Chapter 14, Problem 14.1.1RE

To determine

To select: A random sample of 8 storms by using random numbers.

To find: The average maximum wind speed.

To compare: The results with the population mean.

Answer to Problem 14.1.1RE

The random sample of 8 storms by using random numbers is:

Serial No.	Name
01	Arlene
06	Franklin
12	Lee
13	Maria
14	Nate
18	Stan
19	Unnamed
27	Epsilon

The average maximum wind speed of the sample is 75.

By comparing the sample mean of 75 with the population mean of 87.5, it is clear that there is some difference between the sample mean and the population mean.

Explanation of Solution

Given info:

In the 2005 Atlantic hurricane season the maximum wind speed and classifications for different storms are given in the data.

Calculation:

Answers will vary. One of the possible answers is given below:

First arrange the data according to the serial number,

Serial No.	Name	Max. wind	Classification
1	Arlene	70	Storm
2	Bret	40	S
3	Cindy	75	Hurricane
4	Dennis	150	H
5	Emily	160	H
6	Franklin	70	S
7	Gert	45	S
8	Harvey	65	S
9	Irene	105	H
10	Jose	50	S
11	Katrina	175	H
12	Lee	40	S
13	Maria	115	H
14	Nate	90	H
15	Ophelia	85	H
16	Philippe	70	H
17	Rita	175	H
18	Stan	80	H
19	Unnamed	50	S
20	Tammy	50	S
21	Vince	75	H
22	Wilma	175	H
23	Alpha	50	S
24	Beta	115	H
25	Gamma	55	S
26	Delta	70	S
27	Epsilon	85	H
28	Zeta	65	S

$\text{S}=\text{Storm, H}=\text{Hurricane}$

From the data select a random sample of 8 storms by using random numbers.

Procedure for selecting eight random samples:

• From the figure 14.1(Table of random numbers), select a starting point.
• Here, the selected starting point is ‘18’ which is present in third row and first column. There are 28 names of storms. Hence, select the random numbers in the range of ‘1’ to ‘28. Avoid repetition of numbers and 00.
• The random numbers starting from ‘18’ are:

  18, 19, 14, 29, 01, 55, 84, 62, 66, 48, 94, 100, 46, 77, 81, 40, 41, 52, 13, 82, 57, 12, 27, 75, 95, 62, 57, 13, 31, 06.

• Select 18, 19, 14 which are in the range (01 to 28).
• Next number 29 is not selected because it is not in the range (1 to 28).
• And select 01.
• Repeat the same until eight random numbers are obtained.
• Thus, the random numbers are and corresponding names of the wind are,

Serial No.	Name
01	Arlene
06	Franklin
12	Lee
13	Maria
14	Nate
18	Stan
19	Unnamed
27	Epsilon

The average maximum wind of above selected sample is calculated by using the following formula.

$\begin{array}{c}\text{Sample mean of Max}.\text{wind speed} =\frac{\text{Sum}\text{of}\text{all}\text{sample}\text{values}\text{of}\text{Max}\text{.wind}}{\text{Number}\text{of}\text{samples}}\\ =\frac{70+70+40+115+90+80+50+85}{8}\\ =\frac{600}{8}\\ =75.\end{array}$

Thus, the sample mean is 75.

The population mean of maximum wind is,

$\begin{array}{c}\text{Population mean} =\frac{\text{Sum}\text{of}\text{all}\text{values}\text{of}\text{Max}\text{.wind}}{\text{Number}\text{of}\text{obervations}}\\ =\frac{70+40+75+150+\mathrm{...}+85+65}{28}\\ =\frac{2,450}{28}\\ =\mathrm{87.5.}\end{array}$

Thus, the population mean is 87.5.

Comparison of means:

The sample mean of maximum wind speed is 75 and the population mean of maximum wind speed is 87.5.

That implies that the sample mean is smaller than the population mean. It is clear that there is some difference between the sample mean and the population mean.