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Let's go through the question step by step.

We are given the following dataLet's go through the question step by step.

We are given the following 

| Number of weeks of employment (x) | Number of cars inspected (y) |
|-----------------------------------|------------------------------|
| 1                                 | 12                           |
| 2                                 | 14                           |
| 5                                 | 20                           |
| 7                                 | 22                           |
| 9                                 | 26                           |

### (a) Fill out one of the tables:

We can choose either the first or second table. Let's proceed with the second one because it's simpler:

| \( x \) | \( y \) | \( x^2 \) | \( y^2 \) | \( xy \) |
|--------|---------|-----------|-----------|----------|
| 1      | 12      | \( 1^2 = 1 \)     | \( 12^2 = 144 \)   | \( 1 \times 12 = 12 \)  |
| 2      | 14      | \( 2^2 = 4 \)     | \( 14^2 = 196 \)   | \( 2 \times 14 = 28 \)  |
| 5      | 20      | \( 5^2 = 25 \)    | \( 20^2 = 400 \)   | \( 5 \times 20 = 100 \) |
| 7      | 22      | \( 7^2 = 49 \)    | \( 22^2 = 484 \)   | \( 7 \times 22 = 154 \) |
| 9      | 26      | \( 9^2 = 81 \)    | \( 26^2 = 676 \)   | \( 9 \times 26 = 234 \) |

Now sum up the columns:

| Sum    | 24      | 94      | 160      | 1900      | 528      |

Next, let's calculate the correlation coefficient (r). I'll calculate this step next.The correlation coefficient (\(r\)) is approximately 0.996. This indicates a very strong positive linear relationship between the number of weeks of employment and the number of cars inspected.

Next, let's find the best-fitted line (part c) using the least squares method. The formula for the line is:

\[
y = mx + b
\]

Where:
- \(m\) is the slope, calculated as:
  \[
  m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}
  \]
- \(b\) is the y-intercept, calculated as:
  \[
  b = \frac{\sum y - m \sum x}{n}
  \]

I'll calculate the slope \(m\) and intercept \(b\) now.The equation of the best-fitted line is:

\[
y = 1.71x + 10.57
\]

### (d) Estimate how many cars will be inspected by someone who is employed for 3 weeks:

Using the equation \(y = 1.71x + 10.57\), substitute \(x = 3\):

\[
y = 1.71(3) + 10.57
\]

I'll calculate this next.For someone employed for 3 weeks, approximately 15.71 cars are estimated to be inspected.

Would you like further details or have any questions? Here are some related questions you might find helpful:

1. How does the correlation coefficient influence predictions?
2. What happens to the accuracy of predictions when the correlation coefficient is low?
3. Can the best-fitted line be used to predict values far outside the data range?
4. How is linear regression used in real-life data analysis?
5. What are the limitations of using linear regression for this data?

**Tip:** Always check if your data follows a linear pattern before applying linear regression to ensure valid predictions.

The following sample data shows the dependence between # of weeks employed and # of inspected cars per day. (a) Fill out the table, (b) calculate the correlation coefficient, (c) find the best fitted line, (d) estimate how many cars will be inspected by someone employed for 3 weeks.

Learn how to analyze data for the number of weeks employed vs the number of cars inspected using correlation and linear regression. Calculate the correlation coefficient and determine the best-fitted line for predictive analysis.

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