The ages (in years) of 10 infants and the number of hours each slept in a day

Age, x: 0.1, 0.2, 0.4, 0.7, 0.6, 0.9, 0.1, 0.2, 0.4, 0.9
Hours slept, y: 14.9, 14.5, 13.9, 14.1, 13.9, 13.7, 14.3, 13.9, 14.0, 14.1
Find the equation of the regression line. Then use the regression equation to predict the value of y for the given x, if meaningful. If it is not meaningful, explain why.
a. X-0.3years b. X=3.9years c. X=0.6years d. X=0.8years

Solution

If you need to show the work by hand, you can develop the regression equation in the following format:

predicted y = a + bx
...where a represents the y-intercept and b the slope.

To get to that point, here are some formulas to calculate along the way.

To find a:
a = (Ey/n) - b(Ex/n)

Note: E here means to add up or to find the total.

To find b:
b = SSxy/SSxx

To find SSxy:
SSxy = Exy - [(Ex)(Ey)]/n

To find SSxx:
SSxx = Ex^2 - [(Ex)(Ex)]/n

Once you have the equation, substitute the values for x and solve for predicted y.

I hope this will help get you started.

To find the equation of the regression line, we can use linear regression analysis. This involves finding the best-fitting line that represents the relationship between the age of the infants (x) and the number of hours they slept (y).

Step 1: Calculate the means of x and y.
Mean of x (x̄) = (0.1 + 0.2 + 0.4 + 0.7 + 0.6 + 0.9 + 0.1 + 0.2 + 0.4 + 0.9) / 10 = 0.48
Mean of y (ȳ) = (14.9 + 14.5 + 13.9 + 14.1 + 13.9 + 13.7 + 14.3 + 13.9 + 14.0 + 14.1) / 10 = 14.2

Step 2: Calculate the deviations (differences) of each x and y value from their mean.
differences of x = (0.1 - 0.48, 0.2 - 0.48, 0.4 - 0.48, 0.7 - 0.48, 0.6 - 0.48, 0.9 - 0.48, 0.1 - 0.48, 0.2 - 0.48, 0.4 - 0.48, 0.9 - 0.48)
= (-0.38, -0.28, -0.08, 0.22, 0.12, 0.42, -0.38, -0.28, -0.08, 0.42)
differences of y = (14.9 - 14.2, 14.5 - 14.2, 13.9 - 14.2, 14.1 - 14.2, 13.9 - 14.2, 13.7 - 14.2, 14.3 - 14.2, 13.9 - 14.2, 14.0 - 14.2, 14.1 - 14.2)
= (0.7, 0.3, -0.3, -0.1, -0.3, -0.5, 0.1, -0.3, -0.2, -0.1)

Step 3: Calculate the product of the deviations of x and y.
product of deviations = (-0.38 * 0.7, -0.28 * 0.3, -0.08 * -0.3, 0.22 * -0.1, 0.12 * -0.3, 0.42 * -0.5, -0.38 * 0.1, -0.28 * -0.3, -0.08 * -0.2, 0.42 * -0.1)
= (-0.266, -0.084, 0.024, -0.022, -0.036, -0.21, -0.038, 0.084, 0.016, -0.042)

Step 4: Calculate the squared deviations of x.
squared deviations of x = (-0.38^2, -0.28^2, -0.08^2, 0.22^2, 0.12^2, 0.42^2, -0.38^2, -0.28^2, -0.08^2, 0.42^2)
= (0.1444, 0.0784, 0.0064, 0.0484, 0.0144, 0.1764, 0.1444, 0.0784, 0.0064, 0.1764)

Step 5: Calculate the sum of squared deviations of x (SSx) and the sum of products of deviations (SP).
SSx = Σ(x - x̄)^2 = 0.1444 + 0.0784 + 0.0064 + 0.0484 + 0.0144 + 0.1764 + 0.1444 + 0.0784 + 0.0064 + 0.1764 = 0.8736
SP = Σ(x - x̄)(y - ȳ) = -0.266 + -0.084 + 0.024 + -0.022 + -0.036 + -0.21 + -0.038 + 0.084 + 0.016 + -0.042 = -0.480

Step 6: Calculate the slope (b) of the regression line.
b = SP / SSx = -0.480 / 0.8736 ≈ -0.549

Step 7: Calculate the y-intercept (a) of the regression line.
a = ȳ - b * x̄ = 14.2 - (-0.549 * 0.48) = 14.2 + 0.26352 ≈ 14.4635

Step 8: Write the equation of the regression line.
Therefore, the equation of the regression line is y = -0.549x + 14.4635.

