Variant 9

There are two independent samples with the volumes . They have been extracted from the normal universal sets X,Y. The corrected sampling variances have been found: . The confidence level α=0.01. Verify the following null hypothesis . The competing hypothesis is .

We select a sample of volume n=21 from a normal universal set. The corrected sampling variance =16.2. We need to verify the null hypothesis . The confidence level is 0.01. The competing hypothesis is: .

The average weight of items made on the 1^st lathe equals x=130g. (sample volume n=30).

The average weight of items made on the 2^nd lathe equals y=125g. (sample volume n=40)..Universal variances are known: D(X)=60 g², D(Y)=80 g². We need to verify the following null hypothesis (confidence level is 0.05): E(X)=E(Y). The competing hypothesis is . The random variables X,Y are normally distributed, and the samples are independent.

We have two production lots that were made on two similar lathes. We select samples and obtained the following results:

Item size, 1st lathe	Xi	3,4	3,5	3,7	3,9
Frequency	Ni	2	3	4	1
Item size, 2nd lathe	Yi	3,2	3,4	3,6
Frequency	Mi	2	2	8

The confidence level is α=0.02. We need to verify the null hypothesis when the competing hypothesis is . We suppose X,Y to be normally distributed.

X is normally distributed. Sample volume n=36, sampling average =10.2, the corrected standard deviation s=1.2. Estimate the unknown mean using confident intervals. Reliability =0.95.

Variant 10

There are two independent samples with the volumes . They have been extracted from the normal universal sets X,Y. The corrected sampling variances have been found: . The confidence level α=0.1. Verify the following null hypothesis . The competing hypothesis is .

We select a sample of volume n=17 from a normal universal set. The corrected sampling variance =0.24. We need to verify the null hypothesis . The confidence level is 0.05. The competing hypothesis is: .

The average size of items made on the 1^st lathe equals x=20.1mm. (sample volume n=50). The average size of items made on the 2^nd lathe equals y=19.8mm. (sample volume n=50). Universal variances are known: D(X)=1.75 mm², D(Y)=1.375 mm². We need to verify the following null hypothesis (confidence level is 0.05): E(X)=E(Y). The competing hypothesis is . The random variables X,Y are normally distributed, and the samples are independent.

We need to verify a null hypothesis for normal universal sets X,Y using the competing hypothesis . The confidence level is 0.05. The sampling data are following:

xi	12.3	12.5	12.8	13.0	13.5	yi	12.2	12.3	13.0
ni	1	2	4	2	1	mi	6	8	2

X is normally distributed. Sample volume n=100, sampling average =25, the corrected standard deviation s=2. Estimate the unknown mean using confident intervals. Reliability =0.95.