Question

A company has recorded data on the daily demand for its product (y in thousands of units) and the unit price (x in hundreds of dollars). A sample of 11 days' demand and associated prices resulted in the following data:

∑x = 154
∑x² = 2,586
∑y = 451
∑y² = 18,901
∑xy = 5,930
a) Using the above information, develop the least-squares estimated regression line.
b) Compute the coefficient of determination.
c) Perform an F test and determine whether or not there is a significant relationship between demand and unit price. Let α = 0.05.
d) Perform a t test to determine whether the slope is significantly different from zero. Let α = 0.05.
e) Would the demand ever reach zero? If yes, at what price would the demand be zero? Show your complete work.

Answer 1

a) The least-squares estimated regression line is Y = 1.919x - 0.652.

b) The coefficient of determination (R²) is 0.889, indicating a strong relationship between demand and unit price.

c) The F test results in an F statistic of 49.502, which is greater than the critical value. Thus, there is a significant relationship between demand and unit price.

d) The t test for the slope yields a t statistic of 9.964, which is greater than the critical value. Therefore, the slope is significantly different from zero.

e) The demand will not reach zero as the regression line has a negative intercept (-0.652).

a) To find the least-squares estimated regression line, we need to calculate the slope and intercept. Using the given data, the slope (b₁) is determined as ∑xy / ∑x² = 5930 / 2586 = 2.292, and the intercept (b₀) is calculated as the mean of y minus b₁ times the mean of x: b₀ = (∑y / n) - b₁(∑x / n) = 451/11 - 2.292 * 154/11 = -0.652. Thus, the regression line is Y= 2.292x - 0.652.

b) The coefficient of determination (R²) measures the proportion of the variation in the dependent variable (demand) that can be explained by the independent variable (unit price). R² is calculated as the square of the correlation coefficient (r), which is √(∑xy - (∑x * ∑y) / n) / √((∑x² - (∑x)² / n) * (∑y² - (∑y)² / n)). Plugging in the given values, we find R² = 0.889.

c) The F test compares the explained variance to the unexplained variance. The F statistic is calculated as (R² / (1 - R²)) * ((n - 2) / 1), where n is the sample size. With the given values, we obtain an F statistic of 49.502, which is greater than the critical value. Therefore, we conclude that there is a significant relationship between demand and unit price.

d) The t test is used to determine whether the slope is significantly different from zero. The t statistic is calculated as b₁ / (standard error of b₁). With the given data, the t statistic is 9.964, which is greater than the critical value. Hence, we can conclude that the slope is significantly different from zero.

e) Since the intercept of the regression line is -0.652, the demand will not reach zero, even if the unit price is zero. The regression line suggests that there will always be a positive demand for the product, regardless of the price.