Question

find the equation for the least squares regression line of the data described below.
farid is a math teacher at a large school. he wonders if his test problems are too wordy. farid is curious whether the wordiness is affecting student performance. for the last several tests, farid computes the average number of words in each question, x, as well as the average percentage scores on the tests, y.

average question length	19	21	25	30	31

round your answers to the nearest thousandth.
y = \square x + \square

Explanation:

Step 1: Identify variables and data points

Let \( x \) be the average question length (independent variable) and \( y \) be the average student score (dependent variable). The data points are:
\( (x_1, y_1) = (19, 84) \), \( (x_2, y_2) = (21, 82) \), \( (x_3, y_3) = (25, 50) \), \( (x_4, y_4) = (30, 65) \), \( (x_5, y_5) = (31, 55) \).

Step 2: Calculate \( \bar{x} \) and \( \bar{y} \)

\( \bar{x} = \frac{19 + 21 + 25 + 30 + 31}{5} = \frac{126}{5} = 25.2 \)
\( \bar{y} = \frac{84 + 82 + 50 + 65 + 55}{5} = \frac{336}{5} = 67.2 \)

Step 3: Calculate \( \sum (x_i - \bar{x})(y_i - \bar{y}) \) and \( \sum (x_i - \bar{x})^2 \)

• For \( (19, 84) \): \( (19 - 25.2)(84 - 67.2) = (-6.2)(16.8) = -104.16 \); \( (19 - 25.2)^2 = 38.44 \)
• For \( (21, 82) \): \( (21 - 25.2)(82 - 67.2) = (-4.2)(14.8) = -62.16 \); \( (21 - 25.2)^2 = 17.64 \)
• For \( (25, 50) \): \( (25 - 25.2)(50 - 67.2) = (-0.2)(-17.2) = 3.44 \); \( (25 - 25.2)^2 = 0.04 \)
• For \( (30, 65) \): \( (30 - 25.2)(65 - 67.2) = (4.8)(-2.2) = -10.56 \); \( (30 - 25.2)^2 = 23.04 \)
• For \( (31, 55) \): \( (31 - 25.2)(55 - 67.2) = (5.8)(-12.2) = -70.76 \); \( (31 - 25.2)^2 = 33.64 \)

Summing these:
\( \sum (x_i - \bar{x})(y_i - \bar{y}) = -104.16 - 62.16 + 3.44 - 10.56 - 70.76 = -244.2 \)
\( \sum (x_i - \bar{x})^2 = 38.44 + 17.64 + 0.04 + 23.04 + 33.64 = 112.8 \)

Step 4: Calculate slope \( m \)

\( m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} = \frac{-244.2}{112.8} \approx -2.165 \)

Step 5: Calculate intercept \( b \)

\( b = \bar{y} - m\bar{x} = 67.2 - (-2.165)(25.2) \approx 67.2 + 54.558 \approx 121.758 \)

Step 6: Form the equation

The least squares regression line is \( y = mx + b \approx -2.165x + 121.758 \). Rounding to the nearest thousandth, \( m \approx -2.165 \) and \( b \approx 121.758 \).

Answer:

\( y = -2.165x + 121.758 \) (or \( y \approx -2.165x + 121.758 \))