Detailed Answer

a) Pearson's correlation coefficient can be easily calculated by inputting the given data into the GDC. Then, we can see that the value of \( r = 0.938 \)

b) The regression equation after inputting data into GDC is equal to \( y = 37.897x + 169.122 \)

c) The value can be predicted by replacing \( x \) with 25. Then our solution is equal to \( y = 37.897 \times 25 + 169.122 = 1116.547 \)

Detailed Answer

a) (i) \( \overline{x} = 176.29 \ (cm) \)

a) (ii) \( \overline{y} = 625.71 \ (cm) \)

b) From the GDC we can find that \( y = 6.71x - 557.45 \). So, \( m = 6.71\) and \( a = -557.45\)

c) (i) X-intercept is when \( y = 0 \), so:

\[ 0 = 6.71x - 557.45 \]

\[ x = 83.08 \]

c) (ii) Y-intercept is when \( x = 0 \), so:

\[ y = 6.71 * 0 - 557.45 \]

\[ y = - 557.45 \]

d) The value of \( m \) indicates by how much the length of long jump changes, when height increases by 1cm. The value of 6.71 means, that with 1cm increase in height, the long jump length increases by 6.71cm, on average.

Detailed Answer

a) For Pearson's correlation, plugging in the values into the GDC results in \( r = -0.77 \).

b) To calculate the mean of \( x \), we simply sum up all observations and divide by 7, as there are 7 observations in total. As a result, we get \( \frac{995}{7} = 142.12 \). By doing the same thing with \( y \), we get the mean of \( y \) equal to 20.86.

c) For this task, first, the regression equation needs to be calculated. By plugging the values into the GDC we get that \( y = -8.43x + 317.93 \). It needs to be remembered that temperature is our \( x \) variable, and distance covered is \( y \) (the dependent variable), as the temperature outside will affect the distance travelled in a car.

Then, we know that the temperature outside was 21 degrees, meaning that the estimated distance travelled by car can be calculated using the regression equation and it is equal to: \( -8.43 \times 21 + 317.93 = 140.9 \). Finally, we are also told that Jack covered 15km on foot, resulting in the total distance of \( 140.9 + 15 = 155.9 \).

Detailed Answer

a) After inputting the values into the GDC the obtained correlation coefficient is equal to \( r = 0.90 \)

b) From the GDC it can be found that the regression equation is equal to \( y = 7.40x + 1237.85 \)

c) The value of coefficient \( a \) means that an additional week of studying leads to the increase in score by approximately 7.4 points.

d) By plugging 40 as the number of weeks into the regression equation, we get:

\[ y = 7.40 * 40 + 1237.85 \]

\[ y = 1533.85 \]

e) No, predictions like that should not be taken into consideration, as the sample used to create the regression had the minimum value of 22 for the number of weeks.

Detailed Answer

a)

The points given are \( P(0, 2) \) and \( Q(3, 8) \).

The formula for the gradient \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

\[ m = \frac{8 - 2}{3 - 0} = \frac{6}{3} = 2 \]

So, the gradient of \( L1 \) is \( 2 \).

b)

The gradient is \( 2 \).

\[ y - y_1 = m(x - x_1) \]

Using point \( P(0, 2) \):

\[ y - 2 = 2(x - 0) \]

\[ y - 2 = 2x \]

\[ y = 2x + 2 \]

So, the regression equation is \( y = 2x + 2 \).

c)

At the start of the experiment we have that \( h = 0 \):

\[ m = 200e^{2(0)} \]

\[ m = 200e^0 \]

\[ m = 200 \]

So, the mass of Bacteria B at the start of the experiment is \( 200 \) kilograms.

d) To answer this part we have to equalize both functions and find the intersect, which can be done in the GDC:

So, the masses of both types of bacteria will be equal after 3.78 hours.

Detailed Answer

a) By inputting the values into the GDC we get taht the correlation coefficient is equal to \( r = 0.94 \).

b) From the GDC it can be found that the regression equation is equal to \( y = 0.20x + 8.27 \).

c) By flipping the independent and dependent variables, we get that the regression equation is \( x = 4.33y -32.50 \).

d) We are trying to predict the age, so the regression equation from part (c) should be used:

\[ x = 4.33 * 13.7 - 32.50 \]

\[ x = 26.82 \]

Detailed Answer

a) To find the two regression equations, the respective values need to be plugged into GDC. Then, we get that for boys the regression equation is: \( y = 3.10x + 59.67 \), and for girls it is: \( y = 5.23x + 47.31 \)

b) To find the answer to this part the x-coefficients need to be analyzed for both regressions. For boys, the x-coefficient of 3.10 means that an additional hour of studying leads to an estimated 3.10 increase in the exam score. For girls, the x-coefficient of 5.23 means that an additional hour of studying on average increases their score by 5.23. So, girls benefit more from an additional hour compared to boys.

c) To find the intersection of those two equations we have to equate them. Thus, we get the following system of two equations:

\[ (1) \quad y = 3.10x + 59.67 \]

\[ (2) \quad y = 5.23x + 47.31 \]

Then, by equating them:

\[ 3.10x + 59.67 = 5.23x + 47.31 \]

\[ 12.36 = 2.13x \]

\[ x = 5.80 \]

\[ y = 5.23 * 5.80 + 47.31 = 77.64 \]

So the intersection point is at \( (5.80, 77.64) \).

Detailed Answer

a) (i)

\[ W = 0.3 \times 200 + 2000 \]

\[ W = 60 + 2000 \]

\[ W = 2060 \]

a) (ii)

\[ C = 3.5 \times 50 - 100 \]

\[ C = 175 - 100 \]

\[ C = 75 \]

b)

Using the wood production equation for 500 mm rainfall:

\[ W = 0.3 \times 500 + 2000 \]

\[ W = 150 + 2000 \]

\[ W = 2150 \]

Now, we calculate chairs production:

\[ C = 3.5 \times 2150 - 100 \]

\[ C = 7425 \]

c)

Profit per chair: \( \$25 \)

Cost of wood per kg: \( \$5 \)

Profit formula: \( Profit = 25C - 5w \)

Substitute \( C = 7425 \) and \( w = 2150 \):

\[ Profit = 25 \times 7425 - 5 \times 2150 \]

\[ Profit = 174875 \]

As the profits are smaller than \( \$200,000 \), they won't be able to purchase the new machine in May.

d)

Profit formula: \( Profit = 25C - 5w \)

Given \( C = 3.5w - 100 \), substitute into the profit formula:

\[ Profit = 25(3.5w - 100) - 5w \]

\[ Profit = 87.5w - 2500 - 5w \]

\[ Profit = 82.5w - 2500 \]

Therefore, the formula for Wooden Freaks' profits as a function of \( w \) is \( 82.5w - 2500 \).