Important Questions for IGNOU MAPC MPC006 Exam with Main Points for Answer - Block 4 Unit 4 Chi-Square and Kendall Rank Correlation

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Block 4 Unit 4 Chi-Square and Kendall Rank Correlation

For Numericals, check the questions and examples in your study material.

1) Compare T and r in terms of correlation and state your views?

Kendall's Tau (T) and Pearson's r are both measures of correlation, but they differ in their underlying scales and assumptions.

Pearson's r:
- Measures the linear relationship between two continuous variables.
- Assumes that the data are normally distributed.
- Sensitive to outliers.
Kendall's Tau (T):
- Measures the strength of association between two ordinal variables or ranked data.
- Does not assume a particular distribution of the data (non-parametric).
- Less sensitive to outliers than Pearson's r.

Views:

When data meets the assumptions of Pearson's r, it is generally considered a more powerful measure of correlation. However, when those assumptions are violated (e.g., non-normal data, ordinal data), Kendall's Tau is a more appropriate choice.
While T and r cannot be directly compared numerically due to their different scales, both provide valuable insights into the relationship between variables depending on the nature of the data.

2) Should a chi-Square test be carried out on the following data?

7 1

2 7

Yes, a chi-square test can be conducted on the following data.

This data can be represented as a 2x2 contingency table. The chi-square test assesses the independence of the two categorical variables represented in the table.

3) A (fictitious) Survey shows that. in a sample of 100. 91 people are against the privatisation of health services, whereas 9 support the idea.

a) What test of significance can be performed on this data?

The appropriate test of significance is the chi-square goodness-of-fit test. This test assesses whether the observed proportions (91% against, 9% for) significantly differ from an expected distribution (assuming no preference, a 50%-50% split might be expected).

b) Calculate the chi square value and check it for significance.

Calculating the Chi-Square Value:

Expected Frequencies: Assuming no preference, we expect 50 people to be against and 50 to be for privatisation.

Calculate the Chi-Square Statistic: χ² = Σ [(O - E)² / E] χ² = [(91 - 50)² / 50] + [(9 - 50)² / 50] χ² ≈ 67.24

Degrees of Freedom: df = number of categories - 1 = 2 - 1 = 1

Checking for Significance: With df = 1, the chi-square critical value at α = 0.05 is 3.841. Our calculated χ² (67.24) is much larger, indicating a highly significant difference between the observed and expected frequencies.

c) Could this test be one-tailed?

Since the research question likely implies a direction (are people more likely to be against privatisation?), a one-tailed test could be argued. However, chi-square tests are typically interpreted as two-tailed.

d) If for a large sample, we knew only that 87% of people were against the idea and were for could we carry out the same test to see whether this split is significant.

Large Sample with Percentages: Yes, you can still conduct a chi-square test. Using the given percentages, calculate the expected frequencies based on your sample size and proceed with the same steps as above.

4) What is the difference between chi square goodness of fit test and measure of independence test?

The chi-square test has two main applications:

Goodness-of-fit test: This tests whether observed frequencies in a sample match expected frequencies based on a known distribution or theoretical model.
Test of independence: This tests whether there is an association between two categorical variables, examining if the observed frequencies in a contingency table differ significantly from what would be expected under the assumption of independence.

5) What do you understand by efficiency of T?

The efficiency of a statistical test refers to its ability to correctly detect a true effect (its statistical power) relative to other tests. The efficiency of Kendall's Tau (T) is generally lower than Pearson's r when data meet the assumptions of r. However, when dealing with non-normal or ordinal data, T can be more efficient as Pearson's r may not accurately represent the relationship. In such situations, T provides a more robust measure of association.

6) What questions does correlation coefficient answers?

A correlation coefficient answers these questions:

Does a relationship exist between two variables?
If so, is the relationship positive or negative?
How strong is the relationship?

7) Name any two methods for calculating correlation?

Two methods for calculating correlation are:

Pearson product-moment correlation coefficient (r) - used for continuous variables.
Spearman's rank-order correlation coefficient (rs) - used for ordinal data or when the assumptions of Pearson's r are not met.

8) What are the assumptions of chi-square goodness-of-fit test?

The assumptions of the chi-square goodness-of-fit test are:

The data are categorical and nominal, representing frequencies in mutually exclusive categories.
The data consist of a random sample of independent observations.
The expected frequency of each cell is 5 or greater.

9) Chi square performs two major functions, what are these?

The two major functions of the chi-square test are:

Goodness of fit: Assessing whether observed frequencies match expected frequencies based on a theoretical model or prior knowledge.
Test of independence: Determining whether there is an association between two categorical variables.

Important Points

Scatter plots may reveal a positive correlation (high values of X associated with high values of Y).
Scatter plots may reveal a negative correlation (high values of X associated with low values of Y).
Scatter plots may reveal no correlation (values of X are not at all predictive of values of Y).
Correlation coefficients range from -1.00 to +1.00.
A correlation coefficient of +1.00 tells you that there is a perfect positive relationship between the two variables.
The closer a correlation coefficient is to 0.00, the weaker the relationship.
A correlation coefficient is a single summary number that gives you a good idea about how closely one variable is related to another variable.
The expected frequency of a cell in the chi-square test can be determined using probability theory or pre-existing empirical information.
If several expected frequencies fall below 5, the possibility of a type I error increases.
The chi-square (χ²) test assesses the alignment between two sets of frequency measures.
The assumption of "a random sample of n independent observations" is a cardinal assumption of the chi-square test.

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