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 Rating
17 1
24 4
29 3
30 9
30 8
Round your answers to the nearest thousandth.
y=
x+
Answers
GPT-4o mini
To find the least squares regression line for the given data, we'll follow these steps:
1. **Calculate the mean of \(x\) (sugar grams) and \(y\) (rating).**
2. **Calculate the slope (m) of the regression line using the formula:**
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
3. **Calculate the y-intercept (b) using the formula:**
\[
b = \bar{y} - m\bar{x}
\]
4. **Write the equation of the regression line in the form \(y = mx + b\).**
### Step 1: Calculate the sums and means
Given data:
- \((x_1, y_1) = (17, 1)\)
- \((x_2, y_2) = (24, 4)\)
- \((x_3, y_3) = (29, 3)\)
- \((x_4, y_4) = (30, 9)\)
- \((x_5, y_5) = (30, 8)\)
Calculating:
- \(n = 5\)
Now, calculate the following sums:
- \(\sum x = 17 + 24 + 29 + 30 + 30 = 130\)
- \(\sum y = 1 + 4 + 3 + 9 + 8 = 25\)
- \(\sum xy = (17 \cdot 1) + (24 \cdot 4) + (29 \cdot 3) + (30 \cdot 9) + (30 \cdot 8)\)
Calculating \(\sum xy\):
\[
\sum xy = 17 + 96 + 87 + 270 + 240 = 710
\]
Next, calculate:
- \(\sum x^2 = 17^2 + 24^2 + 29^2 + 30^2 + 30^2\)
- \(17^2 = 289\), \(24^2 = 576\), \(29^2 = 841\), \(30^2 = 900\)
Calculating \(\sum x^2\):
\[
\sum x^2 = 289 + 576 + 841 + 900 + 900 = 3506
\]
Now we have:
- \(\sum x = 130\)
- \(\sum y = 25\)
- \(\sum xy = 710\)
- \(\sum x^2 = 3506\)
### Step 2: Calculate the slope (m)
Using the regression slope formula:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
Substituting in our values:
\[
m = \frac{5(710) - (130)(25)}{5(3506) - (130)^2}
\]
Calculating:
\[
m = \frac{3550 - 3250}{17530 - 16900} = \frac{300}{630} \approx 0.4762
\]
### Step 3: Calculate the y-intercept (b)
Now calculate \(b\):
\[
b = \bar{y} - m\bar{x}
\]
Calculating \(\bar{y}\) and \(\bar{x}\):
\[
\bar{x} = \frac{\sum x}{n} = \frac{130}{5} = 26
\]
\[
\bar{y} = \frac{\sum y}{n} = \frac{25}{5} = 5
\]
Substituting for \(b\):
\[
b = 5 - 0.4762 \cdot 26 \approx 5 - 12.3782 \approx -7.3782
\]
### Step 4: Write the equation of the regression line
Now we can write the regression equation:
\[
y = mx + b
\]
\[
y \approx 0.476x - 7.378
\]
Rounding to the nearest thousandth gives:
\[
y \approx 0.476x - 7.378
\]
Thus, the equation for the least squares regression line is:
\[
y = 0.476x - 7.378
\]
1. **Calculate the mean of \(x\) (sugar grams) and \(y\) (rating).**
2. **Calculate the slope (m) of the regression line using the formula:**
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
3. **Calculate the y-intercept (b) using the formula:**
\[
b = \bar{y} - m\bar{x}
\]
4. **Write the equation of the regression line in the form \(y = mx + b\).**
### Step 1: Calculate the sums and means
Given data:
- \((x_1, y_1) = (17, 1)\)
- \((x_2, y_2) = (24, 4)\)
- \((x_3, y_3) = (29, 3)\)
- \((x_4, y_4) = (30, 9)\)
- \((x_5, y_5) = (30, 8)\)
Calculating:
- \(n = 5\)
Now, calculate the following sums:
- \(\sum x = 17 + 24 + 29 + 30 + 30 = 130\)
- \(\sum y = 1 + 4 + 3 + 9 + 8 = 25\)
- \(\sum xy = (17 \cdot 1) + (24 \cdot 4) + (29 \cdot 3) + (30 \cdot 9) + (30 \cdot 8)\)
Calculating \(\sum xy\):
\[
\sum xy = 17 + 96 + 87 + 270 + 240 = 710
\]
Next, calculate:
- \(\sum x^2 = 17^2 + 24^2 + 29^2 + 30^2 + 30^2\)
- \(17^2 = 289\), \(24^2 = 576\), \(29^2 = 841\), \(30^2 = 900\)
Calculating \(\sum x^2\):
\[
\sum x^2 = 289 + 576 + 841 + 900 + 900 = 3506
\]
Now we have:
- \(\sum x = 130\)
- \(\sum y = 25\)
- \(\sum xy = 710\)
- \(\sum x^2 = 3506\)
### Step 2: Calculate the slope (m)
Using the regression slope formula:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
Substituting in our values:
\[
m = \frac{5(710) - (130)(25)}{5(3506) - (130)^2}
\]
Calculating:
\[
m = \frac{3550 - 3250}{17530 - 16900} = \frac{300}{630} \approx 0.4762
\]
### Step 3: Calculate the y-intercept (b)
Now calculate \(b\):
\[
b = \bar{y} - m\bar{x}
\]
Calculating \(\bar{y}\) and \(\bar{x}\):
\[
\bar{x} = \frac{\sum x}{n} = \frac{130}{5} = 26
\]
\[
\bar{y} = \frac{\sum y}{n} = \frac{25}{5} = 5
\]
Substituting for \(b\):
\[
b = 5 - 0.4762 \cdot 26 \approx 5 - 12.3782 \approx -7.3782
\]
### Step 4: Write the equation of the regression line
Now we can write the regression equation:
\[
y = mx + b
\]
\[
y \approx 0.476x - 7.378
\]
Rounding to the nearest thousandth gives:
\[
y \approx 0.476x - 7.378
\]
Thus, the equation for the least squares regression line is:
\[
y = 0.476x - 7.378
\]