A determination coefficient of 0.725 shows:

How many explanatory variables are in the model?

How many observations are for each variable?

Theoretical value of t Student ratio is (using the Excel function "=tinv(...)" for 5% probability level and degrees freedom of is (with 3 decimals): .

Establish the result of statistical test of the significance for each estimator knowing only this value and their t* in the table.

Based on the following regression table, answer the questions:

a) Write the model (coefficients rounded at 2 decimals):

                           yteo=+x1+x2+x3

b) Which is the number of explanatory variables? .

c) Which is the number of the observations of variables? .

d) The model validity is appreciated based on the value of . Write its value with 4 decimals: .

e) Write the multiple correlation coefficient with 4 decimals: 

f) Looking at the regression table establish if the estimators are significant from 0, at a significance level of 5%.

The following table contains the indicators: GDP per inhabitant and the volume of greenhouse gas emissions in transport activity in Romania during the period 1995-2014.

Using the function =CORREL(array_1,array_2) of Excel, calculate the correlation coefficient and appreciate the intensity and the link nature, for the required period (write with 3 decimals).

The simple correlation coefficient between the two indicators for the period 2000-2014 is (with 3 decimals): .

The nature of correlation between the two indicators is 

The intensity of the correlation is

The simple correlation coefficient between the two indicators for the period 1995-2014 is (with 3 decimals) .

The intensity of the correlation is 

For the period 2000-2014 the correlation is

Use the functions intercept and slope to identify the estimators of the simple regression model for the greenhouse gas emissions depending on the real GDP per capita for the entire period 1995-2014.

The free term (with 3 decimals) is estimated a₀= and the regression coefficient (with 6 decimals) is estimated a₁=.

Without considering the GDP per capita, the greenhouse gas emissions from transports activity, yearly recorded an average quantity of (3 decimals) millions tonnes CO₂ equivalent.

For each euro added to the GDP per capita, the greenhouse gas emissions increased yearly in average with (round at integer; attention to the unit measure) tonnes CO₂ equivalent.

Calculate the theoretical values using the identified linear model and then the residuals.

The maximum positive deviation of the empirical values of greenhouse gas emissions from to the model was the residual value of (3 decimals) millions tonnes CO₂ equivalent, recorded in the year .

The maximum negative deviation of the empirical values of greenhouse gas emissions from to the model was the residual value of (3 decimals) millions tonnes CO₂ equivalent, recorded in the year .

Make the chart of correlation between y and x.

Make the chart of evolution for y variable. On both charts add the theoretical y values.

Specify the number of y values lower and equal than the theoretical coresponding values: .

Specify the number of y values higher than the theoretical coresponding values: .

Make also the chart of x variable evolution.

Looking at the chart of the theoretical y variable evolution (on the same chart with y variable evolution) you may conclude that it looks

About the link between the two variables we can say that:

- is of

- it has a

- the intensity of link is

The dependent variable is

The independent variable is

For each variable there were recorded observations.

Complete the title of the chart (write only the missing words): .

About the link between the two variables we can say that:

- is of

- it has a

- the intensity of link is

The dependent variable is

The independent variable is

For each variable there were recorded observations.

Complete the title of the chart (write only the missing words):

About the link between the two variables we can say that:

- is of

- it has a

- the intensity of link is

The dependent variable is

The independent variable is

For each variable there were recorded observations.

The manager of a supermarket says 50% of buyers can immediately appreciate the price of the products placed in the basket. Considering that this proportion is underestimated, a commercial consultancy makes a survey of 802 buyers, out of which 423 prove that they can really appreciate the correct price of the purchased products.

The ratio (w) of those in the sample setting the correct price is (in %, with 2 decimals): %.

Test for a threshold of 5%, 7% and 10% significance level, if you can accept the null hypothesis, that is, the director's statement that half of the buyers can appreciate the correct price.

The Critical Report (CR) is (with 2 decimals): .

The test is

At a 5% significance threshold, the z coefficient to compare the CR is (use the Excel function =NORMSINV(....) and depending on the nature of the test consider its positive and / or negative value) (write here in absolute value): . Because the CR is

At a significance threshold of 7%, the z coefficient to compare CR is (use the function Excel =NORMSINV(....) and depending on the nature of the test consider its positive and / or negative value) (write here in absolute value): . Because CR is

At a 10% significance threshold, the z coefficient to compare CR is (use the Excel function =NORMSINV(....) and depending on the nature of the test consider its positive and / or negative value) (write here in absolute value): . Because the CR is

Knowing the CR value to be established, using the =NORMSINV(....) function, which is the probability of accepting the hypothesis H0 (in %, with 2 decimals) based on this sample: %? The significance threshold (a) is in this case (in %, with 2 decimals): % and represents the probability from which the H1 hypothesis can be considered true. This level of significance is called "p-value" in English.

From the three analyzed situations, the "p-value" calculated on the basis of the sample is greater than the significance threshold of

A detergent company, packs the products in 3 kg bags.

The average weight distribution of the bags is known to be normal, with a standard deviation of ± 150 grams.

A sample of 16 detergent bags shows an average weight of a 2.95 kg per bag.

At a 5% significance threshold, do the test of the null hypothesis according to which the average weight of a bag is 3 kg.

This statistical test for product quality control is:

The calculated critical report CR (with 2 decimals) is .

The theoretical value of the

The critical ratio CR calculated in absolute value is

We can say that