STATISTICS

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STATISTICS

• Summarizing, Visualizing and Understanding Data

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I. Populations, Variables, and Data
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Populations and Samples

• To a statistician, the population is the set or
  collection under investigation. Individual
  members of the population are not usually of
  interest. Rather, investigators try to infer
  with some degree of confidence the general
  features of the population.

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Examples

• Students currently enrolled at a certain
  university.
• Registered voters in a certain Congressional
  district.
• The population of large-mouthed bass in a certain
  lake.
• The population of all decay times of a
  radioactive isotope.

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Statistical Inference

• Drawing and quantifying the reliability of
  conclusions about a population from observations
  on a smaller subset of the population.
• Sample The subset observed.

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Variables and Data

• A population variable is a descriptive number or
  label associated with each member of a
  population.
• The values of a population variable are the
  various numbers (or labels) that occur as we
  consider all the members of the population.
• Values of variables that have been recorded for a
  population or a sample from a population
  constitute data.

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Types of Data

• Nominal variables are variables whose values are
  labels.
• Ordinal variables are variables whose values have
  a natural order.
• Interval variables have values represented by
  numbers referring to a scale of measurement.
• Ratio variables have values that are positive
  numbers on a scale with a unit of measurement and
  a natural zero point.

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Guess the Type

• Age
• Questionnaire responses 1strongly
  agree,2agree,5strongly disagree
• Letter grades
• Reading comprehension scores
• Gender
• Zip codes
• Molecular velocities

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II. Summarizing Data
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Location Measures (Measures of Central Tendency)

• A location measure or measure of central
  tendency for a variable is a single value or
  number that is taken as representing all the
  values of the variable. Different location
  measures are appropriate for different types of
  data.

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The Mean

• For interval or ratio variables x
• N individuals in the sample or population
• xi value of x for ith individual

The mean of a population variable is denoted by m
(the Greek letter mu).
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The Mean with Repeated Values

• Distinct values of x
• nj frequency of occurrence of

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The Mean with Repeated Values

• Relative frequencies

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Example
-2 1 3 4 6
2 1 3 5 3
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The Median

• Informally, the middle value when all the
  values are arranged in order
• A number m is a median of x if at least half the
  individuals i in the population have
• and at least half of them have

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The Median Example 1

• x 2.0, 1.5, 2.2, 3.1, 5.7 (no repetitions)
• median(x)2.2

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The Median Example 2

• x -2.0, 1.5, 3.1, 3.1, 3.1
• median(x) 3.1

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The Median Example 3

• x -2.0, 1.5, 3.1, 5.7, 5.9, 7.1
• median(x)Any number in 3.1,5.7
• By convention, for an even number of individuals
  choose the midpoint between the smallest and
  largest medians, e.g.,

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Example

• Change 7.1 to 71. What happens to the mean and
  the median?
• The mean changes from 3.55 to 14.2
• No change in the median
• The median is much less sensitive to outliers
  (which may be mistakes in recording data)

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The Median for Ordered Categories
A A- B B B- C C C- D D D- F
8 5 10 18 18 15 14 6 4 1 1 0
N100. The median grade is B-.
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The Mode

• The data value with the greatest frequency
• Not useful for interval or ordinal data if
  recorded with precision
• The only useful location measure for strictly
  nominal data

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Example
A A- B B B- C C C- D D D- F
8 5 10 18 18 15 14 6 4 1 1 0
The modes are B and B-.
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Cumulative Frequencies and Percentiles

• x is an interval or ratio variable.
• Ordered distinct values
• Relative frequencies

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Cumulative Frequencies and Percentiles

• Cumulative Frequencies

• Cumulative Relative Frequencies

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The Weather Persons Prediction Errors x
x'j -2 1 3 4 6
nj 2 1 3 5 3
Nj 2 3 6 11 14
fj .1429 .0714 .2143 .3571 .2143
Fj .1429 .2143 .4286 .7857 1.000
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Exercise

• From the table above, what fraction of the data
  is less than 1? What fraction is greater than 3?
  What fraction is greater than or equal to 3?

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Percentiles

• x an interval or ratio variable
• A number a is a pth percentile of x if at least
  p of the values of x are less than or equal to a
  and at least (100-p) of the values of x are
  greater than or equal to a.
• The 25th percentile is called the first quartile
  of x and the 75th percentile is the third
  quartile of x.
• The 50th percentile is the second quartile or
  median.

