Chapter 9.1, Problem 15P

(a)

To determine

Draw a scatter plot and the line that best fits the data.

Answer to Problem 15P

The scatter plot with the line that best fits the data is,

Explanation of Solution

Calculation:

The variable x denotes lowest pressure (in millibars) as a cyclone approaches and variable y denotes the maximum wind speed (in miles per hour) of the cyclone.

Step by step procedure to obtain scatter plot using MINITAB software is given below:

• Choose Graph > Scatterplot.
• Choose With regression. Click OK.
• In Y variables, enter the column of x.
• In X variables, enter the column of y.
• Click OK.

(b)

To determine

Identify whether the correlation is low, moderate, or strong.

Identify whether the direction of correlation is positive or negative.

Answer to Problem 15P

The correlation is strong.

The direction of correlation is negative.

Explanation of Solution

From the scatter diagram of part (a), it can be observed that, all the data points are closer to the straight line indicating that correlation is strong. Also, as the value of x increases the value of y also has decreased, indicating that the two variables are in opposite direction. That is, the variables are negatively related.

Hence, the correlation is strong and direction of correlation is negative.

(c)

To determine

Verify the values of ${\displaystyle \sum x}$, ${\displaystyle \sum x^2}$, ${\displaystyle \sum y}$, ${\displaystyle \sum y^2}$ and ${\displaystyle \sum xy}$ using the calculator.

Find the value of r.

Explain whether y should tend to increase or decrease, as x increases based on value of r.

Answer to Problem 15P

The value of r is –0.990.

The variable y should tend to decrease, as x increases based on value of r.

Explanation of Solution

Calculation:

The formula for correlation coefficient is,

$r=\frac{n{\displaystyle \sum xy}-\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}{\sqrt{n{\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2}\sqrt{n{\displaystyle \sum y^2}-{\left({\displaystyle \sum y}\right)}^2}}$

In the formula, n is the sample size.

Use Ti 83 calculator to find the sums of x, y, and xy as follows:

• Select STAT > Edit > Enter the values of x and y as L1 and L2.
• Click $\boxed{2^{\text{nd}}}$ button; take the arrow to the MATH menu, and then press ‘5’ numbered key.
• Click $\boxed{2^{\text{nd}}}$ button; then press ‘1’ numbered key to get L1, and close the ‘)’ bracket.
• Click $\boxed{2^{\text{nd}}}$ button; take the arrow to the MATH menu, and then press ‘5’ numbered key.
• Click $\boxed{2^{\text{nd}}}$ button; then press ‘2’ numbered key to get L2, and close the ‘)’ bracket.
• Click $\boxed{2^{\text{nd}}}$ button; take the arrow to the MATH menu, and then press ‘5’ numbered key.
• Click $\boxed{2^{\text{nd}}}$ button; then press ‘1*2’ numbered key to get L1*L2, and close the ‘)’ bracket.
• Click Enter.

Output using Ti 83 calculator is given below:

From the Ti 83 calculator output, the value of ${\displaystyle \sum x}=5,823$, ${\displaystyle \sum y}=580$ and ${\displaystyle \sum xy}=556,315$

Use Ti 83 calculator to find the sums of $x^2$ and $y^2$ as follows:

• Select STAT > Edit > Enter the values of x and y as L1 and L2.
• Click $\boxed{2^{\text{nd}}}$ button; take the arrow to the MATH menu, and then press ‘5’ numbered key.
• Click $\boxed{2^{\text{nd}}}$ button; then press ‘1*1’ numbered key to get L1*L1, and close the ‘)’ bracket.
• Click $\boxed{2^{\text{nd}}}$ button; take the arrow to the MATH menu, and then press ‘5’ numbered key.
• Click $\boxed{2^{\text{nd}}}$ button; then press ‘2*2’ numbered key to get L2*L2, and close the ‘)’ bracket.
• Click Enter.

Output using Ti 83 calculator is given below:

From the Ti 83 calculator output, the value of ${\displaystyle \sum x^2}=5,655,779$ and ${\displaystyle \sum y^2}=65,750$.

Hence, the values are verified using calculator.

The value of correlation r is,

$\begin{array}{c}r=\frac{n{\displaystyle \sum xy}-\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}{\sqrt{n{\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2}\sqrt{n{\displaystyle \sum y^2}-{\left({\displaystyle \sum y}\right)}^2}}\\ =\frac{6\left(556,315\right)-\left(5,823\times 580\right)}{\sqrt{6\left(5,655,779\right)-{\left(5,823\right)}^2}\sqrt{6\left(65,750\right)-{\left(580\right)}^2}}\\ =\frac{3,337,890-3,377,340}{\sqrt{27,345}\sqrt{58,100}}\\ =\frac{-39,450}{165.3632\times 241.0394}\\ =\frac{-39,450}{39,859.0465}\\ =-0.990\end{array}$

Hence, the value of r is –0.990.

The value of correlation coefficient is negative indicating that as the variable x increases the variable y decreases. The value r (= –0.990) is very close to –1, indicating that there is strong relationship between variables x and y. This shows that, when x increases the value of y decreases.

Hence, the variable y should tend to decrease, as x increases based on value of r.