Making Inferences for Associations Between Categorical Variables: Chi Square Chapter 12

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Making Inferences for Associations Between
Categorical Variables Chi SquareChapter 12

• Reading Assignment
• pp. 463-482 485

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Elements of a test of hypotheses 3

• Hypothesis testing Process for finding out
  whether we can generalize about an association
  from a sample to a population
• Null hypothesis (H_0) Represents the status quo
  to the party performing the sampling experiment,
  i.e., will be accepted unless the data provides
  convincing evidence it is false.
• Research hypothesis (H_1) (aka alternative
  hypothesis) Will be accepted only if the data
  provides convincing evidence of its truth
• Homework Skills 1, p. 464

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Process of Hypothesis Testing 5

• Step 1 Specify a research hypothesis and a null
  hypothesis
• Step 2 Compute the value of a test statistic for
  the relationship
• Step 3 Calculate the degrees of freedom for the
  variables involved
• Step 4 Look up the distribution for the test
  statistic to find its critical value at a
  specified level of probability (to determine the
  likelihood that a test stat. of a particular
  value could have occurred by chance alone)
• Step 5 Decide whether to reject the null
  hypothesis

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Null Hypothesis 3

• Null Hypothesis(H_0) speculates there is no
  association between the two variables. Examples
• H_0 men are no different from women in there
  political affiliations
• H_0 There is no relationship between a
  respondents educational level and his or her
  parents
• H_0 Older people are no more likely to be happy
  than younger people
• This is the only hypothesis that can actually be
  tested-we either reject or fail to reject the
  null hypothesis
• EX H_0 There is no association between age and
  happiness among American adults hw/ read p. 466

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2 Statistical Independence

• Statistical Independence Two variables are
  statistically independent when changes in one
  variable (age of respondents) have nothing to do
  with changes in a second (happiness), ie, they
  vary independently of one another
• Conversely, when two variables are statistically
  dependent on one another, changes in one variable
  are associated with changes in a second
  variable.,ie, changes in age(older respondents)
  are associated with changes in levels of
  happiness (more happiness)

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2 Statistical Independence and hypothesis testing

• Ex/ Null Hypothesis Age is statistically
  independent of happiness, ie, differences among
  respondents to the variable age are unrelated to
  any differences in their levels of reported
  happiness
• Hyp. Testing can assess the likelihood that the
  degree of statistical indep found in the sample
  is due to chance
• If we find that the degree of statistical indep
  found in the sample is not likely to be due to
  chance, null hyp is rejected
• If it is likely due to chance, null hyp is
  accepted

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3 Type I and Type II Errors

• Mistakes arising from whether a given sample
  may or may not be representative of a population
• If a Null Hypothesis assumes there is no
  association between two variables, and we reject
  it even though there is no association is a Type
  I error, i.e, we call someone a liar when he is
  telling the truth
• If a Null Hypothesis assumes there is no
  association between two variables, and we accept
  it even though there is an association is a Type
  II error, i.e., we say someone is truthful when
  he is lying

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3 Type I and Type II Errors

Conclusion H_0 true H_1 true
H_0 true Correct decision Type II error
H_1 true Type I error Correct decision
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3 Elements of a Test of Hypothesis

• Null Hypothesis (H_0) a theory about one of the
  population parameters. The theory generally
  represents the status quo, which must be proven
  false
• Research Hypothesis (H_1) a theory that
  contradicts the null hypothesis. The theory
  generally represents the truth that will be
  accepted only if there is evidence
• Test statistic Sample statistic used to decide
  whether to reject the null hypothesis

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3 Elements of a Test of Hypothesis (cont)

• Rejection region The numerical values of the
  test statistic for which the null hypothesis will
  be rejected. The rejection region is chosen so
  that the probability is ? that it will contain
  the test statistic when the null hypothesis is
  true, thereby leading to a Type I error. The
  value of ? chosen is usually small (e.g.,
  0.01,0.05, or 0.1), and is referred to as the
  level of significance of the test. A 0.05 (or 5)
  level of significance indicates that there is a
  5 chance that we would reject the hypothesis
  when we should not, or we have 95 confidence
  that we have made the right decision
• Assumptions Clear statement(s) of any
  assumptions made about the population(s) being
  sampled

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Elements of a Test of Hypothesis (cont)

