Step 1: Understanding the Concept:
We are given the mean and variance of a dataset of 5 numbers, along with three of those numbers. We need to use the definitions of mean and variance to set up a system of two equations to solve for the two unknown numbers.
Step 2: Key Formula or Approach:
1. Mean:
2. Variance:
Step 3: Detailed Explanation:
Let the two unknown values be and . We are given:
Number of values,
Mean,
Variance,
Known values are 1, 7, 3.
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Using the Mean:
Using the Variance:
Now we have a system of two equations with two variables:
1.
2.
From Equation 1, we can write . Substitute this into Equation 2:
Divide the equation by 2:
Factor the quadratic equation:
This gives two possible solutions for : or .
If , then .
If , then .
In either case, the remaining two values are 4 and 5.
Step 4: Final Answer:
The remaining two values are 4 and 5.