The variance of 20 observations is 5. If each observation is multiplied by 2, find the new variance of the resulting observations.
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Step-by-Step Solution
Step 1: Understand the given information
We are given that there are 20 observations, and their variance is 5. We need to find the new variance if each observation is multiplied by 2.
Step 2: Recall the property of variance
A key property of variance states that if each observation in a dataset is multiplied by a constant k, the new variance will be k2 times the original variance. This is because variance measures the spread of data, and multiplying by k scales the spread by k, so the squared spread (variance) scales by k2.
Step 3: Identify the constant multiplier
In this problem, each observation is multiplied by 2. Therefore, the constant multiplier k is 2.
Step 4: Calculate the new variance
Using the property of variance, we substitute the value of k=2 and the original variance of 5 into the formula. This gives us 22×5, which simplifies to 4×5, resulting in a new variance of 20.