y= 0 . 25,35,45,55,65,75,85,95
x= 60,65,70,75,80,85,90,95
Using the trend line, what whole number could you infer the daily sales amount is when the temperature is 68°? Round your answer to the nearest whole number.
To find the whole number value of daily sales amount when the temperature is 68°, we need to determine the equation of the trend line that best fits the given data.
We can use the equation of a straight line, y = mx + b, where y represents the daily sales amount and x represents the temperature.
To find the values of m and b, we can use the least squares method:
1. Calculate the mean of x and y:
mean(x) = (60 + 65 + 70 + 75 + 80 + 85 + 90 + 95) / 8 = 77.5
mean(y) = (0.25 + 35 + 45 + 55 + 65 + 75 + 85 + 95) / 8 = 56
2. Calculate the differences between each x value and the mean of x (77.5):
x - mean(x) = -17.5, -12.5, -7.5, -2.5, 2.5, 7.5, 12.5, 17.5
3. Calculate the differences between each y value and the mean of y (56):
y - mean(y) = -55.75, -21, -11, -1, 9, 19, 29, 39
4. Calculate the product of the differences obtained in steps 2 and 3:
(x - mean(x)) * (y - mean(y)) = 966.25, 262.5, 82.5, 2.5, 22.5, 142.5, 362.5, 682.5
5. Calculate the squared differences between each x value and the mean of x (77.5):
(x - mean(x))^2 = 306.25, 156.25, 56.25, 6.25, 6.25, 56.25, 156.25, 306.25
6. Calculate the sum of the squared differences obtained in step 5:
Σ(x - mean(x))^2 = 1040
7. Calculate the sum of the products obtained in step 4:
Σ(x - mean(x)) * (y - mean(y)) = 2504.5
8. Calculate the slope (m) of the trend line:
m = Σ(x - mean(x)) * (y - mean(y)) / Σ(x - mean(x))^2 = 2504.5 / 1040 = 2.4077
9. Calculate the y-intercept (b) of the trend line:
b = mean(y) - m * mean(x) = 56 - 2.4077 * 77.5 = -93.92
The equation of the trend line is y = 2.4077x - 93.92.
Now we can substitute x = 68 into the equation to find the whole number value of y (daily sales amount):
y = 2.4077 * 68 - 93.92
y ≈ 45.69
Rounding to the nearest whole number, we can infer that the daily sales amount is approximately 46 when the temperature is 68°.