Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Ask your questions and receive detailed answers from professionals with extensive experience in various fields. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
To calculate the standard deviation of the difference [tex]\( D = X - Y \)[/tex], where [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] are independent random variables, we follow several steps. Here is a detailed breakdown:
1. Understand the Given Information:
- The random variable [tex]\( X \)[/tex] represents the number of frogs and has a mean [tex]\( \mu_X = 28 \)[/tex] and a standard deviation [tex]\( \sigma_X = 2.7 \)[/tex].
- The random variable [tex]\( Y \)[/tex] represents the number of koi and has a mean [tex]\( \mu_Y = 15 \)[/tex] and a standard deviation [tex]\( \sigma_Y = 1.6 \)[/tex].
2. Calculate the Mean of [tex]\( D \)[/tex]:
The mean of the difference [tex]\( D = X - Y \)[/tex] is given by:
[tex]\[ \mu_D = \mu_X - \mu_Y \][/tex]
However, for this problem, we are not required to report [tex]\(\mu_D\)[/tex], so we focus on the standard deviation.
3. Calculate the Variance of [tex]\( D \)[/tex]:
Since [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] are independent, their variances add when calculating the variance of the difference [tex]\( D \)[/tex]:
[tex]\[ \sigma_D^2 = \sigma_X^2 + \sigma_Y^2 \][/tex]
Substituting the given values:
[tex]\[ \sigma_D^2 = (2.7)^2 + (1.6)^2 \][/tex]
[tex]\[ \sigma_D^2 = 7.29 + 2.56 \][/tex]
[tex]\[ \sigma_D^2 = 9.85 \][/tex]
4. Calculate the Standard Deviation of [tex]\( D \)[/tex]:
To find the standard deviation, we take the square root of the variance:
[tex]\[ \sigma_D = \sqrt{9.85} \][/tex]
[tex]\[ \sigma_D \approx 3.1384709652950433 \][/tex]
5. Interpret the Result:
The standard deviation [tex]\(\sigma_D \approx 3.14\)[/tex] means that the difference between the number of frogs and koi can be expected to vary by approximately 3.14 from the mean difference.
Thus, the correct answer is:
[tex]\[ \sigma_D = 3.1 \quad \text{(rounded to 1 decimal place)} \][/tex]
Interpretation: This pond can expect the difference of frogs and koi to vary by approximately 3.1 from the mean.
Therefore, the correct answer choice 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]
1. Understand the Given Information:
- The random variable [tex]\( X \)[/tex] represents the number of frogs and has a mean [tex]\( \mu_X = 28 \)[/tex] and a standard deviation [tex]\( \sigma_X = 2.7 \)[/tex].
- The random variable [tex]\( Y \)[/tex] represents the number of koi and has a mean [tex]\( \mu_Y = 15 \)[/tex] and a standard deviation [tex]\( \sigma_Y = 1.6 \)[/tex].
2. Calculate the Mean of [tex]\( D \)[/tex]:
The mean of the difference [tex]\( D = X - Y \)[/tex] is given by:
[tex]\[ \mu_D = \mu_X - \mu_Y \][/tex]
However, for this problem, we are not required to report [tex]\(\mu_D\)[/tex], so we focus on the standard deviation.
3. Calculate the Variance of [tex]\( D \)[/tex]:
Since [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] are independent, their variances add when calculating the variance of the difference [tex]\( D \)[/tex]:
[tex]\[ \sigma_D^2 = \sigma_X^2 + \sigma_Y^2 \][/tex]
Substituting the given values:
[tex]\[ \sigma_D^2 = (2.7)^2 + (1.6)^2 \][/tex]
[tex]\[ \sigma_D^2 = 7.29 + 2.56 \][/tex]
[tex]\[ \sigma_D^2 = 9.85 \][/tex]
4. Calculate the Standard Deviation of [tex]\( D \)[/tex]:
To find the standard deviation, we take the square root of the variance:
[tex]\[ \sigma_D = \sqrt{9.85} \][/tex]
[tex]\[ \sigma_D \approx 3.1384709652950433 \][/tex]
5. Interpret the Result:
The standard deviation [tex]\(\sigma_D \approx 3.14\)[/tex] means that the difference between the number of frogs and koi can be expected to vary by approximately 3.14 from the mean difference.
Thus, the correct answer is:
[tex]\[ \sigma_D = 3.1 \quad \text{(rounded to 1 decimal place)} \][/tex]
Interpretation: This pond can expect the difference of frogs and koi to vary by approximately 3.1 from the mean.
Therefore, the correct answer choice 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]
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
