Question

Show how ( Chi-Square) can be applied, and how it may be used in any industry of your choosing. Provide a brief description of what the “Chi-Square” distribution is used for, then provide the example, and then tell what that example showed.

(Note: The answer has to be typed, not hand written nor in a picture.) Thank you.

Answer 1

We use Chi-Square when we have two categorical variables and want to determine whether there is a significant association between the two variables.

Step 1: The observed frequencies are calculated for the sample.

Step 2: The expected frequencies are obtained from previous knowledge (or belief) or probability theory. In order to proceed to the next step, it is necessary that each expected frequency is at least 5.

Step 3: A hypothesis test is performed:
(i) The null hypothesis H0: the population frequencies are equal to the expected frequencies.
(ii) The alternative hypothesis, Ha: the null hypothesis is false (what does this imply about the population frequencies?).
(iii) α is the level of significance.
(iv) The degrees of freedom: k − 1.
(v) A test statistic is calculated:

  
(vi) From α and k − 1, a critical value is determined from the chi-square table.
(vii) Reject H0 if X² is larger than the critical value (right-tailed test).

Example:A department store, A, has four competitors: B,C,D, and E. Store A hires a consultant to determine if the percentage of shoppers who prefer each of the five stores is the same. A survey of 1100 randomly selected shoppers is conducted, and the results about which one of the stores shoppers prefer are below. Is there enough evidence using a significance level α = 0.05 to conclude that the proportions are really the same?

Store	A	B	C	D	E
No. of shoppers	262	234	204	190	210

Solution:

(i) The null hypothesis H0:the population frequencies are equal to the expected frequencies (to be calculated below).
(ii) The alternative hypothesis, Ha: the null hypothesis is false.
(iii) α = 0.05.
(iv) The degrees of freedom: k − 1 = 5 − 1 = 4.
(v) The test statistic can be calculated using a table:

Preferences	% of Shoppers	E	O	O-E	(O-E)2
A	20	0.2*1100=220	262	42	1764	8.018
B	20	0.2*1100=220	234	14	196	0.891
C	20	0.2*1100=220	204	-16	256	1.163
D	20	0.2*1100=220	190	-30	900	4.091
E	20	0.2*1100=220	210	-10	100	0.455
---------------	--------------------	------------------	----	---------	--------

(vi) From α = 0.05 and k − 1 = 4, the critical value is 9.488.
(vii) Is there enough evidence to reject H0? Since χ2 ≈ 14.618 > 9.488, there is enough statistical evidence to reject the null hypothesis and to believe that customers do not prefer each of the five stores equally.

The most widely used applications of Chi-square distribution are:

1. The Chi-square Test for Association which is a non-parametric test; therefore, it can be used for nominal data too. It is a test of statistical significance widely used bivariate tabular association analysis. Typically, the hypothesis is whether or not two populations are different in some characteristic or aspect of their behavior based on two random samples. This test procedure is also known as the Pearson Chi-square test.
2. The Chi-square Goodness-of-Fit Test is used to test if an observed distribution conforms to any particular distribution. Calculation of this goodness-of-fit test is by comparison of observed data with data expected based on a particular distribution.

One of the disadvantages of some of the Chi-square tests is that they do not permit the calculation of confidence intervals; therefore, determination of the sample size is not readily available.

Example: Consider the following problem: you sample two scores from a standard normal distribution, square each score, and sum the squares. What is the probability that the sum of these two squares will be six or higher?

Solution:Since two scores are sampled, the answer can be found using the Chi Square distribution with two degrees of freedom. A Chi Square calculator can be used to find that the probability of a Chi Square (with 2 df) being six or higher is 0.050.

The mean of a Chi Square distribution is its degrees of freedom. Chi Square distributions are positively skewed, with the degree of skew decreasing with increasing degrees of freedom. As the degrees of freedom increases, the Chi Square distribution approaches a normal distribution. Figure 1 shows density functions for three Chi Square distributions. Notice how the skew decreases as the degrees of freedom increases.

Figure 1. Chi Square distributions with 2, 4, and 6 degrees of freedom.

The Chi Square distribution is very important because many test statistics are approximately distributed as Chi Square. Two of the more common tests using the Chi Square distribution are tests of deviations of differences between theoretically expected and observed frequencies (one-way tables) and the relationship between categorical variables (contingency tables). Numerous other tests beyond the scope of this work are based on the Chi Square distribution.

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