Chi Square and ANOVA Questions and Answers

Q: Which of the following is a Non-parametric Test ?

The non-parametric test is the chi-square test.A non-parametric test makes no assumption about the shape of the population distribution and can be used with nominal or ordinal data. The chi-square test compares observed with expected frequencies in categories, so it works with counts and needs no assumption of normality.The t-test, F-test and Z-test are parametric tests: they assume normally distributed data and test hypotheses about means or variances.

Q: In a multiple regression analysis, an examination of variances revealed that explained sum of squares per degree of freedom and residual sum of squares per degree of freedom were 250 and 100, respectively. What is the F-ratio ?

The F-ratio is 2.5.The F-ratio in regression is the explained mean square divided by the residual mean square, where each mean square is the sum of squares per degree of freedom.\[ F = \frac{250}{100} = 2.5 \]The value 0.4 is the ratio turned upside down, and 6.25 and 5.25 do not follow from the two numbers given.The numerator, 250, is the variance explained by the regression per degree of freedom, and the denominator, 100, is the unexplained variance per degree of freedom; an F-ratio above 1 shows that the model explains more than chance would.

Which of the following is a Non-parametric Test ?

A.T-test
B.Chi-square test
C.F-test
D.Z-test

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

Answer: B

The non-parametric test is the chi-square test.

A non-parametric test makes no assumption about the shape of the population distribution and can be used with nominal or ordinal data. The chi-square test compares observed with expected frequencies in categories, so it works with counts and needs no assumption of normality.

The t-test, F-test and Z-test are parametric tests: they assume normally distributed data and test hypotheses about means or variances.

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In a multiple regression model, the explained variance per d.f. is 50 and unexplained variance per d.f. is 10. The F-ratio is

A.2.5
B.5
C.25
D.0.2

Exam Year(s)

UGC NET - 2012,

Answer: B

The F-ratio is 5, option B. In a regression analysis the F-ratio is the explained variance per degree of freedom divided by the unexplained variance per degree of freedom, that is, the mean square of the regression over the mean square of the residual.

\[ F = \frac{50}{10} = 5 \]

Option A, 2.5, has no basis in the figures. Option C, 25, is 5 squared and is a common slip. Option D, 0.2, is the inverse ratio, 10 divided by 50, which is obtained when the two variances are put the wrong way round. The larger mean square goes on top.

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For 5 degrees of freedom, the variance of \(\chi^{2}\) distribution is

A.10
B.5
C.16
D.4

Exam Year(s)

UGC NET - 2012,

Answer: A

A chi-square distribution with k degrees of freedom has mean k and variance 2k. For 5 degrees of freedom the variance is therefore \(2 \times 5 = 10\). The value 5 is the mean, not the variance, and 16 and 4 do not follow from either formula. The answer is 10.

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Assertion (A) : χ2 distribution is a non-parametric distribution.

Reason (R) : χ2 is a sample statistic having no corresponding population parameter.
Codes :

A.Both (A) and (R) are correct and (R) is the correct explanation of (A).
B.Both (A) and (R) are correct, but (R) is not the correct explanation of (A).
C.(A) is true, but (R) is false.
D.Both (A) and (R) are false.

Exam Year(s)

UGC NET - 2013,

Answer: A

Following the official key, both statements are correct and the reason explains the assertion. The chi-square statistic is used in non-parametric tests, which make no assumptions about population parameters, and it is a sample statistic with no corresponding population parameter.

So the reason gives why the distribution is treated as non-parametric. The options that call either statement false disagree with the key.

Testing each of the remaining options against the same fact shows that every one of them fails on the point the question is asking about, which confirms the keyed answer.

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In a multiple regression analysis, an examination of variances revealed that explained sum of squares per degree of freedom and residual sum of squares per degree of freedom were 250 and 100, respectively. What is the F-ratio ?

A.6.25
B.5.25
C.0.4
D.2.5

Exam Year(s)

UGC NET - 2013,

Answer: D

The F-ratio is 2.5.

The F-ratio in regression is the explained mean square divided by the residual mean square, where each mean square is the sum of squares per degree of freedom.

\[ F = \frac{250}{100} = 2.5 \]

The value 0.4 is the ratio turned upside down, and 6.25 and 5.25 do not follow from the two numbers given.

The numerator, 250, is the variance explained by the regression per degree of freedom, and the denominator, 100, is the unexplained variance per degree of freedom; an F-ratio above 1 shows that the model explains more than chance would.

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A \(\Psi^{2}\) distribution with 10 degrees of freedom has variance

A.10
B.20
C.5
D.40

Exam Year(s)

UGC NET - 2014,

Answer: B

The variance of a chi square distribution is twice its degrees of freedom, so with 10 degrees of freedom the variance is 20. Its mean equals the degrees of freedom, 10.

10 is the mean, not the variance, 5 is half of the mean, and 40 is four times the degrees of freedom, so they are wrong.

A chi square variable is the sum of squares of independent standard normal variables, so its mean equals the number of terms and its variance is twice that number.

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