Testing of Statistical Hypotheses

According to surveys of 29 families conducted in 1968 in the south-west region of England, the sample mean of the weekly rent for a furnished flat was $2,5 £$, and the sample variance was $0,67 £^{2}$. In Wales, the sample mean rent of 16 families was $2,06 £$, and the sample variance was $0,42 £^{2}$. Test whether the difference in rent between these regions is statistically significant, assuming that the samples are generated by Gaussian random variables. Take the significance level equal to 0,05.

The level of histamine in the sputum of seven allergy-prone smokers was (in micrograms): 102,4; 100,0; 67,5; 65,9; 64,7; 39,6; 31,2, and that of ten smokers not prone to allergy: 48,1; 45,5; 41,7; 35,4; 29,1; 18,9; 58,3; 66,8; 71,3; 94,3. Is the assumption correct that the histamine level in smokers susceptible to allergy is higher than in non-allergic ones? Take the significance level equal to 0,05.

To determine the serum iron content, Ramsay used a direct determination method. This method is very laborious, the reactions proceed slowly, and turbidity of the mixture cannot be ruled out. To overcome these difficulties, Jung and Parekh proposed an improved method based on a recently synthesized additive. The procedure for performing the analysis by the new method is clearly better than that of Ramsay's method. However, a suspicion arose that the new method has lower precision than Ramsay's procedure. To compare the precision of the new and the established methods, 20 pairs of analyses were performed, each by both methods. Hyland sera containing 105 micrograms of serum iron per 100 milliliters served as standards. The data are presented in the table.

Can it be considered, at significance level 0,05 , that the new method has lower precision than Ramsay's method?

The following are data of the Federal State Statistics Service on the average size of awarded pensions (rubles) by region of the Central and Siberian Federal Districts in 2008.

Test the hypothesis that the mean values of the awarded pensions in the Central and Siberian Federal Districts are equal, assuming that the presented observations are generated by Gaussian random variables. Take the significance level to be 0,05.

Using formula (7.14) for the ARE $e(W, T)$ of the Wilcoxon test relative to Student's test, compute the values of $e(W, T)$ given in Table 7.4.

A TV repair shop has data on the failure-free (until the first repair) operating time (in weeks) of 9 TV sets of brand A and 8 TV sets of brand B. For the brand A TV sets this time was: $5 ; 12 ; 26 ; 50 ; 57 ; 110 ; 200 ; 230 ; 270$ weeks, and for the brand B TV sets: $1 ; 25 ; 31 ; 42 ; 54 ; 70 ; 250 ; 260$ weeks. Can the mean failure-free operating times of brand A and brand B TV sets be considered equal? Take the significance level equal to 0,05.

The performance of a bank branch is characterized by a certain indicator $X$. For a check, a group of 10 bank branches of the same type was selected at random. The indicator $X$ for these branches was: 258, 588, 477, 577, 619, 614, 641, 543, 517, 593. After the economic crisis, the indicator $X$ for 9 randomly selected branches was: 537, 398, 256, 440, 376, 524, 527, 589, 479. Based on these data, can it be considered that the economic crisis led to a decrease in the indicator $X$ ? Take the significance level equal to 0,05.

To test the hypothesis $H_{0}$ of the form (7.3) - (7.4) against alternatives of the form $H_{1}: \theta < 0, H_{2}: \theta > 0, H_{3}: \theta \neq 0$, Mann and Whitney proposed a test based on the statistic

Show that, in the absence of ties, the Mann-Whitney statistic $U_{m, n}$ and the Wilcoxon statistic $W_{m, n}$ are related by $W_{m, n} = U_{m, n}+\frac{n(n+1)}{2}$.

Two factories manufacture electric lamps of the same type. From the output of factory No. 1, 10 lamps were selected at random, and from the output of factory No. 2, 12 lamps. Tests of the burning duration (in hours) of these lamps gave the following results. For the lamps of factory No. 1: 1243, 1238, 1253, 1243, 1254, 1260, 1251, 1246, 1255, 1237. For the lamps of factory No. 2: 1244, 1255, 1258, 1266, 1249, 1257, 1260, 1247, 1256, 1271, 1252, 1259. Test the hypothesis that the mean burning durations of the electric lamps of factory No. 1 and factory No. 2 are equal. Take the significance level equal to 0,05.

Test the hypothesis of homogeneity of the two samples from Problem 9 using the Kolmogorov-Smirnov test. Take the significance level equal to 0,05.

A study was conducted to assess the role of pure motivation (knowledge of the goal of the work) in performing boring monotonous operations when turning a metal workpiece. Eighteen workers of equal qualification were randomly divided into three groups. Group $A$ had no information about the required labor productivity, group $B$ received only general information, and group $C$ had exact information about the task. Table 8.7 gives the number of processed workpieces. Can it be considered, at significance level 0,05, that labor productivity increases with awareness?

Table 8.7

Prove that the statistic (8.15) $\hat{\gamma }$ is an unbiased estimator of the parameter $\gamma$ from Definition 8.1.

Find the distribution of the statistic (8.16) and prove that the statistic (8.16) is a pivotal statistic for the parameter contrast $\gamma$ from Definition 8.1.

The following are data on the subsistence minimum for the working-age population (rubles per capita per month) established in the constituent entities of the Russian Federation for the 4th quarter of 2008 (Table 8.8).

Table 8.8

Can it be considered that the subsistence minimum is, on average, the same in all the presented federal districts? Choose the significance level equal to 0,05.

Let the subsistence-minimum values for the working-age population $X_{i j}$, where $j = 1,2,3$ is the number of the federal district and $i = 1, \ldots , n_{j}$ is the number of the region in the corresponding federal district, given in Problem 8.3 , be described by model (8.12), in which the $\varepsilon_{i j}$ have the distribution $\mathcal{N}\left(0 ; \sigma^{2}\right)$. Construct point estimates and confidence intervals with confidence level 0,95 for the contrasts $\gamma_{1} = \theta_{1}-\theta_{2}, \gamma_{2} = \theta_{1}-\theta_{3}$ and $\gamma_{3} = \theta_{2}-\theta_{3}$ in model (8.12).

Three groups of randomly selected people were taught speed-reading skills by three different methods. At the end of the training, a test was held at which the reading speed was assessed. The trainees showed the following results (pages per ten minutes). First group: $20,23,24,24,25,26$, $28,30,31,32$. Second group: $38,42,42,44,47,48,49,50,51,52$. Third group: $29,32,33,35,35,37,38,39,40,42$. Can it be considered, at significance level 0,05 , that the proposed teaching methods differ in effectiveness?

The time (in sec.) of a chemical reaction for different catalyst contents was distributed as follows (Table 8.9)

Table 8.9

Can one assume, at the significance level 0,01, that increasing the catalyst content decreases the chemical reaction time on average?

Prove the equivalence of formulas (8.3) and (8.4).

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.