Transcript Lecture 10

Statistics 111 - Lecture 10
Midterm review
Chapters 1-5
June 11, 2008
Stat 111 - Lecture 10 - Review
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Administrative Notes
• Homework 3 is due Monday
– Covers material from Chapter 5, so worth doing as
practice for the midterm!
• Exam on Monday
– Starts exactly at 10:40 – get here early
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Stat 111 - Lecture 10 - Review
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Some Topics Not Covered on Midterm
• Continuity correction for binomial
calculations (chapter 5)
• Normal quintile plots(chapter 1)
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Stat 111 - Lecture 10 - Review
3
Experiments
•
Used to examine effect of a treatment eg. medical
trials, education interventions
Treatment Group
Treatment
Result
1
Experimental
Units
Population
2
3
Control Group
•
4
No Treatment
Result
Different from an observational study, where no
treatment is imposed
Observational studies can only examine associations
between variables, whereas experiments try to
establish causal effects
•
•
Experiments can still be biased though!
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Stat 111 - Lecture 10 - Review
4
Sampling and Surveys
Population
?
Parameter
Sampling
Sample
Inference
Estimation
Statistic
• Just like in experiments, we must be cautious of
potential sources of bias in our sampling results
• Voluntary response samples, undercoverage, nonresponse, untrue-response, wording of questions
• Simple Random Sampling: less biased since each
individual in the population has an equal chance of
being included in the sample
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Stat 111 - Lecture 10 - Review
5
Distributions
• A distribution describes what values a variable
takes and how frequently these values occur
• Boxplots are good for center and spread, but don’t
indicate shape of a distribution
• Histograms much more effective at displaying the
shape of a distribution
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Stat 111 - Lecture 10 - Review
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Numerical Measures of Center
• Mean:
x1  x 2 
X
n
 xn
n
 1n  x i
i1
• Median: “middle number in distribution”
• Mean is more affected by large outliers and
asymmetry than the median

• Symmetric: Mean ≈ Median
• Skewed Left: Mean<Median
• Skewed Right: Mean>Median
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Numerical Measures of Spread
• Variance: average of the squared deviations of each
2
observation
(x

x
)

i
2
s 
n 1
• Standard Deviation =

• Inter-Quartile Range: IQR = Q3 - Q1
• First Quartile (Q1) is the median of the smaller half of data
• Third Quartile (Q3) is the median of the larger half of data
• With outliers or asymmetry, median and IQR are
better but we will use mean and SD more since most
distributions we use (eg. normal distribution) are
symmetric with no outliers
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Stat 111 - Lecture 10 - Review
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Scatterplots of two variables
• Positiveassociation vs Negative association
• Some associations are not just positive or negative,
but also appear to be linear
• Correlation is a measure of the strength of linear
relationship between variables X and Y
• r near 1 or -1 means strong linear relationship
• r near 0 means weak linear relationship
• Negative r means negative association
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Stat 111 - Lecture 10 - Review
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Linear Regression
• Best fit line between X and Y:
Y = a + b·X
• The slope b(
): average change you get in
the Y variable if you increased the X variable by one
• The intercept a (
):average value of the Y
variable when the X variable is equal to zero
• Regression equation used to predict response
variable Y for a value of our explanatory variable X
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Probability
• Random process: outcome not known exactly, but
have probability distribution of possible outcomes
• Event: outcome of random process with prob. P(A)
• Additive Rule for Disjoint Events:
P(A or B) = P(A) + P(B) if A and B are disjoint
• Multiplication Rule for Independent Events:
P(A and B) = P(A) x P(B) if A and B are independent
• Need to combine different rules (Eg. Lecture 8)
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Probability and Random Variables
• Conditional Probability:
• Random variable: numerical outcome or summary of
a random process
• A discrete random variable has a finite number of
distinct values
• Continuous random variables can have a noncountable number of values
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Discrete vs. Continuous RV’s
• Probability histogram for distribution of discrete r.v.
• Calculate probabilities by adding up bars of histogram
• Density curve used for distribution of continuous r.v.
• Calculate probabilities by integrating area under curve
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Stat 111 - Lecture 10 - Review
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Linear Transformations of Variables
• Same rules for both data and random variables:
mean(a·X + c) = a·mean(X) + c
variance(a·X + c) = a2 ·variance(X)
SD(a·X + c) = |a|· SD(X)
• Adding constants does not change spread measures
• Can also do combinations of more than one variable:
If X and Y are variables and Z = a·X + b·Y + c
mean(Z) = a·mean(X) + b·mean(Y) + c
If X and Y are also independent then
Variance(Z) = a2·Variance(X) + b2·Variance(Y)
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The Normal Distribution
• The Normal distribution has the shape of a “bell
curve” with parameters  and 2,denoted N(,2)
N(0,1)
N(2,1)
N(-1,2)
N(0,2)
• StandardNormal:  = 0 and 2 = 1
• Normal distribution follows the 68-95-99.7 rule:
• 68% of observations are between  -  and  + 
• 95% of observations are between  - 2 and  + 2
• 99.7% of observations are between  - 3 and  + 3
• Have tables for any probability from the standard
normal distribution
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Stat 111 - Lecture 10 - Review
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Standardization
• For non-standard normal probabilities, need to
transform to a standard normal distribution
• If X has a N(,2) distribution, then we can convert to
Z which follows a N(0,1) distribution:
• Can then calculate P(Z < k) using table
• Reverse standardization: converting a standard
normal Z into a non-standard normal X
X = σZ + μ
• Practice makes perfect!
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Inference for Continuous Data
• Continuous data is summarized by sample mean
• Sample mean is used as our estimate of the
population mean, but how does sample mean vary
between samples?
Population
Parameters:
 and 2
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Sample 1 of size n
Sample 2 of size n
Sample 3 of size n
Sample 4 of size n
Sample 5 of size n
Sample 6 of size n
.
.
.
Stat 111 - Lecture 10 - Review
x
x
x
x
x
x
Distribution
of these
values?
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Sampling Distribution of Sample Mean
• The center of the sampling distribution of the sample
mean is the population mean:
• Over all samples, the sample mean will, on average, be
equal to the population mean (no guarantees for 1 sample!)
• The spread of the sampling distribution of the sample
mean is
• As sample size increases, variance of the sample mean
decreases!
• Central Limit Theorem: if the sample size is large
enough, then the sample mean X has an
approximately Normal distribution
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Inference for Count Data
• Goal for count data is to estimate the population proportion p
• From a sample of size n, we can calculate two statistics:
1. sample count Y
2. sample proportion
• Use sample proportion as our estimate of population proportion p
• Sampling Distribution of the Sample Proportion
• how does sample proportion change over different samples?
Population
Parameter: p
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Sample 1 of size n
Sample 2 of size n
Sample 3 of size n
Sample 4 of size n
Sample 5 of size n
Sample 6 of size n
.
.
Stat 111 - Lecture 10 - Review
.
Distribution
of these
values?
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Sampling Distribution for Proportion
• For small samples, use the Binomial distribution to calculate
probabilities for the sample count or sample proportion
• Definition of “small”: n·p < 10 or n·(1-p) < 10
• For large samples, we use the Normal approximation to the
Binomial distribution for the sample count or sample proportion
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Next Week - Lecture 11
• Chapter 6
• Good luck on midterm next Monday!
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Stat 111 - Lecture 10 - Review
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