Question

List the characteristics of the chi-square distribution, and provide an example. What are the limitations of the chi-square? What is the difference between the goodness-of-fit test for equal expected frequencies and unequal expected frequencies? Provide an example of each. Kindly no pictures and no handwritten materials. Thank you.

Answer 1

1)Tests for Different Purposes. Chi square test for testing goodness of fit is used to decide whether there is any difference between the observed (experimental) value and the expected (theoretical) value. For example given a sample, we may like to test if it has been drawn from a normal population.

2)There is a family of Chi-Square distributions. There is aChi-Square distribution for 1 degree of freedom, another for 2 degrees of freedom, another for 3 degrees of freedom, and so on.

3)The shape of the Chi-Square distribution does not depend on the size of the sample. It does depend upon the number of categories.

4)The Chi-Square distribution is positively skewed. However, as the number of degrees of freedom increases, the distribution begins to approximate the normal distribution.

Limitations:-

First, chi-square is highly sensitive to sample size. As sample size increases, absolute differences become a smaller and smaller proportion of the expected value. What this means is that a reasonably strong association may not come up as significant if the sample size is small, and conversely, in large samples, we may find statistical significance when the findings are small and uninteresting., i.e., the findings are not substantively significant, although they are statistically significant.

Chi-square is also sensitive to small frequencies in the cells of tables. Generally when the expected frequency in a cell of a table is less than 5, chi-square can lead to erroneous conclusions. The rule of thumb here is that if either (i) an expected value in a cell is less than 5 or (ii) more than 20% of the expected values in cells are less than 5, then chi-square should not and usually is not computed.

Equal expected frequency:-

Chi-Square goodness of fit test is a non-parametric test that is used to find out how the observed value of a given phenomena is significantly different from the expected value. In Chi-Square goodness of fit test, the term goodness of fit is used to compare the observed sample distribution with the expected probability distribution. Chi-Square goodness of fit test determines how well theoretical distribution (such as normal, binomial, or Poisson) fits the empirical distribution. In Chi-Square goodness of fit test, sample data is divided into intervals. Then the numbers of points that fall into the interval are compared, with the expected numbers of points in each interval.

The chi-square goodness-of-fit test is a single-sample nonparametric test, also referred to as the one-sample goodness-of-fit test or Pearson's chi-square goodness-of-fit test. It is used to determine whether the distribution of cases (e.g., participants) in a single categorical variable (e.g., "gender", consisting of two groups: "males" and "females") follows a known or hypothesised distribution (e.g., a distribution that is "known", such as the proportion of males and females in a country; or a distribution that is "hypothesised", such as the proportion of males versus females that we anticipate voting for a particular political party in the next elections). The proportion of cases expected in each group of the categorical variable can be equal or unequal (e.g., we may anticipate an "equal" proportion of males and females voting for the Republican Party, or an "unequal" proportion, with 70% of those voting for the Republican Party being male and only 30% female).

Unequal expected frequency:-

Suppose that it was claimed that the car colors in your area were present in the following proportions: 40% silver, 25% red, 15% blue, 10% green and other colors 10%. If you decided to test this claim and went out and took a random sample of 100 cars you might end up with the following resuls: 35 silver, 22 red, 21 blue, 6 green and 16 other colors.