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1 The Czech state commissioned a survey of how people perceive the change of winter and summer time. The survey included larger cities (Prague, Brno), smaller cities (Znojmo, Tišnov) and municipalities (Paseky, Horní Lomná, Dolní Věstonice). In the survey, they found out what suits people better - whether alternating summer and winter time, only winter time or only summer time. You can see the respondents' answers in the table:

Praha Brno Znojmo Tišnov Paseky Horní Lomná Dolní

Věstonice okolí studenta

number of

respondents 1327 915 681 587 284 176 215 23

winter time 510 324 302 257 147 66 87 3

summer time 352 284 185 178 87 58 65 2

alternation of

times 257 178 124 78 44 33 31 11

he has no opinion 208 129 70 74 6 19 32 7

At the α = 0.05 significance level (α = 0.05 is the total type 1 error for a) to e)), test the hypotheses:

a) In cities, towns and the student's surroundings (8th surveys) there is the same percentage representation of residents who prefer winter time.

b) In cities, towns and the student's surroundings (8 surveys) there is the same percentage representation of residents who prefer summer time.

c) In cities, towns and the student's neighborhood (8 surveys) there is the same percentage representation of the population that prefers to alternate time.

d) In larger cities, smaller cities and municipalities (3 surveys) there is the same percentage representation of residents who prefer winter time.

e) In larger cities, smaller cities and municipalities (3 surveys) there is the same percentage representation of undecided residents.

f) Based on the answers from the student's surroundings, try to determine from the data whether the student conducted research in a larger city, a smaller city or a village. Compare the result with reality and comment.

2:

The data consists of 70 realizations of 3 random variables. The first two columns in the table (Excel) contain explanatory variables X and Y (regressors – the same for all entries), the third column – see entry number - indicates the values of the dependent (explained) variable Z. Perform tests at a significance level of 0.05%, calculate interval estimates with a confidence level of 95%. Mark the individual steps to make the text clearer.

Perform the following regression analysis in data file processing::

a) Determine the appropriate model using the backward method and regression diagnostics. Consider a polynomial–quadratic (in both variables) model.

So start from the regression function: Z X Y X 2 Y 2 X Y up to Z . Compare the suitability of the found models using the coefficient of determination R2. Verify the possibility of simplifying one model to its submodel (the model obtained by omitting some column of the plan matrix) using a suitable test of the nullity of the regression parameters.

b) For the model obtained in this way (sufficient submodel), list in one table the estimates of the regression parameters by the method of least squares and their 95% confidence intervals.

c) Estimate the variance of the dependent variable in an unbiased way.

d) Use a suitable test to determine that the two regression parameters you have chosen are simultaneously zero.

e) Use a suitable test to determine that the two regression parameters you have chosen are the same

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