Question

the table represents the results of survey that asked people their age and the percent of their budget that they spend on entertainment.
age | % of budget for entertainment
27 | 10
34 | 7
68 | 5
25 | 9.5
41 | 6
39 | 9
52 | 4
70 | 2
50 | 6.8
write the function that best represents this data (do not enter spaces and round to nearest hundredths place).
f(x) =

Explanation:

Step1: Identify the type of model

We can assume a linear or exponential model, but looking at the data (age vs. % of budget for entertainment), it seems to have a negative correlation, so we can try a linear regression model. Let \( x \) be age and \( y \) be % of budget. The linear model is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step2: Calculate the slope \( m \)

First, we need the mean of \( x \) (\( \bar{x} \)) and mean of \( y \) (\( \bar{y} \)).
The data points are:
\( (27,10), (34,7), (68,5), (25,9.5), (41,6), (39,9), (52,4), (70,2), (50,6.8) \)

Number of points \( n = 9 \)

Calculate \( \sum x \): \( 27 + 34 + 68 + 25 + 41 + 39 + 52 + 70 + 50 = 406 \)

\( \bar{x} = \frac{406}{9} \approx 45.11 \)

Calculate \( \sum y \): \( 10 + 7 + 5 + 9.5 + 6 + 9 + 4 + 2 + 6.8 = 59.3 \)

\( \bar{y} = \frac{59.3}{9} \approx 6.59 \)

Calculate \( \sum (x_i - \bar{x})(y_i - \bar{y}) \) and \( \sum (x_i - \bar{x})^2 \)

For each point:

• \( (27,10) \): \( (27 - 45.11)(10 - 6.59) \approx (-18.11)(3.41) \approx -61.75 \); \( (27 - 45.11)^2 \approx 328.0 \)
• \( (34,7) \): \( (34 - 45.11)(7 - 6.59) \approx (-11.11)(0.41) \approx -4.55 \); \( (34 - 45.11)^2 \approx 123.4 \)
• \( (68,5) \): \( (68 - 45.11)(5 - 6.59) \approx (22.89)(-1.59) \approx -36.40 \); \( (68 - 45.11)^2 \approx 524.0 \)
• \( (25,9.5) \): \( (25 - 45.11)(9.5 - 6.59) \approx (-20.11)(2.91) \approx -58.52 \); \( (25 - 45.11)^2 \approx 404.4 \)
• \( (41,6) \): \( (41 - 45.11)(6 - 6.59) \approx (-4.11)(-0.59) \approx 2.42 \); \( (41 - 45.11)^2 \approx 16.9 \)
• \( (39,9) \): \( (39 - 45.11)(9 - 6.59) \approx (-6.11)(2.41) \approx -14.72 \); \( (39 - 45.11)^2 \approx 37.3 \)
• \( (52,4) \): \( (52 - 45.11)(4 - 6.59) \approx (6.89)(-2.59) \approx -17.85 \); \( (52 - 45.11)^2 \approx 47.4 \)
• \( (70,2) \): \( (70 - 45.11)(2 - 6.59) \approx (24.89)(-4.59) \approx -114.25 \); \( (70 - 45.11)^2 \approx 619.5 \)
• \( (50,6.8) \): \( (50 - 45.11)(6.8 - 6.59) \approx (4.89)(0.21) \approx 1.03 \); \( (50 - 45.11)^2 \approx 23.9 \)

Sum of \( (x_i - \bar{x})(y_i - \bar{y}) \): \( -61.75 -4.55 -36.40 -58.52 +2.42 -14.72 -17.85 -114.25 +1.03 \approx -304.59 \)

Sum of \( (x_i - \bar{x})^2 \): \( 328.0 + 123.4 + 524.0 + 404.4 + 16.9 + 37.3 + 47.4 + 619.5 + 23.9 \approx 2124.8 \)

Slope \( m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \approx \frac{-304.59}{2124.8} \approx -0.143 \)

Step3: Calculate the y-intercept \( b \)

Using \( \bar{y} = m\bar{x} + b \)

\( 6.59 = -0.143(45.11) + b \)

\( 6.59 = -6.45 + b \)

\( b = 6.59 + 6.45 = 13.04 \)

Step4: Write the function

So the linear function is \( f(x) = -0.14x + 13.04 \) (rounded to nearest hundredths)

Answer:

\( f(x)=-0.14x + 13.04 \)