Thefollowingregressionequationforquantitysuppliedwasestimatedusinga
sampleoffifty observations.
Standard errors are in the brackets. The total sum of squares was 132 and theresidual
ii.Testthehypothesisthatslopecoefficient(ß1)=0fallswithinthisinterval. Can
i.Establisha99%confidenceintervalforslopeandinterceptcoefficient.
we say the price has no effect on the quantity supplied? Usethe test of significant
sumofsquares was 19.5
variationsinthequantitysupplied?
iii.CalculateandinterpretR-square.Whatotherfactorscouldpossiblyexplain
Q = 2.2 + 0.104P.
(3.4) (0.005)
approach at 1% significance level to test the abovehypothesis?
iv.From(c) canwe say themodel best fit in the data set?
To test the hypothesis that the slope coefficient (ß1) equals 0, you can use t-test approach. Here's how you can do it step by step:
i. To establish a 99% confidence interval for the slope and intercept coefficient, you can use the following formulas:
For the slope coefficient:
Confidence interval = ß1 ± t(α/2, n-2) * standard error of ß1
For the intercept coefficient:
Confidence interval = ß0 ± t(α/2, n-2) * standard error of ß0
In this case, the regression equation is:
Q = 2.2 + 0.104P
The standard error of ß1 is 0.005 and the standard error of ß0 is 3.4. We need to find the t-statistic for a 99% confidence level with n-2 degrees of freedom. Since the sample size is 50, the degrees of freedom would be 50-2 = 48.
Using a t-table or a statistical software, you can find the t-value for a 99% confidence level with 48 degrees of freedom. Let's say the t-value is 2.678.
Now we can calculate the confidence intervals:
For the slope coefficient:
Confidence interval for ß1 = 0.104 ± 2.678 * 0.005
For the intercept coefficient:
Confidence interval for ß0 = 2.2 ± 2.678 * 3.4
ii. To test the hypothesis that the slope coefficient (ß1) equals 0, you can use the t-test approach. The null hypothesis (H0) is ß1 = 0, and the alternative hypothesis (Ha) is ß1 ≠ 0.
To perform the t-test, you need to calculate the t-statistic using the formula:
t = (ß1 - 0) / standard error of ß1
In this case, the standard error of ß1 is 0.005, and the estimated value of ß1 is 0.104. So the t-statistic would be:
t = (0.104 - 0) / 0.005
Next, you can compare the calculated t-statistic to the critical t-value at the desired significance level (let's say 1%). If the calculated t-statistic is greater than the critical t-value or less than the negative of the critical t-value, you can reject the null hypothesis and conclude that there is a significant relationship between the price (P) and the quantity supplied (Q).
iii. R-square is a measure of how well the regression model explains the variation in the dependent variable (quantity supplied in this case). It ranges from 0 to 1, where a higher R-square value indicates a better fit of the model.
To calculate R-square, you need to use the total sum of squares (TSS) and the residual sum of squares (RSS) from the regression model. In this case, the TSS is given as 132 and the RSS is given as 19.5.
R-square can be calculated using the formula:
R-square = 1 - (RSS / TSS)
Substituting the given values:
R-square = 1 - (19.5 / 132)
Interpreting R-square: R-square tells us the proportion of the total variation in the quantity supplied (Q) that is explained by the variation in the independent variable (price, P). In this case, R-square would be a value between 0 and 1. If R-square is close to 1, it means that a large proportion of the variation in the quantity supplied is explained by the variation in price, indicating that price has a significant effect on the quantity supplied. If R-square is close to 0, it means that only a small proportion of the variation in the quantity supplied is explained by the variation in price, indicating that price does not have a significant effect on the quantity supplied.
Apart from price, there could be other factors that could possibly explain the variations in the quantity supplied. These factors could include variables like input costs, technology, consumer preferences, market demand, etc.
iv. To determine if the model best fits the dataset, you can assess the goodness of fit using various measures such as R-square, adjusted R-square, F-statistic, etc.
In this case, we have calculated R-square in the previous step. If R-square is close to 1, it indicates that the model is a good fit for the dataset, meaning that a large proportion of the variation in the quantity supplied is explained by the variation in the independent variable (price).
However, it's important to note that R-square alone may not be sufficient to determine the best fit of the model. It is recommended to also consider other measures such as adjusted R-square, F-statistic, and diagnostic tests for model assumptions to make a more comprehensive assessment of the model's fit.