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21 Cards in this Set

  • Front
  • Back

A Goodness of fit test

Assesses the hypothesis that an observed frequency distribution fits (or conforms) to some claimed distribution.

Requirements for Conducting a Multinomial Experiment

1. The number of trials (or observations) must be fixed.


2. The trials (or observations) must be independent.


3. Each trial must result in exactly one of several mutually exclusive categories.


4. The probability of a trial resulting in a particular categorical outcome must remain the same from trial to trial.

The Chi-Square (χ2) Probability Distribution

There is a family of chi-square distributions; each of these curves is slightly different depending on the number of degrees-of-freedom that are associated with it. An illustration appears below.

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Note that the χ2 distribution is not symmetric or bell-shaped; instead, it is skewed to the right. Note also that a χ2 value can be zero or positive; however, it cannot be negative.

Mode

The distribution occurs at a χ2 value equal to the number of degrees-of-freedom minus 2 (provided that the number of degrees-of-freedom is ≥ 3).

Mean

Equals the number of degrees-of-freedom (provided that the number of degrees-of-freedom is ≥ 3).

Median

Equals the number of degrees-of-freedom minus (provided that the number of degrees-of-freedom is ≥ 3).

Variance

Equals two times the number of degrees-of-freedom (provided that the number of degrees-of-freedom is ≥ 3).

Standard deviation

Equals the square root of two times the number of degrees-of-freedom (provided that the number of degrees-of-freedom is ≥ 3).

Assumptions Underlying the χ2 Goodness-of-Fit Test

1. The data has been randomly selected.


2. The data consists of frequency counts for different non-overlapping categories.


3. For each category, the expected frequency is at least 5. (Note there is no such requirement for the observed frequencies.)

The Null and Alternate Hypotheses

Ho: the proportions in the categories conform to some claimed distribution


H1; at least one of the proportions is different from its claimed value

The Test Statistic

The chi-square test statistic is given by the following formula:

Where “observed” is the observed frequency count in each category and “expected” is expected frequency count in each category.

If all of the expected frequencies are equal, then the expected frequency in each category is the sum of all the observations divided by the number of categories. Thus, E (the expected frequency in each category) equals (n / k), where n is the total number of observations and k is the number of categories.

If all of the expected frequencies are not equal, then the expected frequency in each category is the sum of all the observations multiplied by the probability for each categories.

Thus, E (the expected frequency in each category) equals (n) (p), where n is the total number of observations and p is the probability for each of the categories.

Degrees of Freedom for the Critical Value

The number of degrees-of-freedom for the χ2 goodness-of-fit test equals the number of categories (k) minus one (k – 1).

The Hypothesis Test

Note that the χ2 goodness-of-fit test is a right-tailed test only. If the test statistic is greater in value than the critical value, one would reject the null hypothesis.

The Chi–Square Test of Independence

A test of independence tests the null hypothesis that the levels (or categories) of one categorical variable are unrelated the levels (or categories) of a second categorical variable. That is, the null hypothesis states that the row variable and column variable in the contingency table are independent.

The Contingency Table

The contingency table cross classifies two categorical variables each of which has two or more levels (or categories). The cells represent the intersections betweenthe levels of one categorical variable and the levels of the other. The count of observations in each cell is recorded. The total number of observations in each of the levels is recorded as the various column and row totals.

The Null and Alternate Hypotheses

Ho: the two categorical variables are independent.HA: the two categorical variables are dependent.orHo: the levels of one categorical variable are unrelated to the levels of the other.HA: the levels of one categorical variable are related to the levels of the other.

The Degrees of Freedom

The number of degrees-of-freedom for the χ2 test of independence equals the number of rows minus one (r – 1) times the number of columns minus one (c – 1). That is, the number of degrees-of-freedom equals (r – 1)(c – 1).

The Hypothesis Test

Note that the χ2 goodness-of-fit test is a right-tailed test only. If the test statistic is greater in value than the critical value, one would reject the null hypothesis.