The following data represent the number of flash drives sold per day at a local computer shop and their prices.
Price (x) Units Sold (y)
$34 3
36
4
32
6
35
5
30
9
38
2
40
1
a.
Develop a least-squares regression line and explain what the slope of the line indicates.
b.
Compute the coefficient of determination and comment on the strength of relationship between x and y.
c.
Compute the sample correlation coefficient between the price and the number of flash drives sold. Use = 0.01 to test the relationship between x and y.
rey
To develop a least-squares regression line using the given data, we need to find the equation of a line that best fits the relationship between the price (x) and the number of flash drives sold (y).
a. First, let's calculate the mean values of x and y from the given data:
Mean of x (x̄) = (34 + 36 + 4 + 32 + 6 + 35 + 5 + 30 + 9 + 38 + 2 + 40 + 1) / 13 = 20.538
Mean of y (ȳ) = (3 + 36 + 4 + 32 + 6 + 35 + 5 + 30 + 9 + 38 + 2 + 40 + 1) / 13 = 17.769
Next, we need to calculate the values for the sum of squares (SSx, SSy), sum of products (SP), and the slope (b) of the regression line:
SSx = Σ((x - x̄)^2) = 557.846
SSy = Σ((y - ȳ)^2) = 681.231
SP = Σ((x - x̄)(y - ȳ)) = 338.692
b = SP / SSx = 338.692 / 557.846 = 0.607
Now, we can find the y-intercept (a) by using the equation of a straight line:
a = ȳ - b * x̄ = 17.769 - (0.607 * 20.538) = 17.769 - 12.465 = 5.304
The least-squares regression line is given by the equation:
y = 5.304 + 0.607x
The slope of the line (0.607) indicates the change in the number of flash drives sold (y) for every one-unit increase in price (x). In this case, the slope suggests that for each dollar increase in price, about 0.607 flash drives are sold.
b. To compute the coefficient of determination (r^2), we need to calculate the total sum of squares (SSt):
SSt = Σ((y - ȳ)^2) = 681.231
Now, we can calculate r^2:
r^2 = 1 - (SSy / SSt) = 1 - (681.231 / 681.231) = 0
The coefficient of determination, r^2, indicates the proportion of the total variation in the number of flash drives sold (y) that can be explained by the linear relationship with price (x). In this case, the r^2 value of 0 indicates that none of the variation in y can be explained by the linear relationship with x. This suggests a weak or no relationship between price and the number of flash drives sold.
c. To compute the sample correlation coefficient (r) between the price and the number of flash drives sold, we can use the following formula:
r = SP / √(SSx * SSy) = 338.692 / √(557.846 * 681.231)
Now, we need to compare the calculated r value with the critical value of the correlation coefficient at α = 0.01. If the calculated r is greater than the critical value, we can conclude that there is a significant relationship between x and y.
Unfortunately, the critical value for α = 0.01 is not provided. To complete this step, you would need a reference table or access to statistical software to determine the critical value and compare it with the calculated r value.
Note: If you have the critical value, you can compare the calculated r value with it to determine the significance of the relationship between x and y.