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Example

• For the weather persons errors, the 25th
  percentile is 3. The 50th percentile and third
  quartile are both 4.

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Measures of Variability

• Statisticians are not only interested in
  describing the values of a variable by a single
  measure of location. They also want to describe
  how much the values of the variable are dispersed
  about that location.

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Population Variance and Standard Deviation

• x an interval or ratio variable.
• Nnumber of individuals in population.
• Variance of x
• Standard deviation of x

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Sample Variance and Standard Deviation

• n the number of individuals in a sample from a
  population
• Sample variance
• Sample standard deviation

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Alternative Formulas for the Variance

• Using frequencies
• Using relative frequencies

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The Interquartile Range

• Q1, Q3 1st and 3rd quartiles, respectively
• Interquartile range
• Not influenced by a few extremely large or small
  observations (outliers)

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The Range

• The difference between the largest data value and
  the smallest
• Range of sample values is not a reliable
  indicator of the range of a population variable

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III. Graphical Methods
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Pie Charts (Circle Graphs)
Sources ATT (1961) The Worlds Telephones R A
language and environment for statistical
computing, the R core development team.
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Bar Charts (Bar Graphs)
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Pros and Cons

• Bar chart has a scale of measurement more
  precise information
• Pie chart gives more vivid impression of relative
  proportions, e.g., obvious at a glance that N.
  America had more than half the telephones in the
  world.

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Stemplots (Stem and Leaf Diagrams)

• StemLeaves Cumulative Frequency
• 4 7 1
• 5 448889 7
• 6 34789 12
• 7 012234455666888889999 33
• 8 0022234457799 46
• 9 0457 50

Grades of 50 students on a test
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Find the Median

• StemLeaves Cumulative Frequency
• 4 7 1
• 5 448889 7
• 6 34789 12
• 7 012234455666888889999 33
• 8 0022234457799 46
• 9 0457 50

25th and 26th leaves circled. Median 78
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Exercise

• StemLeaves Cumulative Frequency
• 4 7 1
• 5 448889 7
• 6 34789 12
• 7 012234455666888889999 33
• 8 0022234457799 46
• 9 0457 50

The 1st quartile is 70 and the 3rd quartile is 82.
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Boxplots (Box and Whisker Diagrams)
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Elements of a Boxplot
largest
outlier
box
whisker
quartiles
median
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Boxplot Shows Distribution Skewed to the Left
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Histograms

• For interval or ratio data
• Data is grouped into class intervals
• Superficially like a bar chart

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Frequency Histogram
Heightbin frequency
Class interval (bin)
Source R A language and environment for
statistical computing, the R core development
team.
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Probability Histogram
Area of bar relative bin frequency E.g.,
.01125.275
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Ogives(Cumulative Frequency Polygons)

• Related to probability histograms
• Examples of cumulative distribution functions
• Probability histograms are examples of density
  functions

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Example Ogive
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Relationship Between Probability Histogram and
Ogive

• The height of the ogive is the cumulative area
  under the histogram

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Estimating Percentiles from Ogives

• Horizontal line has height .75
• Vertical line intersects horizontal axis at 60
• Estimated 3rd quartile is 60
• True 3rd quartile is 62

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Scatterplots (Scatter Diagrams)

• Used for jointly observed interval or ratio
  variables
• Example Heights and weights of individuals
• Example State per capita spending on secondary
  education and state crime rate
• Example Wind speed and ozone concentration

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Example Scatterplot
centroid
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Fitting a Line

• Relationship between variables x and y is
  approximately linear.
• Approximately, y a bx.
• Find a and b so that data comes closest to
  satisfying the equation.
• Least squares a formal mathematical technique
  to be shown later.

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Line Fitted by Least Squares
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IV. Sampling
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Why Sample?

• Because the population is too large to observe
  all its members.
• The population may be partly inaccessible.
• The population may even be hypothetical.

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Statistical Inference

• Drawing conclusions about the population based on
  observations of a sample.
• Reliability of inferences must be quantifiable.
• Random sampling allows probability statements
  about the accuracy of inferences.

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Sampling With Replacement

• Population has N members.
• n population members chosen sequentially.
• Once chosen, a member of the population may be
  chosen again.
• At each stage, all members of the population are
  equally likely to be chosen.
• Random experiment with possible equally
  likely outcomes.

