Computers, One and TwoSample Analyses, and Chisquared Tests

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Computers, One- and Two-Sample Analyses, and
Chi-squared Tests
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Using personal computers in this course

• Your use will be important and extensive
• We will be using two main things
• Excel - to do statistical analyses
• World Wide Web - to obtain course data and
  information

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Using Excel

• Excel is a spreadsheet program - the interface
  is divided into a large number of rectangular
  cells
• Each cell can contain one of four things
• A number
• A label
• A formula
• A built-in Excel function

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Using Excel, cont.

• There are two additional parts of Excel that we
  will be using extensively
• The Data Analysis Tools (does the statistical
  number crunching for us)
• The Solver optimization routine (solves linear
  programming problems for us)
• If you own your own computer, you should make
  sure you have these

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Checking for impt. add-ins

• From the Tools menu, select Add-Ins . . .
  then make sure the following are on the list
• Analysis ToolPak
• Analysis ToolPak - VBA
• Solver Add-In
• and that they are checked
• Get my handout if these three are not on your list

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Using the course Web site

• The URL is on the course syllabus
• Five parts
• Homework assignments (which problems will be due
  on which Thursdays)
• Homework answers
• Exam answers
• Recommended practice problems
• Access to the course data sets

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Accessing the course data sets

• Right click on the data set link (or Shift Left
  click)
• Select the Save As . . . option
• Expand the compressed file once you have
  downloaded it

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Using Excel, cont.

• Saving a file - use Save As (not Save) under
  the File menu
• Printing and page setup
• Opening text files (like the MBS files) -
  learning to use the Text Import Wizard (Dont
  forget to remove the End-of-file character!)

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Using Excel, cont.

• Excels built-in statistical functions - for the
  moment, two of many
• AVERAGE( Some range )
• STDEV( Some range )
• Excels Data Analysis Tools - again, two of
  several
• Descriptive Statistics (calculates lots of
  different numerical summaries of the data)
• Histogram

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A Diagram of the Statistical Process
The Population The individuals have at least
one characteristic, e.g., age, which varies over
them. Hence,this characteristic has a
probability distribution
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Lets Look at Your Results

• What are the values of your three statistics?
• What sampling method did you use to obtain your
  sample?
• Did you sample with or without replacement?

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I Now Have Three Sets of Values

• Why are they different?
• Which ones are correct? Why?
• In this process, what is the
• Population?
• Sample outcome?
• Statistic?
• Parameter?

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Summary

• Statistics are random variables -- as such they
• have a probability distribution. Like all
  probability distributions, this distribution has
• a mean
• a standard deviation
• a characteristic shape

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Summary, cont.

• The probability distribution of a statistic is
  called the sampling distribution
• Be careful to distinguish the sampling
  distribution from the population distribution
  (See, for example, Fig. 6.10 on p. 255 of MBS)

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Summary, cont.

• Statisticians know the most about the sampling
  distribution of the sample mean
• The mean of this sampling distribution is the
  same as the mean of the population distribution
• The standard deviation of this sampling
  distribution is the standard deviation of the
  population distribution divided by the square
  root of the sample size.

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Samp. Dist. Of Mean, cont.

• If the sample size is large enough, no matter
  what the shape of the population distribution,
  the sampling distribution is approximately
  Normally distributed. (This is the Central Limit
  Theorem.)

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Samp. Dist. Of Mean, cont.

• If the population distribution is Normal, no
  matter what the sample size, the sampling
  distribution is
• Normally distributed if the value of s is known
• Students t distributed if the value of s is
  unknown

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The Sampling Distribution of the Sample Mean,
cont.
Sampling Distribution Normal
Sampling Distribution Students t
Punt!
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Samp. Dist. Of Mean, cont.

• Note what is not known
• The sampling distribution of the sample mean when
  the population distribution is non-Normal and the
  sample size is small
• The sampling distribution of any statistic other
  than the sample mean. (You will learn about some
  of these as you further study statistics.)

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A Diagram of the Statistical Process
The Population The individuals have at least
one characteristic, e.g., age, which varies over
them. Hence,this characteristic has a
probability distribution
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Overview of Statistical Inference

• Closing the gap between statistics, whose
  numerical values we know, and parameters, whose
  numerical values we want to know

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Recall Two Important Concepts

• Statistics vary in value from sample to sample
• Statistics rarely exactly coincide with their
  corresponding parameter
• The lack of coincidence means more complex
  techniques will be needed for inferring parameter
  values

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Two Kinds of Statistical Inference

• Confidence intervals
• Hypothesis testing
• Each is based on the notion that a statistic is a
  random variable
• and consequently heavily depends on the sampling
  distribution of the statistic

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Bala -- What is the difference between hypothesis
testing and confidence intervals?
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Confidence Intervals

• You have no idea what the value of m is
• Use as the best estimate
• Construct a confidence interval around
  which is likely to contain m

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Hypothesis Testing

• You have some belief about the value of m
• Then ask the question - If that value of m is
  correct, how likely would it be that the value of
  which I have obtained would occur?

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Hypothesis Testing, cont.

