If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then a + b + ab is equal to:
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Step-by-Step Solution
Step 1: Calculate the sum of observations using the mean
The mean of a set of observations is calculated by dividing the sum of all observations by the total number of observations. We are given the mean as 9 and the number of observations n=8. We can use this to find the sum of the observations.
Step 2: Formulate the first equation for a and b
Substitute the given values into the mean formula. The sum of the known numbers is 6+4+8+12+10+13=53. So, 53+a+b=72. This gives us our first equation relating a and b.
Step 3: Calculate the sum of squares using the variance
The variance is given as 9.25. We can use the formula for variance to find the sum of the squares of the observations. We know the mean is 9 and n=8.
Step 4: Formulate the second equation for a and b
Substitute the known values into the variance formula. The sum of the squares of the known numbers is 36+16+64+144+100+169=529. So, 529+a2+b2=8×(9.25+81)=8×90.25=722. This gives us our second equation.
Step 5: Solve the system of equations for a and b
From the first equation, a+b=72−53=19. From the second equation, a2+b2=722−529=193. We know that (a+b)2=a2+b2+2ab. Substitute the values to find ab.
Step 6: Calculate the value of ab
Using the identity (a+b)2=a2+b2+2ab, we can solve for 2ab=(a+b)2−(a2+b2). Substituting the values, 2ab=(19)2−193=361−193=168. Therefore, ab=84.
Step 7: Calculate a + b + ab
Now that we have the values for a+b and ab, we can find the required expression a+b+ab.