Use and interpretation of Statistical Techniques MCQ Quiz in मल्याळम - Objective Question with Answer for Use and interpretation of Statistical Techniques - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

we generally make use of parametric and non-parametric tests for making inferences about various population values (parameters). Many methods and techniques are used in statistics. These have been grouped under parametric and non-parametric statistics.

Variable:

• A variable, as the name implies, is something that varies. This is the simplest way of defining a variable.
• A variable is “a thing that is changeable” or “a quantity that may have a number of different values.”
• True, a variable is something that has at least two values: however, it is also important that the values of the variable be observable. Thus, if what is being studied is a variable, it has more than one value and each value can be observed. For example, the outcome of throwing dice is a variable. That variable has six possible values (each side of the dice having one to six dots on it), each of which can be observed.
• The rel ationship between variables are measured by correlation of coefficient.

Correlation coefficient 

• It is a numerical term used to show the strength of the relationships between two variables. It is ranges from -1 to 1
• Types of correlational coefficients:
  • Positive Correlation – when the value of one variable increases with respect to another.
  • Negative Correlation – when the value of one variable decreases with respect to another.
  • No Correlation – when there is no relation between the two variables.

Effect size is a statistical concept that measures the strength of the relationship between two variables on a numerical scale.

Effect sizes are often measured in terms of the proportion of variance (R2)explained by a variable.

Hence, we can the strength of the relationship between X and Y is sometimes expressed by proportion of variance which is obtained by squaring the correlation coefficient and multiplying by 100.

It is given by formula, (R 2) = (r 2) ×  100

Example: a correlation of 0.6 means (0.6²) × 100 = 36% of the variance in Y is "explained" or predicted by the X variable.

Important Points

Given: Correlation coefficient (r) = - 0.48

To find: proportion of variance (R2)

Formula: proportion of variance (R2) = (r 2) ×  100

Calculation:

proportion of variance (R2) = (r 2) × 100

R2 =  (-0.48 2) ×  100 = 0.2304 ×  100

R2 = 23.04% 

R2 = 23.04%

R2 = 23.04/100

R2 = 0.23

The following diagram explains the T-score corresponding to a Z-score.

T = 10 × Z + 50

T = 10 × 2 + 50

T = 20 + 50

T = 70

The t-test

• A t-test is a type of statistical test that is used to compare the means of two groups. 
• It is used for testing the significance of the difference between the means of two small samples. 
• There are two types of statistical inference: parametric and nonparametric methods.
• Parametric methods refer to a statistical technique in which one defines the probability distribution of probability variables and makes inferences about the parameters of the distribution. In cases in which the probability distribution cannot be defined, nonparametric methods are employed.
• T-tests are a type of parametric method; they can be used when the samples satisfy the conditions of normality, equal variance, and independence.
• It is a two-tailed test.
• If the absolute value of the t-value is greater than the critical value, you reject the null hypothesis.
• If the absolute value of the t-value is less than the critical value, you accept the null hypothesis.
• A critical value is a line on a graph that splits the graph into sections. 

​Critical values in the given question with the t value = 3 and N = 300:

We will check for the degree of freedom (df) 0.95, and the significance level (\(\alpha\)) is 0.05.

Critical values for two-tailed t-test: ±cdf_t,d-1 (1-α/2)

Critical region: (-∞, -1.985] ∪ [1.985, ∞)
Decision: The t-score belongs to the critical region, so we reject H₀ and accept H₁.

In conducting One-way Analysis of Variance (ANOVA), the appropriate test statistic to be used is option 4) F.

Important Points

F-test:

• The F-test is used in ANOVA to compare the variance between multiple groups to the variance within those groups.
• It determines whether the means of different groups are significantly different from each other.
• ANOVA assesses whether there are statistically significant differences among the means of three or more independent (unrelated) groups.

The other options are used for different types of statistical tests:

 Z-test:

• The Z-test is used for comparing sample means to a known population mean when the population standard deviation is known.

 t-test:

• The t-test is used for comparing means of two groups to determine if they are significantly different.
• It's used when the sample size is small and the population standard deviation is unknown.

 χ2-test:

• The χ2-test (chi-squared test) is used for categorical data analysis, such as comparing observed and expected frequencies in contingency tables.

So, the correct answer is option 4) F for One-way ANOVA

The correct answer is A- II, B- IV, C- I, D- III.

Key Points A. Kruskal-Wallis test - II. Non-parametric test to compare means of more than two population groups: The Kruskal-Wallis test is a non-parametric statistical test used to compare the means of more than two independent population groups. It is applicable when the assumptions of normality and equal variances are not met. The test determines whether there are any significant differences among the groups based on the ranks of the observations.

B. Z-test - IV. Testing the difference between means of two sample groups: The Z-test is a statistical test used to determine if there is a significant difference between the means of two sample groups. It is a parametric test that assumes the data follows a normal distribution and requires information about the population standard deviation or a large sample size.

C. ANOVA test - I. Parametric test to compare means of more than two population groups: The ANOVA (Analysis of Variance) test is a parametric statistical test used to compare the means of more than two population groups. It assesses whether there are any statistically significant differences between the group means by analyzing the variances within and between the groups.

