In the distributions of two populations combined mean of fourteen observations is 14 and if the sum of the first seven observations for population one and mean of population two are 90 and 16 respectively ,then find the last (eighth)observation of population one and the mean of population one.
To solve this problem, we'll use a combination of algebra and arithmetic.
Let's denote the last (eighth) observation of population one as x and the mean of population one as μ1. Here are the given facts:
1. The sum of the first seven observations for population one (denoted as S1) is 90.
2. The mean of population two (denoted as μ2) is 16.
3. The combined mean of the two populations is 14.
We can use these facts to set up equations:
1. Equation 1: S1 = 90
2. Equation 2: (7*S1 + x) / 8 = μ1
3. Equation 3: (7*S1 + 14*x) / 15 = 14
4. Equation 4: μ2 = 16
Now, let's solve these equations step-by-step:
1. From Equation 1, we know that S1 = 90.
2. Substituting this value into Equation 2:
(7*90 + x) / 8 = μ1
3. Simplifying Equation 2:
630 + x = 8*μ1
4. Now, let's proceed to Equation 3:
(7*90 + 14*x) / 15 = 14
5. Simplifying Equation 3:
630 + 14*x = 15*14
6. Solving Equation 3 for x:
630 + 14*x = 210
14*x = -420
x = -420 / 14
x = -30
7. Now, substituting the value of x into Equation 2:
630 + (-30) = 8*μ1
600 = 8*μ1
μ1 = 600 / 8
μ1 = 75
Therefore, the last (eighth) observation of population one is -30, and the mean of population one is 75.