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Sampling With Replacement (continued)

• x is a population variable.
• X1 value of x for 1st sampled individual, X2
  value of x for 2nd sampled individual, etc.
• Each Xi is a random variable. The random
  variables are independent.
• The sequence is a random
  sample of values of x, or a random sample from
  the distribution of x.

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Sampling Without Replacement

• Population has N individuals.
• n members chosen sequentially.
• Once chosen, an individual may not be chosen
  again.
• At each stage, all of the remaining members are
  equally likely to be chosen next.
• Random experiment with
  possible equally likely outcomes.

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Sampling Without Replacement (continued)

• Sample without replacement.
• Ignore the order of the sequence of individuals
  in the sample.
• Random experiment whose outcomes are subsets of
  size n.
• Experiment has possible
  equally likely outcomes.
• Common meaning of random sample of size n

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Random Number Generators

• Calculators and spreadsheet programs can generate
  pseudorandom sequences.
• Press the random number key of your calculator
  several times.
• Simulates a random sample with replacement from
  the set of numbers between 0 and 1 (to high
  precision).

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Generating a Sample with Replacement

• Number the individuals from 1 to N.
• Generate a pseudorandom number R.
• Include individual i in the sample if
• Repeat n times. Individuals may be included more
  than once.

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Exercise

• Suppose you have 30 students in your class.
  Use the procedure just described to obtain a
  sample of size 10 (a) with replacement, (b)
  without replacement.

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V. Estimation
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The Sample Mean and Standard Deviation

• is a random sample from the
  distribution of a population variable x.
• The sample mean is
• The sample variance is

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The Sample Mean and Standard Deviation (continued)

• The sample standard deviation is
• The sample mean, variance and standard
  deviation are all random variables because they
  depend on the outcome of the random sampling
  experiment.

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Estimators

• The sample mean, variance, and standard deviation
  have distributions derived from the distribution
  of values of the population variable x.
• They are estimators of the population mean m, the
  population variance s2, and the population
  standard deviation s of x.

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Unbiased Estimators

• The theoretical expected values of the sample
  mean and sample variance are equal to their
  population counterparts, i.e.,
• and S2 are said to be unbiased estimators
  of m and s2, respectively
• S is biased.

and
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The Distribution of the Random Variable

• The mean of is m, the same as the mean of
  the population variable x.
• The standard deviation of is
• These are the theoretical mean and standard
  deviation.

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Density Functions

• A density function is a nonnegative function
  such that the total area between the graph of the
  function and the horizontal axis is 1.
• A probability histogram is a density function.
• Other density functions are limits of
  histograms as the number of data elements grows
  without bound.

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The Standard Normal Density Function
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Percentiles of the Standard Normal Distribution
za is the 100(1-a) percentile of the distribution
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Symmetry About the Vertical Axis
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Probabilities Related to the Standard Normal
Distribution
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Other Normal Distributions

• Let Z be a random variable with the standard
  normal distribution.
• The mean of Z is 0 and the standard deviation of
  Z is 1.
• Let m and s be any numbers, sgt0.
• Let Y sZm
• Y has the normal distribution with mean m and
  standard deviation s.

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Other Normal Distributions Example
m 1 and s 1.5
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Standardizing The Inverse Operation

• Let Y be normally distributed with mean m and
  standard deviation s.
• Let . This is the z-score of
  Y.
• Then Z has the standard normal distribution and

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The Central Limit Theorem

• Let be the sample average of a random sample
  of n values of a population variable x.
• The population variable x has mean m and standard
  deviation s.
• Standardize by subtracting its mean and
  dividing by its standard deviation

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The Central Limit Theorem (continued)

• Get Ready for the Central Limit Theorem!

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The Central Limit Theorem(continued)

• The Central Limit Theorem
• As the sample size n grows without bound, the
  distribution of Z approaches the standard
  normal distribution. This is true no matter what
  the distribution of values of the population
  variable x.

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Another Statement of the CLT

• For sufficiently large sample sizes n and for
  all numbers a and b,
• In almost all applications, n50 is large
  enough.

