SLIDES PREPARED

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STATISTICS for the Utterly Confused, 2nd ed.

• SLIDES PREPARED
• By
• Lloyd R. Jaisingh Ph.D.
• Morehead State University
• Morehead KY

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Chapter 14

• Chi-Square Procedures

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Outline

• Do I Need to Read This Chapter? You should read
  the Chapter if you would like to learn about
• 14-1 Properties of the chi-square
  distribution.
• 14-2 The chi-square test for
  goodness-of-fit.
• 14-3 The chi-square test for
  independence.
• 14-4 Benfords Law.

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Objectives

• To introduce you to the chi-square
  distribution.
• To use the chi-square distribution to perform
  tests for goodness-of-fit and independence.

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Objectives

• To introduce you to Benfords Law.
• To introduce technology integration for
  chi-square tests.

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14-1 The Chi-Square ( ?2 ) Distribution
- Properties

• It is a continuous distribution.
• It is not symmetric.
• It is skewed to the right.
• The distribution depends on the degrees of
  freedom, df n 1, where n is the sample size.

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14-1 The Chi-Square ( ?2 ) Distribution
- Properties

• The value of a ?2 random variable is always
  nonnegative.
• There are infinitely many ?2 distributions, since
  each is uniquely defined by its degrees of
  freedom.

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14-1 The Chi-Square ( ?2 ) Distribution
- Properties

• For small sample size, the ?2 distribution is
  much skewed to the right.
• As n increases, the ?2 distribution becomes more
  and more symmetrical.

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14-1 The Chi-Square ( ?2 ) Distribution
- Properties
Family of ?2 distributions.
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14-1 The Chi-Square ( ?2 ) Distribution
- Properties

• Since we will be using the ?2 distribution for
  the tests in this chapter, we will need to be
  able to find critical values associated with the
  distribution.

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Quick Tip

• Extensive tables of critical values are
  available for use in solving confidence
  intervals and hypothesis testing problems that
  are associated with the ?2 distribution.

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14-1 The Chi-Square ( ?2 ) Distribution
- Properties

• Notation ?2?, n-1
• Explanation of the notation ?2?, n -1 ?2?, n
  -1 is a ?2 value with n - 1 degrees of freedom
  such that ? area is to the right of the
  corresponding ?2 value.

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14-1 The Chi-Square ( ?2 ) Distribution
- Properties
Diagram explaining the notation ?2?, n-1
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14-1 The Chi-Square ( ?2 ) Distribution
- Properties

• Values for the random variable with the
  appropriate degrees of freedom can be obtained
  from the tables in the appendix of the text
  (Table 4).
• Example What is the value of ?20.05,10?

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14-1 The Chi-Square ( ?2 ) Distribution
- Properties

• Solution From Table 4 in the appendix, ?20.05,10
  18.307. (Verify).
• Example What is the value of ?20.95,20?
• Solution From Table 4 in the appendix, ?20.95,20
  10.851. (Verify).

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14-2 The Chi-Square test for Goodness of Fit

• Have you ever wondered whether a sample of
  observed data (frequency distribution or
  proportions) fits some pattern or distribution?
• We should not expect the pattern to exactly fit a
  given distribution, so we can look for
  differences and make conclusions as to the
  goodness-of-fit of the data.

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14-2 The Chi-Square test for Goodness of Fit

• From the Figure on the next slide, one can
  clearly see that the pattern of the sample data
  does not quite follow the distribution of the
  population.
• As a matter of fact, the sample data deviates
  quite severely from the population distribution.

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14-2 The Chi-Square test for Goodness of Fit
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14-2 The Chi-Square test for Goodness of Fit

• Hence one may intuitively conclude in this case
  that the sample data did not come from the
  population to which it is compared because of the
  large deviations from the sample distribution to
  the population distribution.

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14-2 The Chi-Square test for Goodness of Fit

• From the Figure on the next slide, one can
  observe that the sample distribution follows
  quite closely to the population distribution.
• In this case, one may intuitively conclude that
  the sample data did come from the population to
  which it is compared because of the very small
  deviation of the sample distribution from the
  population distribution.

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14-2 The Chi-Square test for Goodness of Fit
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14-2 The Chi-Square test for Goodness of Fit

• Generally, we can assume that a good fit exists.
  
• That is, we can propose a hypothesis that a
  specified theoretical distribution is appropriate
  to model the pattern.
• Below is a summary of the tests for
  goodness-of-fit.

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14-2 The Chi-Square test for Goodness of Fit
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Quick Tip

• The chi-square goodness of fit test is always a
  right-tailed test.

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Quick Tip

• For the chi-square goodness-of-fit test, the
  expected frequencies should be at least 5.
• When the expected frequency of a class or
  category is less than 5, this class or category
  can be combined with another class or category so
  that the expected frequency is at least 5.

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EXAMPLE

• Example There are 4 TV sets that are located in
  the student center of a large university. At a
  particular time each day, four different soap
  operas (1, 2, 3, and 4) are viewed on these TV
  sets. The percentages of the audience captured
  by these shows during one semester were 25
  percent, 30 percent, 25 percent, and 20 percent,
  respectively. During the first week of the
  following semester, 300 students are surveyed.

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EXAMPLE (Continued)

• (a) If the viewing pattern has not changed, what
  number of students is expected to watch each soap
  opera?
• Solution Based on the information, the expected
  values will be 0.25?300 75, 0.30?300 90,
  0.25?300 75, and 0.20?300 60.

