Econometrics Practical Option 9

The economist, studying the dependence of the level of distribution costs y (thousand rubles) on the volume of trade x (thousand rubles), examined 10 stores selling the same assortment of goods in different regions. The data obtained are reflected in table No. 1.

Х 80 60 100 130 120 50 90 150 70 125
У 4,2 4,9 7,2 9,1 6,4 3,9 5,1 8,4 3,5 8,7

According to the table for your variant, complete the following tasks.

Task 1. Build a scatter plot and form a hypothesis about the type of connection.

Task 2. Determine the parameters of linear regression and write down the linear dependence function.

Task 3. Plot a regression plot on the scatterplot field and draw a conclusion about the adequacy of the constructed model.
Task 4.
Find the correlation coefficient and the coefficient of determination of this dependence. Whether the conclusions about the type of connection and the adequacy of the model have been confirmed.
Task 5.
Determine the reliability of the regression coefficients by the Student's test.

Let us check the hypothesis H0 about the equality of individual regression coefficients to zero (with the alternative H1 is not equal) at the significance level α = 0.05.
If the main hypothesis turns out to be wrong, we accept the alternative. To test this hypothesis, the Student's t-test is used.
The value of the t-test found from the observational data (also called the observed or actual) is compared with the tabular (critical) value determined from the Student's distribution tables (which are usually given at the end of textbooks and workshops on statistics or econometrics).
The tabular value is determined depending on the level of significance (α) and the number of degrees of freedom, which in the case of linear pairwise regression is (n-2), n is the number of observations.
If the actual value of the t-criterion is greater than the tabular one (in absolute value), then the main hypothesis is rejected and it is considered that with probability (1-α) the parameter or statistical characteristic in the general population differs significantly from zero.
If the actual value of the t-criterion is less than the tabular one (in absolute value), then there is no reason to reject the main hypothesis, i.e. parameter or statistical characteristic in the general population does not differ significantly from zero at the significance level α.
tcrit (nm-1; α / 2) = (8; 0.025) = 2.306
Task 6.
Examine the regression for reliability by Fisher's test at a significance level of 0,05.

The significance of the regression model is checked using the Fisher's F-test, the calculated value of which is found as the ratio of the variance of the initial series of observations of the studied indicator and the unbiased estimate of the variance of the residual sequence for this model.
If the calculated value with k1 = (m) and k2 = (nm-1) degrees of freedom is greater than the tabular value for a given significance level, then the model is considered significant
Task 7.
Find the coefficient of elasticity y with respect to x at the mean value of x.

The average coefficient of elasticity E shows how many percent on average across the population the result y will change from its average value when the factor x changes by 1% from its average value.
The coefficient of elasticity is found by the formula
Task 8.
Calculate the expected average cost for a turnover of 115 thousand rubles.
Task 9.
Estimate the forecast error at a significance level of 0,05.

Let us calculate the boundaries of the interval in which 95% of the possible values ​​of Y will be concentrated with an unlimited number of observations and Xp = 115
Task 10.
Plot forecast confidence intervals for a significance level of 0,05.