Question

1. the table below gives the number of hours spent studying for a science exam (x) and the final exam grade (y).

x | 2 | 5 | 1 | 0 | 4 | 2 | 3
y | 77 | 92 | 70 | 63 | 90 | 75 | 84
a) find the correlation coefficient, r, for the table below and describe the relationship between x and y.
b) write the equation of the line of best fit.
c) predict the exam grade of a students that studied for 6 hours.

1. true /false the age of a child and his/her shoe size is an example of causation.
2. true/false the number of miles driven and the amount of gas used is an example of causation.

Explanation:

Part a)

Step 1: Calculate necessary sums

First, we need to find \( n \), \( \sum x \), \( \sum y \), \( \sum xy \), \( \sum x^2 \), and \( \sum y^2 \).
Given \( n = 7 \) (number of data points).

• \( \sum x = 2 + 5 + 1 + 0 + 4 + 2 + 3 = 17 \)
• \( \sum y = 77 + 92 + 70 + 63 + 90 + 75 + 84 = 551 \)
• \( \sum xy = (2\times77)+(5\times92)+(1\times70)+(0\times63)+(4\times90)+(2\times75)+(3\times84) = 154 + 460 + 70 + 0 + 360 + 150 + 252 = 1446 \)
• \( \sum x^2 = 2^2 + 5^2 + 1^2 + 0^2 + 4^2 + 2^2 + 3^2 = 4 + 25 + 1 + 0 + 16 + 4 + 9 = 59 \)
• \( \sum y^2 = 77^2 + 92^2 + 70^2 + 63^2 + 90^2 + 75^2 + 84^2 = 5929 + 8464 + 4900 + 3969 + 8100 + 5625 + 7056 = 44043 \)

Step 2: Use correlation coefficient formula

The formula for the correlation coefficient \( r \) is:

$$ r=\frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} $$

Substitute the values:

• Numerator: \( 7\times1446 - 17\times551 = 10122 - 9367 = 755 \)
• Denominator part 1: \( 7\times59 - 17^2 = 413 - 289 = 124 \)
• Denominator part 2: \( 7\times44043 - 551^2 = 308301 - 303601 = 4700 \)
• Denominator: \( \sqrt{124\times4700}=\sqrt{582800}\approx763.42 \)
• Then \( r=\frac{755}{763.42}\approx0.989 \)

Since \( r \approx 0.989 \), which is close to 1, there is a strong positive linear relationship between \( x \) (hours studied) and \( y \) (exam grade).

Part b)

Step 1: Calculate slope (\( m \)) and y - intercept (\( b \)) of the line of best fit \( y = mx + b \)

The formula for the slope \( m \) is:

$$ m=\frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2} $$

We already know from part a that \( n\sum xy - (\sum x)(\sum y)=755 \) and \( n\sum x^2 - (\sum x)^2 = 124 \), so \( m=\frac{755}{124}\approx6.09 \)

The formula for the y - intercept \( b \) is:

$$ b=\frac{\sum y - m\sum x}{n} $$

Substitute \( \sum y = 551 \), \( m\approx6.09 \), \( \sum x = 17 \), \( n = 7 \)
\( b=\frac{551 - 6.09\times17}{7}=\frac{551 - 103.53}{7}=\frac{447.47}{7}\approx63.92 \)

So the equation of the line of best fit is \( y\approx6.09x + 63.92 \)

Part c)

Step 1: Use the line of best fit equation

We have the equation \( y = 6.09x + 63.92 \). Substitute \( x = 6 \) (hours studied)
\( y = 6.09\times6+63.92 = 36.54 + 63.92 = 100.46 \approx 100 \) (we can round to a reasonable grade value)

Part 9)

Answer:

False

Part 10)