Testing the Hypothesis of Independence of Random Variables

Two attributes were measured for each of 6800 respondents. Attribute $A$ (eye color) has three levels: brown, gray, green. Attribute $B$ (hair color) has the levels: dark-haired, brown-haired, fair-haired. The data are given in Table 9.12.

Table 9.12

Is there a statistical dependence between the attributes $A$ and $B$ at the significance level 0,05? Estimate Guttman's measures of prediction $\lambda_{A}$ and $\lambda_{B}$.

Athletes whose ranks when lined up by height were $1,2, \ldots , 13$ showed the following long-jump results (in m): 6,94; 7,12; 7,01; 6,98; 7,24; 7,42; 7,13; 6,95; 7,34; 7,82; 7,23; 7,05; 7,15. Is there a dependence between an athlete's height and the length of his jump? Take the significance level to be 0,05.

Prove equalities (9.8).

Prove that the rank correlation coefficient $\widehat{\rho }_{X Y}(n)$ defined by formula (9.12) can be transformed to the form (9.13).

Prove the validity of formula (9.34).

Prove that $\mathbb {E}\left[\hat{\tau }_{X Y}\right] = \tau_{X Y}$.

In a children's clinic, 40 newborns of a certain category were examined. The sample correlation coefficient $\widehat{r}_{X Y}(n)$ between the length ($X$) and the weight $(Y)$ of these newborns was 0,76. At the significance level 0,05, test the hypothesis of independence of the indicators $X$ and $Y$. Using Fisher's $z$-transformation, construct an asymptotic confidence interval for the correlation coefficient $r_{X Y}$ with confidence level 0,95.

There are 12 disks of the same size whose coloring differs in tone — from light blue to dark blue. To assess a fashion designer's ability to distinguish color shades, he is asked to arrange the disks in order of increasing color intensity. The designer established the following order of the disks: $1,4,7,2,3,5$, $8,12,10,6,11,9$. The objective assessment of color intensity, obtained by a colorimetric test: $1,2, \ldots , 12$. Based on these data, characterize the designer's ability to distinguish color shades.

Table 9.13 gives 2663 cases classified by two attributes: $A$ is the presence of a vaccination against cholera, $B$ is the absence of the disease. Test the hypothesis of independence of the attributes $A$ and $B$ at the significance level 0,05.

Table 9.13

A panel of six judges evaluates the performances of five figure skaters who reached the final of a competition. The results are given in Table 9.14.

Table 9.14

Compute Kendall's coefficient of concordance. Can one assume, at the significance level 0,05, that this panel of judges has a common system of preferences?

From the available data on the economic indicators $X_{1}, X_{2}$ and $X_{3}$ for 20 regions of Russia, the sample correlation coefficients $\widehat{r}_{X_{1} X_{2}} = 0,746, \widehat{r}_{X_{1} X_{3}} = 0,507, \widehat{r}_{X_{2} X_{3}} = 0,432$ were computed. The observations are assumed to have a Gaussian distribution. At the significance level 0,05, test the hypothesis of independence of the indicators $X_{1}$ and $X_{3}$ using information only about the coefficient $\widehat{r}_{X_{1} X_{3}}$. Compute the partial correlation coefficient $\widehat{r}_{13 ; 2}$ between $X_{1}$ and $X_{3}$ with the indicator $X_{2}$ held fixed. Test the hypothesis $H_{0}: r_{13 ; 2} = 0$.

The Center for Studies of Civil Society and the Nonprofit Sector of the National Research University Higher School of Economics formed a representative sample of 2000 respondents. The respondents were asked whether they were satisfied with their income. Of the 1097 women who took part in the survey, 18 found it difficult to answer, 839 answered negatively and 240 — positively. Of the 903 men, 13 found it difficult to answer, 652 answered negatively and 238 — positively. Based on these data, can one claim that satisfaction with one's income and gender are dependent? Take the significance level to be 0,05.