Question

A company sells name plates for doors. An employee takes a random sample of name plates, measures the length of each name (in inches), and counts the number of letters in each name. She generates the scatterplot and trend line equation shown below.



Name Plates
A graph has letters on the x-axis and length (inches) on the y-axis. A trend line goes through points (2, 4) and (9, 16). The regression equation is y = 1.75 x + 0.5.


A customer would like the company to build him a name plate that is 22 inches long. What is the best estimate of the number of letters the company can have on the name plate?
9
10
12
39

Answer 1

To estimate the number of letters that would correspond to a name plate length of 22 inches, we can use the trend line equation provided, which is:

\[ y = 1.75x + 0.5 \]

In this equation:
- \( y \) represents the length of the name in inches
- \( x \) represents the number of letters in the name

To find the estimated number of letters for a name plate length of 22 inches, we can set \( y = 22 \) and solve for \( x \):

\[
22 = 1.75x + 0.5
\]

Subtract 0.5 from both sides:

\[
22 - 0.5 = 1.75x
\]

\[
21.5 = 1.75x
\]

Now divide both sides by 1.75 to solve for \( x \):

\[
x = \frac{21.5}{1.75}
\]

Calculating this gives:

\[
x = 12.2857
\]

Since the number of letters must be a whole number, we can round this to the nearest whole number, which is 12.

Thus, the best estimate of the number of letters the company can have on the name plate is **12**.