Now, let's use the regression equation to predict the value of y for each given x.

a. X = 0.3 years
y = -0.549(0.3) + 14.4635
y ≈ 14.3038

b. X = 3.9 years
y = -0.549(3.9) + 14.4635
y ≈ 12.2911

c. X = 0.6 years
y = -0.549(0.6) + 14.4635
y ≈ 14.1365

d. X = 0.8 years
y = -0.549(0.8) + 14.4635
y ≈ 13.814

Therefore, the predicted values of y are:
a. X = 0.3 years: y ≈ 14.3038
b. X = 3.9 years: y ≈ 12.2911
c. X = 0.6 years: y ≈ 14.1365
d. X = 0.8 years: y ≈ 13.814

The predictions are meaningful for each value of x.

To find the equation of the regression line, we can use linear regression analysis. The equation of a linear regression line is represented by the formula:

y = mx + b

where:
- y is the dependent variable (hours slept)
- x is the independent variable (age)
- m is the slope of the regression line
- b is the y-intercept of the regression line

To calculate the regression line's equation, we need to find the values of m and b. Here's how we can do that:

Step 1: Calculate the means of x and y.

x̄ = (Sum of x) / (Number of data points)
ȳ = (Sum of y) / (Number of data points)

For the given data, we can use the following values:
n = 10 (since we have 10 data points)

Sum of x = 0.1 + 0.2 + 0.4 + 0.7 + 0.6 + 0.9 + 0.1 + 0.2 + 0.4 + 0.9 = 5.5
Sum of y = 14.9 + 14.5 + 13.9 + 14.1 + 13.9 + 13.7 + 14.3 + 13.9 + 14.0 + 14.1 = 141.3

x̄ = 5.5 / 10 = 0.55
ȳ = 141.3 / 10 = 14.13

Step 2: Calculate the slope (m).

m = Σ[(x - x̄)(y - ȳ)] / Σ[(x - x̄)²]

To find m, we need to calculate the sums of the products (x - x̄)(y - ȳ) and the sum of squares (x - x̄)².

(x - x̄)(y - ȳ) = (0.1 - 0.55)(14.9 - 14.13) +
(0.2 - 0.55)(14.5 - 14.13) +
(0.4 - 0.55)(13.9 - 14.13) +
(0.7 - 0.55)(14.1 - 14.13) +
(0.6 - 0.55)(13.9 - 14.13) +
(0.9 - 0.55)(13.7 - 14.13) +
(0.1 - 0.55)(14.3 - 14.13) +
(0.2 - 0.55)(13.9 - 14.13) +
(0.4 - 0.55)(14.0 - 14.13) +
(0.9 - 0.55)(14.1 - 14.13)

= -0.221

(x - x̄)² = (0.1 - 0.55)² +
(0.2 - 0.55)² +
(0.4 - 0.55)² +
(0.7 - 0.55)² +
(0.6 - 0.55)² +
(0.9 - 0.55)² +
(0.1 - 0.55)² +
(0.2 - 0.55)² +
(0.4 - 0.55)² +
(0.9 - 0.55)²

= 0.363

Now we can calculate the slope:

m = -0.221 / 0.363 ≈ -0.608

Step 3: Calculate the y-intercept (b).

b = ȳ - (m * x̄)

b = 14.13 - (-0.608 * 0.55)

b = 14.13 + 0.3344

b ≈ 14.464

Therefore, the equation of the regression line is:

y ≈ -0.608x + 14.464

To predict the value of y for the given x, we substitute the value of x into the equation. Let's go through each option:

a. x = 0.3 years
y ≈ -0.608(0.3) + 14.464
y ≈ 14.2616

b. x = 3.9 years
This is beyond the range of the given data, so it is not meaningful to predict the hours slept for this value of x.

c. x = 0.6 years
y ≈ -0.608(0.6) + 14.464
y ≈ 14.1348

d. x = 0.8 years
y ≈ -0.608(0.8) + 14.464
y ≈ 14.0352

Therefore, the predicted values are:
a. For x = 0.3 years: y ≈ 14.2616
b. For x = 3.9 years: Not meaningful
c. For x = 0.6 years: y ≈ 14.1348
d. For x = 0.8 years: y ≈ 14.0352