Chapter 3.2, Problem 69E

To determine

To find: the accurate of DR.G’s forecasts and the number of tropical storms and the effect of the disastrous

Answer to Problem 69E

DR.G’s forecasts are not very accurate.

There are 16tropical storms

The effect of the disastrous has a little effect.

Explanation of Solution

Given:

year	forecast	actual	year	forecast	actual
1984	10	12	1997	11	7
1985	11	11	1998	10	14
1986	8	6	1999	14	12
1987	8	7	2000	12	14
1988	11	12	2001	12	15
1989	7	11	2002	11	12
1990	11	14	2003	14	16
1991	8	8	2004	14	14
1992	8	6	2005	15	27
1993	11	8	2006	17	9
1994	9	7	2007	17	14
1995	12	19	2008	15	16
1996	10	13

Calculation:

Let's start by making a scatterplot with forecast (X) as a explanatory variable and actual number of storm as the response variable. If the graph shows a linear form, fit a least-squares line to the data. Then make a residual plot. The residuals $r^2$ and s will tell how well the line fits the data and how large our prediction errors will be.

The below figure shows a scatterplot of the data. The scatterplot shows a positive association That is, higher number of forecast tends to have higher number of actual storms. The overall pattern is moderately linear (a calculator gives r = 0.5478). There is one outlier on the scatterplot in the Y direction.

  

Using the MINITAB, the regression equation is shown below

  

The least-square equation is $\widehat{Y}=1.69+0.915X$

Using the MINITAB, the residual plot is shown below:

  

The slope suggests that for every added forecast, actual forecast increased by 0.915. Y intercept is the value of Y when X = 0, here if forecast is O then number of actual storm will be 1. From the residual plot. The graph shows a fairly "random" scatter points around the "residual = 0" line with one very large positive residual (point for year 2005). Most of the prediction errors (residuals) are 10 points or fewer on the forecast scale. It is calculated the standard error of the residuals to be s = 3.9983. This is roughly the size of an average prediction error using the regression line. Since $r^2$ = 0.3001, 30.01% of the variation in actual number of storm is accounted for by the least-squares regression of actual number of storms on forecast. That leaves 69.99% of the variation in actual number of storms unaccounted for by the linear relationship for these data.

Use the equation

Actual storm = 1.69+0.915(forecast)

To predict an actual number of storm from the forecast. Our predictions may not be very accurate, though. It is hesitated to use this model to make predictions. If Dr. Gray's preseason forecast call is 16 then;

Actual storm $\begin{array}{l}=1.69+0.915\left(\text{forecast}\right)\\ =1.69+0.915\left(16\right)\\ =16.33\end{array}$

Consider the below figure:

  

The above figure shows the result of removing 2005 season on the correlation and regression line. The graph adds one more regression line calculated after leaving 2005 season. It is clearthat removing this point has little effect on the regression line.

Conclusion:

Therefore,

DR.G’s forecasts are not very accurate.

There are 16tropical storms

The effect of the disastrous has a little effect.