Find the equation for the least squares regression line of the data described below.A food scientist wants to understand how much sugar children like in their desserts. He sets up a study where children taste cups of vanilla pudding with varying sweetness and rate each cup on a scale of 1 to 10, where 10 is excellent.For each tasting, the food scientist writes down the grams of sugar in the pudding cup, x, as well as its rating, y.Grams of sugar Rating10 318 221 924 627 9Round your answers to the nearest thousandth.
Question
Find the equation for the least squares regression line of the data described below.A food scientist wants to understand how much sugar children like in their desserts. He sets up a study where children taste cups of vanilla pudding with varying sweetness and rate each cup on a scale of 1 to 10, where 10 is excellent.For each tasting, the food scientist writes down the grams of sugar in the pudding cup, x, as well as its rating, y.Grams of sugar Rating10 318 221 924 627 9Round your answers to the nearest thousandth.
Solution
To find the equation for the least squares regression line, we first need to calculate the means of x (grams of sugar) and y (rating), the standard deviations of x and y, and the correlation coefficient (r).
The means of x and y are calculated as follows:
mean of x = (10 + 18 + 21 + 24 + 27) / 5 = 20 mean of y = (3 + 2 + 9 + 6 + 9) / 5 = 5.8
The standard deviations of x and y are calculated as follows:
standard deviation of x = sqrt[((10-20)^2 + (18-20)^2 + (21-20)^2 + (24-20)^2 + (27-20)^2) / 4] = sqrt[100 + 4 + 1 + 16 + 49] / 4 = sqrt[34] standard deviation of y = sqrt[((3-5.8)^2 + (2-5.8)^2 + (9-5.8)^2 + (6-5.8)^2 + (9-5.8)^2) / 4] = sqrt[7.84 + 14.44 + 10.24 + 0.04 + 10.24] / 4 = sqrt[8.56]
The correlation coefficient (r) is calculated as follows:
r = [1/(n-1)] * Σ[((xi - mean of x) / standard deviation of x) * ((yi - mean of y) / standard deviation of y)] = [1/4] * [((10-20)/sqrt[34] * (3-5.8)/sqrt[8.56]) + ((18-20)/sqrt[34] * (2-5.8)/sqrt[8.56]) + ((21-20)/sqrt[34] * (9-5.8)/sqrt[8.56]) + ((24-20)/sqrt[34] * (6-5.8)/sqrt[8.56]) + ((27-20)/sqrt[34] * (9-5.8)/sqrt[8.56])] = [1/4] * [-1.71 + -0.23 + 0.37 + 0.17 + 1.01] = 0.15
The equation for the least squares regression line is y = a + bx, where a is the y-intercept and b is the slope.
The slope (b) is calculated as follows:
b = r * (standard deviation of y / standard deviation of x) = 0.15 * (sqrt[8.56] / sqrt[34]) = 0.22
The y-intercept (a) is calculated as follows:
a = mean of y - b * mean of x = 5.8 - 0.22 * 20 = 1.4
So, the equation for the least squares regression line is y = 1.4 + 0.22x.
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