Math Problems Vault

Problem 7.1

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.

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Problem 7.2

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.

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Problem 7.3

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.

Ramsay's method	111	107	100	99	102	106	109	108	104	99
J.--P. method	107	108	106	98	105	103	110	105	104	100
Ramsay's method	101	96	97	102	107	113	116	113	110	98
J.--P. method	96	108	103	104	114	114	113	108	106	99

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

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Problem 7.4

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.

Central
Federal District	Pension
amount	Siberian
Federal District	Pension
amount
Belgorod Oblast	4297,3	Republic of Buryatia	4281,2
Bryansk Oblast	4244,3	Tyva Republic	4426,5
Ivanovo Oblast	4386,3	Kemerovo Oblast	4571,7
Kursk Oblast	4085,4	Krasnoyarsk Krai	4896,0
Smolensk Oblast	4321,5	Omsk Oblast	4339,5
Ryazan Oblast	4290,5	Altai Republic	4278,0
Moscow Oblast	4771,7
City of Moscow	4809,9

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.

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Problem 7.5

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.

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Problem 7.6

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.

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Problem 7.7

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.

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Problem 7.8

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

U_{m, n} = \sum _{i = 1}^{m} \sum _{j = 1}^{n} \varphi \left(X_{i}, Y_{j}\right), \text{ where } \varphi (z, v) = \begin{cases} 1, & \text{ if } z < v, \\ 0, & \text{ if } z \geqslant v .\end{cases}

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

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Problem 7.9

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.

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Problem 7.10

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.

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