30 Questions
Which test statistic is used to test the hypothesis for the Spearman rank correlation?
t-distribution
For a two-tailed test with a 5 percent significance level and 33 degrees of freedom, what are the critical values for the t-test?
±2.0345
What is the null hypothesis in the test of the Spearman rank correlation?
H0: rS = 0
What is the formula for the test statistic in the chi-square test of independence?
χ 2 = ∑i=1 m (Oij − Eij) _ 2 Eij
How many degrees of freedom does the chi-square test of independence have in a contingency table with r rows and c columns?
(r - 1)(c - 1)
What is the expected number of observations in each cell of a contingency table assuming independence?
Eij = (Total row i)× (Total column j) / Overall total
What is the rejection region for the test of independence using a contingency table?
Right side
How is the chi-square statistic calculated in the chi-square test of independence?
By summing the values of (Oij − Eij) _ 2 Eij for each of the m cells
What is the number of cells in a contingency table with r rows and c columns?
r × c
What is the name of the graphic that reflects the comparison between the observed and expected frequencies in a contingency table?
Mosaic
Which of the following best describes the interpretation of the standardized residual in the ETF example?
The standardized residual indicates the difference between the observed and expected values, scaled by the expected values.
What does the width of the rectangles in Exhibit 10 represent?
The proportion of ETFs that are small, medium, and large.
What does the height of the rectangles in Exhibit 10 represent?
The proportion of ETFs that are value, growth, and blend.
What does the darker shading in Exhibit 10 indicate?
The number of observations that are more than expected under the null hypothesis of independence.
What does the lighter shading in Exhibit 10 indicate?
The number of observations that are less than expected under the null hypothesis of independence.
What is the formula for calculating the standardized residual?
Standardized residual = Oij − Eij √ Eij
What does a positive standardized residual indicate in the ETF example?
There are more observations than expected.
What does a negative standardized residual indicate in the ETF example?
There are fewer observations than expected.
What is the interpretation of the standardized residual in the ETF example?
There are more medium-size growth ETFs and fewer large-size growth ETFs than expected.
What is another name for the standardized residual in the ETF example?
Pearson residual
Which of the following statements is true about the Spearman rank correlation coefficient, rS?
It is calculated on the ranks of the two variables within their respective samples.
What is the formula for calculating the Spearman rank correlation coefficient, rs?
$rs = 1 - \frac{6\sum_{i=1}^n d_i^2}{n(n^2 - 1)}$
What are the hypotheses for testing Spearman rank correlations?
H0: rS = 0 and Ha: rS ≠ 0
What is the first step in calculating the Spearman rank correlation coefficient?
Rank the observations on X from largest to smallest.
What is the second step in calculating the Spearman rank correlation coefficient?
Calculate the difference, di, between the ranks for each pair of observations on X and Y.
What is the third step in calculating the Spearman rank correlation coefficient?
Calculate the squared difference, di^2, between the ranks for each pair of observations on X and Y.
When would the use of the Spearman rank correlation coefficient be appropriate?
When the population under consideration departs from normality.
What does the Spearman rank correlation coefficient measure?
The strength and direction of the monotonic relationship between two variables.
What is the sample size, n, in the formula for the Spearman rank correlation coefficient?
The number of pairs of observations on X and Y.
What does it mean if the Spearman rank correlation coefficient, rs, is equal to 1?
There is a perfect monotonic relationship between the two variables.
Study Notes
Hypothesis Testing for Spearman Rank Correlation
- The test statistic used to test the hypothesis for the Spearman rank correlation is not specified.
- The null hypothesis in the test of the Spearman rank correlation is that there is no correlation between the two variables.
t-test
- For a two-tailed test with a 5% significance level and 33 degrees of freedom, the critical values for the t-test are not specified.
Chi-Square Test of Independence
- The formula for the test statistic in the chi-square test of independence is not specified.
- The chi-square test of independence has (r-1)(c-1) degrees of freedom in a contingency table with r rows and c columns.
- The expected number of observations in each cell of a contingency table assuming independence is calculated by multiplying the row total and column total, then dividing by the grand total.
- The rejection region for the test of independence using a contingency table is not specified.
- The chi-square statistic is calculated by finding the difference between the observed and expected frequencies in each cell, squaring the difference, dividing by the expected frequency, and summing the results.
- The number of cells in a contingency table with r rows and c columns is r*c.
Contingency Tables
- The graphic that reflects the comparison between the observed and expected frequencies in a contingency table is a mosaic plot.
- The width of the rectangles in Exhibit 10 represents the expected frequency.
- The height of the rectangles in Exhibit 10 represents the observed frequency.
- The darker shading in Exhibit 10 indicates the observed frequency is greater than the expected frequency.
- The lighter shading in Exhibit 10 indicates the observed frequency is less than the expected frequency.
- The formula for calculating the standardized residual is (observed frequency - expected frequency) / sqrt(expected frequency).
Standardized Residual
- A positive standardized residual indicates that the observed frequency is greater than the expected frequency.
- A negative standardized residual indicates that the observed frequency is less than the expected frequency.
- The interpretation of the standardized residual in the ETF example is that it measures the difference between the observed and expected frequencies in a contingency table.
- Another name for the standardized residual in the ETF example is the Pearson residual.
Spearman Rank Correlation Coefficient
- The Spearman rank correlation coefficient, rS, measures the strength and direction of the correlation between two ranked variables.
- The formula for calculating the Spearman rank correlation coefficient, rs, is rs = 1 - (6 * Σd^2) / (n * (n^2 - 1)), where d is the difference between the ranks and n is the sample size.
- The hypotheses for testing Spearman rank correlations are H0: rs = 0 and H1: rs ≠ 0.
- The first step in calculating the Spearman rank correlation coefficient is to rank the data.
- The second step in calculating the Spearman rank correlation coefficient is to calculate the differences between the ranks.
- The third step in calculating the Spearman rank correlation coefficient is to calculate the sum of the squared differences.
- The Spearman rank correlation coefficient would be appropriate to use when the data is ordinal or ranked.
- The sample size, n, in the formula for the Spearman rank correlation coefficient is the number of observations.
- If the Spearman rank correlation coefficient, rs, is equal to 1, it means that there is a perfect positive correlation between the two variables.
Test your knowledge on the Spearman Rank Correlation Coefficient, a non-parametric test of correlation used when the population departs from normality. Discover how it differs from the Pearson correlation coefficient.
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