerically.
Graphically:
Exploring Data: Describing Patterns and Departures from Patterns
DotplotStemplot
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More graphs…
Histogram
Cumulative Frequency
Will need to read, interpret and answer questions of graphs.
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When asked to describe data based on the graph, focus on SOCS
Shape: Mound, Skewed (left or right? positive or negative?), Bimodal, Unimodal, Uniform, approximately normal…
Outliers: Are there any potential? If you are only asked to describe the graph, you don’t need to calculate, just mention any potential outliers
Center: Where (approx) is the median? the mean? based on the shape of the data, which is a better choice for center?
Spread: Range, IQR
SOCS
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While a plot provides a nice visual description of a dataset, we often want a more detailed numeric summary of the center and spread.
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Mean
Median or Q2
Describing Center
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Range: Max-Min InterQuartile Range: IQR=Q3-Q1 Standard Deviation:
Measures of Variability: When describing the “spread” of a set of data, we can use:
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Use a bar graph to display categorical data. Make sure graph is labeled, bars are equal width and evenly spaced.
A pie chart may be
used to display categorical data if the data is parts
of a whole.
Analyzing Categorical Data
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Conditional Distribution
Describes the values of a variable amongindividuals who have a specific value ofanother variable.
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Many distributions of data and many statistical applications can be described by an approximately normal distribution.
Symmetric, Bell-shaped Curve Centered at Mean μ Described as N(μ, )Empirical Rule: 68% of data within 1 of µ 95% within 2 of µ 99.7% within 3 of µ
Normal Distribution
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If the data does not fall exactly 1, 2, or 3 from µ, we can standardize the value using a z-score:
Standardizing Data
You can find the or percentile by finding the area left of the z-score.
The area under a distribution curve = 1
Use table A to find percentiles or p-values
To Get area above: To Get area below:normalcdf(z, 100, 0, 1) normalcdf(-100, z, 0, 1)
Betweeen: normalcdf (z, z, 0, 1 )
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Shape Normal Probability Plot – Linear Plot =
Normal
distribution
Assessing Normality
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Bivariate Data (That means 2 variables)
The study of bivariate data is the studyof the relationship between quantitativevariables.
• D O F S (Direction, Outliers, Form, Strength)• Least Squares Regression Line• Residuals (observed – predicted)• Correlation (r) –Correlation Coefficient• r2 – Coefficient of Determination
Calculator Steps:Make sure Diagnostics are onEnter data in L1, L2STAT CALC LinReg (a + bx)
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Correlation “r”We can describe the strength of a linear relationship with the Correlation Coefficient, r
-1 < r < 1
The closer r is to 1 or -1, the strongerthe linear relationship between x and y.
r alone is not enough to say thereis a linear relationship between 2variables.
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Least Squares Regression Line
When we observe a linear relationship between x and y, we often want to describe it with a “line of best fit” y=a+bx.
We can find this line by performing least-squares regression.
We can use the resulting equation to predict y-values for given x- values.
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Assessing FitIf we hope to make useful predictions of y we must assess whether or not the LSRL is indeed the best fit. If not, we may need to find a different model.
Use the residual plot to help determine linearity.
Plots should be scattered with no obvious patterns or curvature.
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If you are satisfied that the LSRL provides an appropriate model for predictions, you can use it to predict a y-hat for x’s within the observed range of x-values.
Predictions for observed x-values can be assessed by noting the residual. Residual =
Making Predictions
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Bivariate Relationship – Non Linear
If data is not best described by a LSRL, we may be able to find a Power or Exponential model that can be used for more accurate predictions.
Power Model (ln x , ln y ) or ( log x , log y )
Exponential Model ( x , ln y ) or ( x , log y )
If (x,y) is non-linear, we can transform it to try to achieve a linear relationship.
If transformed data appears linear, we can find a LSRL and then transform back to the original terms of the data
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Our goal in statistics is often to answer a question about a population using information from a sample.
Observational Study vs. Experiment We must be sure the sample is
representative of the population in question.
Sampling and Surveys
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If you are performing an observational study, your
sample can be obtained in a number of ways: ConvenienceCluster SystematicSimple Random Sample Stratified Random Sample
Observational Studies
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In an experiment, we impose a treatment with the hopes of establishing a causal relationship.
Experiments exhibit 3 Principles:
RandomizationControlReplication
Experimental Study
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Like Observational Studies, Experiments can take a number of different forms:
Completely Controlled Randomized Comparative Experiment Blocked Matched Pairs
Experimental Designs
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Scope of Inference