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To divide: The 28 storms into 4 subgroups.
To select: A random sample of 3 storms from each group.
To compute: The means for maximum wind speeds.
To compare: The means to the population
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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:
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 |
Procedure for dividing the 28 storms into 4 subgroups:
Step1:
Divide the population into two groups. From the 28 storms, divide the observations according to classification- Storm and Hurricane. There are 15 Hurricanes and 13 Storms:
Group 1 (H) | Group 2 (S) | ||
Name | Max. wind | Name | Max. wind |
Cindy | 75 | Arlene | 70 |
Dennis | 150 | Bret | 40 |
Emily | 160 | Franklin | 70 |
Irene | 105 | Gert | 45 |
Katrina | 175 | Harvey | 65 |
Maria | 115 | Jose | 50 |
Nate | 90 | Lee | 40 |
Ophelia | 85 | Unnamed | 50 |
Philippe | 70 | Tammy | 50 |
Rita | 175 | Alpha | 50 |
Stan | 80 | Gamma | 55 |
Vince | 75 | Delta | 70 |
Wilma | 175 | Zeta | 65 |
Beta | 115 | ||
Epsilon | 85 |
Step2:
Divide each group into two subgroups, based on the
For Group 1 (H), the median value of max. wind is 105. Thus, the 1st subgroup will contain all storms with max. wind less than or equal to 105 and the 2nd subgroup will contain all storms with max. wind greater than 105.
For Group 2 (S), the median value of max. wind is 50. Thus, the 3rd subgroup will contain all storms with max. wind less than or equal to 50 and the 4th subgroup will contain all storms with max. wind greater than 50.
The subgroups are:
SNO | Subgroup1 | Subgroup2 | Subgroup3 | Subgroup4 |
1 | Cindy | Dennis | Bret | Arlene |
2 | Irene | Emily | Gert | Franklin |
3 | Nate | Katrina | Jose | Harvey |
4 | Ophelia | Maria | Lee | Gamma |
5 | Philippe | Rita | Unnamed | Delta |
6 | Stan | Wilma | Tammy | Zeta |
7 | Vince | Beta | Alpha | |
8 | Epsilon |
From the s select a random sample of 3 storms by using random numbers.
Procedure for selecting 3 random samples from subgroup 1:
- From the figure 14.1(Table of random numbers), select a starting point.
- Here, the selected starting point is ‘7’ which is present in 1st row and 1st column. There are 8 names of storms in subgroup 1. Hence, select the random numbers in the
range of ‘1’ to ‘8. Avoid repetition of numbers and 0. - The random numbers starting from ‘7’ are:
7, 2, 1.
Thus, the random numbers are and corresponding names of the wind are,
Serial No. | Name |
7 | Vince |
2 | Irene |
1 | Cindy |
The average maximum wind of above selected sample is calculated by using the following formula.
Thus, the sample mean for subgroup 1 is 85.
Procedure for selecting 3 random samples from subgroup 2:
- From the figure 14.1(Table of random numbers), select a starting point.
- Here, the selected starting point is ‘4’ which is present in 1st row and 2nd column. There are 7 names of storms in subgroup 2. Hence, select the random numbers in the range of ‘1’ to ‘7’. Avoid repetition of numbers and 0.
- The random numbers starting from ‘4’ are:
4, 5, 1.
Thus, the random numbers are and corresponding names of the wind are,
Serial No. | Name |
4 | Maria |
5 | Rita |
1 | Dennis |
The average maximum wind of above selected sample is calculated by using the following formula.
Thus, the sample mean for subgroup 2 is 146.67.
Procedure for selecting 3 random samples from subgroup 3:
- From the figure 14.1(Table of random numbers), select a starting point.
- Here, the selected starting point is ‘7’ which is present in 5th row and 13th column. There are 7 names of storms in subgroup 3. Hence, select the random numbers in the range of ‘1’ to ‘7’. Avoid repetition of numbers and 0.
- The random numbers starting from ‘7’ are:
7, 4, 2.
Thus, the random numbers are and corresponding names of the wind are,
Serial No. | Name |
4 | Alpha |
5 | Lee |
1 | Gret |
The average maximum wind of above selected sample is calculated by using the following formula.
Thus, the sample mean for subgroup 3 is 45.
Procedure for selecting 3 random samples from subgroup 4:
- From the figure 14.1(Table of random numbers), select a starting point.
- Here, the selected starting point is ‘5’ which is present in 6th row and 9th column. There are 6 names of storms in subgroup 4. Hence, select the random numbers in the range of ‘1’ to ‘6’. Avoid repetition of numbers and 0.
- The random numbers starting from ‘5’ are:
5, 2, 6.
Thus, the random numbers are and corresponding names of the wind are,
Serial No. | Name |
5 | Delta |
2 | Franklin |
6 | Zeta |
The average maximum wind of above selected sample is calculated by using the following formula.
Thus, the sample mean for subgroup 4 is 68.3.
The population mean of maximum wind is,
Thus, the population mean is 87.5.
Comparison of means:
The sample mean for subgroup 1 is 85 and the population mean of maximum wind speed is 87.5.
The sample mean for subgroup 2 is 146.67 and the population mean of maximum wind speed is 87.5.
The sample mean for subgroup 3 is 45 and the population mean of maximum wind speed is 87.5.
The sample mean for subgroup 4 is 68.3 and the population mean of maximum wind speed is 87.5.
That implies that the sample means are smaller than the population mean for subgroup 3 an 4 whereas the sample mean is greater than the population mean for subgroup 2. But, for subgroup 1 sample mean is slightly smaller than population mean.
It is clear that there is some difference between the sample mean and the population mean.
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Chapter 14 Solutions
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