• If it is likely that value of would occur,
  given m, we conclude that our belief about m
  must be correct
• If it is not likely that value of would
  occur, given m, we conclude that our belief
  about m must not be correct

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Three Ways to do Hypothesis Testing

• Method 1 Standardize the value of the statistic
• and compare this to the z-value or t-value which
  is the boundary of the rejection region

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Three Ways, cont.

• Method 2 Convert the boundary (z or t value)
  of the rejection region to the scale of the
  statistic
• and compare this value to the value of the
  statistic

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Three Ways, cont.

• Method 3 Using the value of the statistic,
  calculate the p-value and compare this p-value to
  the value of a
• Remember The p-value must always match the
  rejection region!

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Rules when using the p-value

• If p-value gt a accept H0
• If p-value lt a reject H0

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Steps in Hypothesis Testing

• Determine the null and alternative hypotheses
• Specify the value of a
• Calculate the standardized value of the statistic

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Steps in Hypothesis Testing, cont.

• Draw a picture of the sampling distribution which
  includes
• Identification of the shape ( z or t )
• Location of the rejection region (Upper tail,
  lower tail, or both)
• Position of the standardized statistic (Above or
  below the mean)
• Use the standardized value of the statistic to
  calculate the p-value

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Steps in Hypotesis Testing, cont.

• Compare the p-value to the value of a to draw a
  conclusion (Reject or accept the null hypothesis)
• Important - Translate your statistical conclusion
  into the practical terms of the problem

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Using Excel to Calculate p-values

• To calculate Normal probabilities, use
• NORMSDIST( z )
• To calculate t probabilities, use
• TDIST( t, d.f., tails )

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Comparing the Means of Two Populations

• Three ways to do the analysis
• When the samples are independent of each other
• Both samples large (30 or more)
• One or both samples small (less than 30)
• When the samples can be paired - this situation
  reduces to a one-sample test

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Comparing Two Means with Independent Samples

• Statistic
• Sampling distribution
• Normal if both samples are 30 or more (p. 370 of
  MBS)
• t if one or both samples smaller than 30 (pp.
  373-374 of MBS)
• Important - maintain order between your
  hypotheses and your statistic (pp. 13-14 of the
  coursepack)

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Comparing Two Means with Independent Samples,
cont.

• In Excel, for the large sample test
• Use VAR( data range ) to calculate the sample
  variance of each data set
• Write down these variances!
• Then use the Data Analysis Tool
• z-Test Two-Sample for Means
• For the small sample test use
• t-Test Two-Sample Assuming Equal Variances

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Comparing Two Means when the data can be paired

• Think of pairing data like you would think of a
  pair of shoes
• Most aspects the same (size, color, style, etc.)
• One aspect different (left vs. right)

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Comparing Two Means using paired data, cont.

• Calculate the difference between the paired
  values
• Analyze these differences as you would analyze a
  one-sample set of data
• Again, be careful about maintaining order
  between your calculations of the differences and
  your hypotheses!

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Comparing Two Means using paired data, cont.

• In Excel - Data Analysis Tool
• t-Test Paired Two Sample for Means

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The Crest toothpaste test

• Pairs of identical twins living in the same
  household were recruited
• One twin was given Crest without stannous
  fluoride, the other Crest with stannous fluoride
• A paired test was conducted
• P G spent 1 M promoting the results, and
  Crest grew from a 3 share brand to the market
  leader at about a 35 share

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Chi-squared Analyses of Categorical Data

• We will discuss two kinds of chi-squared analyses
• Analysis of contingency tables - testing two
  variables to see whether they are related to one
  another
• Goodness-of-fit tests - checking some data
  against a probability distribution

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Analysis of Contingency Tables

• Hypotheses
• H0 There is no relationship between the two
  variables
• Ha There is a relationship between the two
  variables

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Analysis of Contingency Tables, cont.

• Using the probability law
• P(A and B) P(A) P(B)
• if A and B are independent
• create a table of Expected Counts
• Compare the table of actual data (Observed) with
  the Expected table using the chi-squared statistic

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The Chi-squared Statistic

• is Chi-squared distributed with
• ( rows - 1) ( columns - 1) d.f.

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Analysis of Contingency Tables, cont.

• Small values of the chi-squared statistic mean
  the counts in the Observed table match those in
  the Expected table. This means that H0 is
  accepted.
• Large values of the chi-squared statistic mean
  the counts in the two tables dont match. This
  means H0 is rejected.

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Analysis of Contingency Tables using Excel

• There is no Data Analysis Tool to do this
• Hence, we create a spreadsheet which mimics the
  calculations we do by hand

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Analysis using Excel, cont.

• Start with the table of observed data, then
• Calculate a table of expected values,
• Calculate a table of chi-squared values,
• Calculate the overall chi-squared value,

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Analysis using Excel, cont.

• Calculate the p-value of the test, using either
• CHIDIST(c2 value, d.f.)
• or
• CHITEST(obs. range, exp. range)

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Goodness-of-fit Tests

• Similar to contingency table analysis, except the
  hypotheses postulate a probability distribution
• H0 The data come from a specified probability
  distribution
• Ha The data do not come from a specified
  probability distribution
• For this analysis
• d.f. categories - 1

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Goodness-of-Fit Example
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Quality Control at M Ms