Question

Problem Set 4.8: Critical Values

Criterion: Explain changes in critical value based on calculations.

Instructions: Read the following and answer the questions.

The chi-square table. The degrees of freedom for a given test are listed in the column to the far left; the level of significance is listed in the top row to the right. These are the only two values you need to find the critical values for a chi-square test.

Work through the following exercise and write down what you see in the chi-square table. This will help familiarize you with the table.

Increasing k and a in the chi-square table:

1. Record the critical values for a chi-square test, given the following values for k at each level of significance:

	.10	.05	.01
k = 10	___	___	___
k = 16	___	___	___
k = 22	___	___	___
k = 30	___	___	___

Note: Because there is only one k given, assume this is a goodness-of-fit test and compute the degrees of freedom as (k − 1).

2. As the level of significance increases (from .01 to .10), does the critical value increase or decrease? Explain. ___________________________________
3. As k increases (from 10 to 30), does the critical value increase or decrease? Explain your answer as it relates to the test statistic. ___________________________________

Answer 1

0.10 0.05 0.01

k=10 14.684 16.919 21.666

k=16 22.307 24.996 30.578

k=22 29.615 32.671 38.932

k=30 39.087 42.557 49.558

all the value are on the basis of k-1 degree of freedom

as the significance level increases from 0.01 to 0.10 the critical value decreases as the level of significance is nothing but the area under the curve (type 1 error ) as the level of significance increases we move to left from the tail of the curve as the area under the curve is greater , you can understand it more clearly by this image

now from the visual above we can see that critical value of 2.398 will enclose more area under the curve to the right as compared to that of 6.251

critical values increases as the degrees of freedom increases for particular level of significance

because of the shift in the tale , now what i really mean by that is , we know chi sqaure distribution is sum of squared standard normal variates and degree of freedom constitutes from no. of variable , as the number of number of variables increases the degree of freedom increases and it is less likely for us to get 0 and adding something positive will only contribute to the probability of getting larger value and hence the shift in the graph below

beacuse of this shift in the right tail the critical values increases with the degree of freedom