Let's approach each part of the problem step-by-step.
### (b) (i) Show that the mean of [tex]\( v \)[/tex] is 3060
Given:
[tex]\[ y = \frac{v}{100} - 20 \][/tex]
and the mean of [tex]\( y \)[/tex] is 10.6.
To find the mean of [tex]\( v \)[/tex], we need to reverse the transformation from [tex]\( y \)[/tex] back to [tex]\( v \)[/tex].
1. Start with the equation:
[tex]\[ y = \frac{v}{100} - 20 \][/tex]
2. Multiply both sides by 100 to isolate [tex]\( v \)[/tex]:
[tex]\[ 100y = v - 2000 \][/tex]
3. Add 2000 to both sides to solve for [tex]\( v \)[/tex]:
[tex]\[ v = 100y + 2000 \][/tex]
4. Now, substitute the given mean of [tex]\( y \)[/tex] (10.6) into the equation:
[tex]\[ \text{Mean}(v) = 100 \times 10.6 + 2000 \][/tex]
[tex]\[ \text{Mean}(v) = 1060 + 2000 \][/tex]
[tex]\[ \text{Mean}(v) = 3060 \][/tex]
Therefore, the mean of [tex]\( v \)[/tex] is 3060.
### (b) (ii) Find the standard deviation of [tex]\( v \)[/tex]
To find the standard deviation of [tex]\( v \)[/tex], we use the property of linear transformations. If [tex]\( y = ax + b \)[/tex], then:
[tex]\[ \text{Std}(y) = |a| \times \text{Std}(x) \][/tex]
Here:
[tex]\[ y = \frac{v}{100} - 20 \][/tex]
The linear transformation coefficient [tex]\( a \)[/tex] is [tex]\( \frac{1}{100} \)[/tex].
Given the standard deviation of [tex]\( y \)[/tex] is 12.6, we need to determine the standard deviation of [tex]\( v \)[/tex]:
[tex]\[ \text{Std}(v) = |100| \times \text{Std}(y) \][/tex]
Substitute the given standard deviation of [tex]\( y \)[/tex] (12.6) into the equation:
[tex]\[ \text{Std}(v) = 100 \times 12.6 \][/tex]
[tex]\[ \text{Std}(v) = 1260 \][/tex]
Therefore, the standard deviation of [tex]\( v \)[/tex] is 1260.
### (c) Using your knowledge of the large data set:
#### (i) State the units for [tex]\( v \)[/tex]
The Daily Mean Visibility ([tex]\( v \)[/tex]) is measured in:
[tex]\[ \text{meters (m)} \][/tex]
#### (ii) State, giving a reason, a location in the large data set that Edward could not use for his Daily Mean Visibility calculations
In the large data set provided, Perth in Australia could not be used for Daily Mean Visibility calculations. This is because the visibility data for Perth is not available, making it impossible to include it in visibility-related analyses.