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The CLT in Action

• Sample n30 from the population variable
  COUNTS whose distribution is tabulated.
  Calculate the sample average. Repeat this 500
  times and construct a histogram of the z-scores
  of the 500 sample averages. Note The
  distribution of COUNTS is very far from normal.

xj? 0 1 2 3 4 5 6
fj .36 .33 .19 .08 .02 .01 .01
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Distribution of COUNTS
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Result-500 Averages of 30 Samples from COUNTS
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Estimating a Population Mean

• The sample mean is an unbiased estimator of
  the population mean m.
• For large sample sizes n, has
  approximately a normal distribution with mean m
  and standard deviation
• For large n, the sample mean is an accurate
  estimator of the population mean with high
  probability.

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Example

• Suppose s 2 and we want to estimate m
  with an error no greater than 0.05.
• Assume is exactly normally distributed.
  Standardize.

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Probabilities of 1-place Accuracys 2
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Confidence Intervals for the Population Mean
Review of
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100(1-a) Confidence Interval

• By the CLT
• Rearranging the inequalities

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A Difficulty

• s is probably unknown, so the confidence
  interval
• cant be used. What to do?

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Enhanced Central Limit Theorem

• Define the modified z-score for as
• As n grows without bound, the distribution of Z
  approaches the standard normal distribution.

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A More Useful Confidence Interval

• By the enhanced CLT
• An approximate 100(1-a) confidence interval is

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Example

• n50 from COUNTS (m 1.14)
• 1.32
• S 1.39
• 1-a .95
• 1.320.39
• 95 confidence interval (0.93, 1.71)
• Dont say .95P0.93ltmlt1.71

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Confidence Intervals for Proportions

• x is a population variable with only two values,
  0 and 1.
• Numerical code for two mutually exclusive
  categories, e.g., male and female, or
  approves and disapproves.
• prelative frequency of x1.
• mp s2p(1-p)

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Confidence Intervals for Proportions (continued)

• Sample n values of x, with replacement. Result is
  a sequence of 1s and 0s.
• Sample mean is the relative frequency in the
  sample of 1s, e.g., the relative frequency of
  females in the sample of individuals.
• Denote the sample mean by since it is an
  estimator of p.

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Confidence Intervals for Proportions(continued)

• By the enhanced CLT, is
  approximately standard normal.
• An approximate 100(1-a) confidence interval is

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Example

• A public opinion research organization polled
  1000 randomly selected state residents. Of
  these, 413 said they would vote for a 1 sales
  tax increase dedicated to funding higher
  education. Find a 90 confidence interval for
  the proportion of all voters who would vote for
  such a proposal.

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Solution

• n 1000
• 1-a .90
• 0.413 1.645
• (0.387, 0.489)

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Linear Regression and Correlation

• x and y are jointly observed numeric variables,
  i.e., defined for the same population or arising
  from the same experiment.
• Have observations for n individuals or outcomes.
• Data

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Examples

• (An observational study) Let x be the height and
  y the weight of individuals from a human
  population.
• (A designed experiment) Let x be the amount of
  fertilizer applied to a plot of cotton seedlings
  and let y be the weight of raw cotton harvested
  at maturity.

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Data on Fertilizer and Cotton Yield
x 2 2 2 4 4 4 6 6 6 8 8 8
y 2.3 2.2 2.2 2.5 2.9 2.7 3.4 2.7 3.4 3.5 3.4 3.3
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Scatterplot of Fertilizer vs. Yield
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Assumptions of Linear Regression

• There is a population or distribution of values
  of y for any particular value of x.
• There are unknown constants a and b so that for
  any particular value of x, the mean of all the
  corresponding values of y is
• The standard deviation s of the values of y
  corresponding to a value of x is the same for all
  values of x.

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The Method of Least Squares

• Estimate a and b by choosing them to minimize the
  sum of squared differences between the observed
  values yi and their putative expected values
• In symbols, minimize

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The Least Squares Estimates

• Let and be the means of the observed
  xs
• and ys. Let be the sample variance of
  the xs.
• The covariance between the xs and the ys is
• The least squares estimate of the slope is
• The least squares estimate of the intercept is
  

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Least Squares Line for Cotton Yield
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Correlation

• The correlation between the xs and ys is
• r is related to the slope b of the least squares
  regression line by
• r is always between -1 and 1. r measures how
  nearly linear the relationship between x and y
  is. If r 0, then x and y are uncorrelated.

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Examples