To fit the linear model (y=a_0 +a _1 x) to the given data, we need to have the data first. Once we have the data, we can use any of the linear regression calculators available online to find the value of y at x=0.5.
Here is an example of how to fit a linear model using Khan Academy's linear regression review:
Suppose we have the following data:
| x | y |
|---|---|
| 10 | 35 |
| 20 | 45 |
| 30 | 55 |
| 40 | 65 |
| 50 | 75 |
Step 1: Plot the points on a graph.
Step 2: Find the slope of the line.
Slope = (y2 - y1) / (x2 - x1)
Slope = (75 - 35) / (50 - 10)
Slope = 40 / 40
Slope = 1
Step 3: Find the y-intercept.
y = mx + b
35 = (1)(10) + b
b = 25
Step 4: Write the equation in y=mx+b form.
y = mx + b
y = x + 25
Now that we have our equation, we can find the value of y at x=0.5 by substituting x=0.5 into our equation:
y = (0.5) + 25
y = 25.5
Therefore, when x=0.5, y=25.5.