D. Chi-square test - III. Non-parametric test to test the goodness of fit: The Chi-square test is a statistical test used to determine if there is a significant difference between the observed frequencies and the expected frequencies in a categorical data set. It is a non-parametric test that is commonly used to test the goodness of fit, such as assessing whether observed data fits a specific distribution or expected proportions.

Therefore, the correct matching is:

A. Kruskal-Wallis test - II. Non-parametric test to compare means of more than two population groups.
B. Z-test - IV. Testing the difference between means of two sample groups.
C. ANOVA test - I. Parametric test to compare means of more than two population groups.
D. Chi-square test - III. Non-parametric test to test the goodness of fit.

The correct answer is B and D only.

Key Points 

B. 't'-test: The 't'-test is used to compare the means of two groups and determine if they are significantly different from each other. It is commonly employed when the sample sizes are relatively small and assumptions of normality and equal variances are met.

D. Mann-Whitney 'U' test: The Mann-Whitney 'U' test, also known as the Wilcoxon rank-sum test, is a non-parametric test used to compare the medians of two independent groups. It can also be used as a test for comparing the means of two groups when the assumptions of normality and equal variances are not met.

Additional Information A. F-test
The F-test is utilized for comparing the variances of two or more groups. It is often employed as part of analysis of variance (ANOVA) to determine if there are significant differences between the means of multiple groups.

C. Chi-square (χ2) test
The Chi-square test is a non-parametric test used to assess the association between categorical variables. It is not suitable for comparing means directly.

E. Kolmogorov-Smirnov test
The Kolmogorov-Smirnov test is a non-parametric test used to compare the distribution of a sample to a specified distribution or to compare the distributions of two samples. It does not focus on means specifically.

Key Points

• ANCOVA does not transform a quasi-experiment into a true experiment. A quasi-experiment is a study in which the researcher does not have full control over the independent variable. For example, a researcher might be interested in the effect of a new teaching method on student achievement. However, the researcher cannot randomly assign students to different teaching methods. Instead, the researcher might have to use existing classes or schools as the different treatment groups. This means that there may be other factors, such as the quality of the teachers or the students' prior knowledge, that could also affect the results.
• ANCOVA also does not have two dependent variables. ANCOVA is a statistical method that is used to compare the means of two or more groups on a single dependent variable, while controlling for the effects of one or more covariates.
• ANCOVA does not control variance at the time of structuring research design. ANCOVA is a statistical method that is used to control for the effects of covariates at the analysis stage.
• A and D are true for ANCOVA. ANCOVA uses partial correlation principles to control for the effects of covariates. ANCOVA also controls variance at the analysis stage.

Therefore the correct answer is B, C, and E.

The correct answer is F-ratio belongs to [-∞, ∞].

 Key Points

• The F-ratio in ANOVA (analysis of variance) is a statistical test used to determine if there are significant differences between the means of two or more groups.
• The F-ratio is calculated as the ratio of two mean square values, which are estimated variances based on the sample data.
• The F-ratio follows the F-distribution, which is a continuous probability distribution that is always positive and has its support on the interval (0, ∞).

 Additional Information

• ANOVA is a powerful tool for comparing means and testing hypotheses about population means.
• The null hypothesis (H0) in ANOVA is usually that all population means are equal, which is symbolically expressed as H0: μ1 = μ2 = ... = μn.
• If the F-ratio calculated from the sample data is large enough, we can reject the null hypothesis and conclude that there are significant differences between some of the population means.
• The F-ratio is commonly used in a variety of fields, including psychology, economics, biology, and engineering, to compare group means and make inferences about population parameters.

The correct answer is -0.80

​ Key Points

• Pearson's correlation coefficient measures the linear relationship between two variables.
• The value of the Pearson's correlation coefficient ranges from -1 to 1
  • ​-1 indicates a perfect negative correlation
  • 0 indicates no correlation
  • 1 indicates a perfect positive correlation.
• To calculate Pearson's correlation coefficient, we first standardize the observations by subtracting the mean and dividing by the standard deviation, then we calculate the product of the standardized variables.
• Pearson's correlation coefficient only measures linear relationships, and cannot detect non-linear relationships or complex relationships between variables.
• Other measures of association, such as Spearman's rank correlation or Kendall's tau, can be used to capture non-linear relationships.
• When interpreting Pearson's correlation coefficient, it is important to consider the strength and direction of the relationship, as well as the sample size and the presence of outliers.

Additional Information 

• Pearson's correlation coefficient measures the linear relationship between two variables. It can range from -1 to 1, where -1 indicates a perfect negative correlation, 0 indicates no correlation, and 1 indicates a perfect positive correlation.
• Let's consider the original values of X and Y:
  • ​\( X = [1, 2, 3, 4], Y = [3, 4, 2, 1]\)
• After halving X and doubling Y:
  • ​ \(X = [0.5, 1, 1.5, 2], Y = [6, 8, 4, 2]\)
• To calculate Pearson's correlation coefficient, we first find the mean of X and Y, then standardize the observations by subtracting the mean and dividing by the standard deviation. Finally, we calculate the product of the standardized X and Y.
• Using the formula for Pearson's correlation coefficient
  • :\( r = sum((X - mean(X)) * (Y - mean(Y))) / (n-1) * (std(X) * std(Y))\)
• We can find that the new Pearson's correlation coefficient equals to \(-0.80.\)
• So, the Pearson's correlation coefficient between the halved X and doubled Y observations is \(-0.80\)