To solve for the standard deviation of the difference [tex]\( D = X - Y \)[/tex] where [tex]\( X \)[/tex] is the number of frogs and [tex]\( Y \)[/tex] is the number of koi in a pond, follow these steps:
1. Identify the given statistics:
- The mean of [tex]\( X \)[/tex] (number of frogs) is [tex]\( \mu_X = 28 \)[/tex].
- The standard deviation of [tex]\( X \)[/tex] is [tex]\( \sigma_X = 2.7 \)[/tex].
- The mean of [tex]\( Y \)[/tex] (number of koi) is [tex]\( \mu_Y = 15 \)[/tex].
- The standard deviation of [tex]\( Y \)[/tex] is [tex]\( \sigma_Y = 1.6 \)[/tex].
2. Understand the question:
- We need to find the standard deviation of the difference [tex]\( D = X - Y \)[/tex]. When dealing with independent random variables, the variance of their difference is the sum of their variances.
- This implies that the variance of [tex]\( D \)[/tex], denoted [tex]\( \sigma_D^2 \)[/tex], is given by:
[tex]\[
\sigma_D^2 = \sigma_X^2 + \sigma_Y^2
\][/tex]
3. Calculate the variance of the difference:
- Plug in the values of the standard deviations of [tex]\( X \)[/tex] and [tex]\( Y \)[/tex]:
[tex]\[
\sigma_X^2 = (2.7)^2 = 7.29
\][/tex]
[tex]\[
\sigma_Y^2 = (1.6)^2 = 2.56
\][/tex]
Therefore,
[tex]\[
\sigma_D^2 = 7.29 + 2.56 = 9.85
\][/tex]
4. Calculate the standard deviation of the difference:
- Taking the square root of the variance gives the standard deviation:
[tex]\[
\sigma_D = \sqrt{\sigma_D^2} = \sqrt{9.85} \approx 3.1384709652950433
\][/tex]
5. Interpret the result:
- The standard deviation of the difference [tex]\( D = X - Y \)[/tex] is approximately 3.1.
Thus, the correct interpretation and calculation is:
[tex]\[
\dot{\sigma}_0 = 3.1; \text{ this pond can expect the difference of frogs and koi to vary by approximately 3.1 from the mean.}
\][/tex]
So, the correct answer choice is:
[tex]\[
\boxed{\dot{\sigma}_0=3.1 \text{; this pond can expect the difference of frogs and koi to vary by approximately 3.1 from the mean.}}
\][/tex]