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EXAMPLE (Continued)

• (b) Suppose that the actual observed numbers of
  students viewing the soap operas are given in the
  following table, test whether these numbers
  indicate a change at the 1 percent level of
  significance.

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EXAMPLE (Continued)

• Solution Given ? 0.01, n 4, df 4 1
  3, ?20.01, 3 11.345. The observed and
  expected frequencies are given below

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EXAMPLE (Continued)

• Solution (continued) The ?2 test statistic is
  computed below.

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EXAMPLE (Continued)

• Solution (continued)

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EXAMPLE (Continued)

• Solution (continued)

Diagram showing the rejection region.
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14-3 The Chi-Square test for Independence

• The chi-square independence test can be used to
  test for the independence between two variables.

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EXAMPLE

• Example A survey was done by a car manufacturer
  concerning a particular make and model. A group
  of 500 potential customers were asked whether
  they purchased their current car because of its
  appearance, its performance rating, or its fixed
  price (no negotiating). The results, broken down
  by gender responses, are given on the next slide.

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EXAMPLE (Continued)
Question Do females feel differently than males
about the three different criteria used in
choosing a car, or do they feel basically the
same?
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EXAMPLE (Continued)

• One way of answering this question is to
  determine whether the criterion used in buying a
  car is independent of gender.

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EXAMPLE (Continued)

• That is, we can do a test for independence.
• Thus the null hypothesis will be that the
  criterion used is independent of gender, while
  the alternative hypothesis will be that the
  criterion used is dependent on gender.

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Quick Tips

• When data are arranged in tabular form for the
  chi-square independence test, the table is called
  a contingency table.
• Here the table on slide 35 has 2 rows and 3
  columns, so we say we have a 2 by 3 (2?3)
  contingency table.

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Quick Tips

• The degrees of freedom for any contingency table
  is given by (number of rows 1)?(number of
  columns 1). In this example,
• df (2 1)?(3 1) 2.

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EXAMPLE (Continued)

• In order to test for independence using the
  chi-square independence test, we must compute
  expected values under the assumption that the
  null hypothesis is true.
• To find these expected values, we need to compute
  the row totals and the column totals.

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EXAMPLE (Continued)

• The table on the next slide shows the observed
  frequencies with the row and column totals.
• These row and column are called marginal totals.

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EXAMPLE (Continued)
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EXAMPLE (Continued)

• Computation of the expected values (example)-
• The total for the first row (male) is 185, and
  the total for the first column (appearance) is
  180. The expected value for the cell in the
  table where the first row (male) and first column
  (appearance) intersect will be (185?180)/500
  66.6.

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EXAMPLE (Continued)

• The table on the next slide shows the expected
  frequencies with the marginal totals.

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EXAMPLE (Continued)
Let us use ? 0.01. So df (2 1)(3 1) 2
and ?20.01, 2 9.210.
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EXAMPLE (Continued)

• Solution (continued) The ?2 test statistic is
  computed in the same manner as was done for the
  goodness-of-fit test.

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EXAMPLE (Continued)

• Solution (continued)

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EXAMPLE (Continued)

• Solution (continued) Diagram showing the
  rejection region.

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14-4 Benfords Law

• Frank Benford, in the 1930s, noticed that
  logarithm tables (these were used by scientists
  long before the common use of computers and
  calculators) tended to be worn out on the early
  pages where the numbers started with the digit 1.
  

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14-4 Benfords Law

• Based on this observation and many others, he
  discovered that more numbers in the real world
  started with the digit 1 rather than with 2, and
  that more started with the digit 2 rather than
  with 3, and so on.
• He later published a formula which describes the
  proportion of times a number will begin with the
  digit 1, 2, 3, etc.

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14-4 Benfords Law

• This formula is now called Benfords Law.
• The Table on the next slide shows the
  distribution of the proportions, to three decimal
  places, for the leading digits of numbers based
  on Benfords Law.

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14-4 Benfords Law
The next slide shows a graphical Depiction of
Benfords Law.
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14-4 Benfords Law
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14-4 Benfords Law

• Example Students who attend college and apply
  for student loans must submit a FAFSA (Free
  Application for Federal Student Aid) form. Part
  of the information that is required is the annual
  income of the parent or parents. A sample of
  3,633 forms was sampled from a college records
  and the proportion, to three decimal places, of
  the leading digits for the total annual income
  for the parents were recorded. This information
  is presented on the next slide.

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14-4 Benfords Law

• Test at the 5 percent significance level whether
  the distribution of the first digits for the
  reported total salaries for the parents follow
  Benfords Law.

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14-4 Benfords Law

• Solution Plots of the proportions of the leading
  digits for both Benfords Law and the parents
  salaries are shown below.

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14-4 Benfords Law

• Solution (continued) The Table on the next slide
  shows the computations needed to compute the ?2
  test statistic.
• The value of the test statistic is equal to
  507.527.
• To obtain the expected frequencies based on
  Benfords Law one should multiply the total of
  3,633 by Benfords proportions.
• For example, from the table, the expected
  frequency value of 639.408 is obtained from
  3,6330.176 639.408, etc.

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14-4 Benfords Law
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14-4 Benfords Law
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EXAMPLE (Continued)

• Solution (continued) Diagram showing the
  rejection region.