Chapter 4.1, Problem 109AYU

(a)

To determine

To draw: a scatter diagram of the given data.

Answer to Problem 109AYU

  

Explanation of Solution

Given:

Decade, x	Major Hurricanes Striking United States, H
1921-1930, 1	5
1931-1940, 2	8
1941-1950, 3	10
1951-1960, 4	8
1961-1970, 5	6
1971-1980, 6	4
1981-1990, 7	5
1991-2000, 8	5

Calculation:

Figure 1 shows the scatter diagram. A cubic relation may exist between the two variables x and H.

Figure 1

  

Conclusion:

Therefore, the scatter diagram is drawn.

(b)

To determine

To find: the cubic function of best fit

Answer to Problem 109AYU

The cubic equation of best fit to the data is

  $H\left(x\right)=0.1591x^3-2.32x^2+9.33x-2.214$

Explanation of Solution

Calculation:

Upon executing the CUBIC Regression Program, obtained results are shown in Figure 2. The output that the utility provides shows us the equation $y=a{x}^3+b{x}^3+cx+d$ .

The cubic equation of best fit to the data is

  $H\left(x\right)=0.1591x^3-2.32x^2+9.33x-2.214$

Figure 2

  

Conclusion:

The cubic equation of best fit to the data is

  $H\left(x\right)=0.1591x^3-2.32x^2+9.33x-2.214$

(c)

To determine

To predict: the number of major hurricanes that struck the United States

Answer to Problem 109AYU

The no. of major hurricane is 6.

Explanation of Solution

Calculation:

The cubic equation of best fit to the data is $H\left(x\right)=0.1591x^3-2.32x^2+9.33x-2.214$

In Figure 1, point $x=5$ corresponding to 1961 through1970.

The no. of major hurricane that struck the United States between 1961 through 1970 can be calculated as

  $\begin{array}{c}H\left(x\right)=0.1591x^3-2.32x^2+9.33x-2.214\\ H\left(5\right)=0.1591{\left(5\right)}^3-2.32{\left(5\right)}^2+9.33\left(5\right)-2.214\\ =6.3235\\ \cong 6\end{array}$

The no. of major hurricane is 6.

Conclusion:

The no. of major hurricane is 6.

(d)

To determine

To draw: a graph of the cubic function of best fit

Answer to Problem 109AYU

  

Explanation of Solution

Calculation:

Figure 3 shows the graph of the cubic function of best fit on the scatter diagram. The function fits the data reasonably well. The cubic function of best fit passes through all the point very nearly.

Figure 3

  

Conclusion:

Therefore, the required graph is drawn.

(e)

To determine

To predict: the number of major hurricanes and explain whether the result appear to agree with what the scientists believe.

Answer to Problem 109AYU

The no. of major hurricane is 10.

The result appears to agree with what these scientists believe.

Explanation of Solution

Calculation:

In Figure 1 point $x=9$ corresponding to 2001 through 2009.

The no. of major hurricane that struck the United States between 2001 and 2009 can be calculated as

  $\begin{array}{c}H\left(x\right)=0.1591x^3-2.32x^2+9.33x-2.214\\ H\left(9\right)=0.1591\left(9^3\right)-2.32\left(9^2\right)+9.33\left(9\right)-2.214\\ =9.8199\\ \cong 10\end{array}$

Therefore the no. of major hurricane that struck the United States between 2001 through 2009 is equal to 10.

From Figure 3 at the point $x=9$ , H is approximately near to 10 and also the scatter diagram and cubic equation of best fit appear very nearby.

So, the result appears to agree with what these scientists believe.

Conclusion:

Therefore the no. of major hurricane is 10 and the result appears to agree with what these scientists believe.

(f)

To determine

To find: whether the given statement support or contradict the prediction

Answer to Problem 109AYU

The prediction supports our result in part €

Explanation of Solution

Given:

From 2001 through 2010,10 major hurricanes struck the United States.

From 2001 through 2009, 10 major hurricanes struck the United States.

Our prediction in part (e) shows that the no. of major hurricane in United States is equal to 10.

The answer is yes, this prediction supports our result in part (e).

Conclusion:

Therefore, theprediction supports our result in part €