• Experiment and calculation of test statistic
• Conclusion
• If the numerical value of the test statistic
  falls in the rejection region, we reject the null
  hypothesis and conclude that the research
  hypothesis is true. We know that hypothesis
  testing will led to this conclusion incorrectly
  (Type I Error) 100? of the time when H_0 is
  true.
• If the test statistic does not fall in the
  rejection region, we do not reject H_0. Thus we
  reserve judgment about which hypothesis is true.
  We do not conclude that the null hypothesis is
  true because we do not, in general, know the
  probability that our test procedure will lead to
  an incorrect failure to reject H_0 (Type II Error)

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5 Chi-Square

• Formula 12.1
• Observed vs. Expected Roll a die 6 times, get
  three 3sobserved expected one 3
• Pp.469-71 skills Filling in the table of
  expected values
• Skills 3,4 Excel
• Generally, the greater the value of chi-square,
  the more statistical dependence between two
  variables

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3 Chi-Square /degrees of freedom

• We are using observations from a sample as well
  as certain population parameters. If these
  parameters are unknown,they must be estimated
  from the sample.
• Degrees of Freedom (?) the number N of
  independent observations in the sample (ie,
  sample size) minus the number k of population
  parameters which must be estimatede from sample
  observations
• ? N k
• When working with a contingency table,
  df(r-1)(c-1), where r and c are the number of
  rows and columns (resp) in the contingency table

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Chi squared examplegenerate random digits 250
times
digit 0 1 2 3 4 5 6 7 8 9
obs freq 17 31 29 18 14 20 35 30 20 36
Exp freq 25 25 25 25 25 25 25 25 25 25
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Chi squared examplegenerate random digits 250
times

• Question Does the observed frequency differ from
  the expected distribution in a significant way?

digit 0 1 2 3 4 5 6 7 8 9
obs freq 17 31 29 18 14 20 35 30 20 36
Exp freq 25 25 25 25 25 25 25 25 25 25
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3 Chi-Square random digit example

• ?2 (17-25)2/25 (31-25)2/25 (29-25)2/25
  (36-25)2/25 excel
• 23.3
• Degrees of freedom 10-19
• Table, p. 545
• ?2 at .99 is 21.7 23.3gt 21.7, so the observed
  frequency differs from the expected frequency at
  the 0.01 level of significance, so the table of
  random numbers is somewhat doubtful

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3 Chi-Square question

• 200 tosses of a fair coin, 115 heads, 85 tails.
  Test the hypothesis that the coin is fair using
  (a) 0.05, (b) 0.01 levels of significance
• Ans
• Df2-11 (2 for H,T)
• O1115, O285 E1E2100
• ?2(115-100)2/100 (85-100)2/100 4.5
• (a) ?2 table for .95 is 3.84 4.5gt3.84, so reject
  hyp that coin is fair at the 0.05 level of
  significance
• (b) ?2 table for .99 is 6.63 4.5lt6.63, so cannot
  reject hyp that coin is fair at the 0.01 level of
  significance

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Interpreting Chi Square 4

• When hypothesizing about an association between
  two variables, chi-square tells the likelihood
  that the degree of statistical dependence
  observed is simply the luck of the draw
• A p value of 0.05 tells that there are no more
  than 5 chances in 100 that the statistical
  dependence is due to chance. Thus, there are 95
  chances in 100 that the statistical dependence
  found is not due to chance, so the null
  hypothesis, ie., no association between
  variables, is rejected
• The higher the value of p, the less likely we are
  to make a Type I error
• bility

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Interpreting Chi Square 4

• When hypothesizing about an association between
  two variables, chi-square tells the likelihood
  that the degree of statistical dependence
  observed is simply the luck of the draw
• A p value of 0.05 tells that there are no more
  than 5 chances in 100 that the statistical
  dependence is due to chance. Thus, there are 95
  chances in 100 that the statistical dependence
  found is not due to chance, so the null
  hypothesis, ie., no association between
  variables, is rejected
• The higher the value of p, the less likely we are
  to make a Type I error
• bility

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Interpreting Chi Square 4

• P. 480-81 Table 12.4 (p. 472) has ?2 15.487,
  ?6
• The higher the ?2 value, the less likely it is
  that the value obtained is due to chance. (read
  table 12.9, p. 481)
• Rule of thumb reject null hypothesis when ?2
  reaches 0.05only 5 chances in 100 that the
  dependence is due to chance
• Skills7, p. 481
• Skills 8, p. 485 (following their example, p.
  484)

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• Homework/ p. 492/ 1,3
• P 494/ spss 1,2