Question

a) The arithmetic mean of 10 numbers is 4, when an eleventh number, x is added so that the overall mean is changed to 5. When a twelfth number, y is added the mean changes to 4. Determine the values of x and y (5 Marks)

Answer 1

Answer to Question #86177 - Math - Statistics and Probability

Question: The arithmetic mean of 10 numbers is 4, when an eleventh number, x is added so that the overall mean is changed to 5. When a twelfth number, y is added the mean changes to 4. Determine the values of x and y.

Solution: Let $z_{1},\cdots,z_{10}$ be the initial 10 numbers. Given that, their arithmetic mean is 4.

Therefore,

$\frac{1}{10}\sum_{i=1}^{10}z_{i}=4$

$\Rightarrow\sum_{i=1}^{10}z_{i}=40$

When $x$ is added in this list, the overall mean is 5.

So we have,

$\frac{1}{11}(z_{1}+\cdots+z_{10}+x)=5$

$\Rightarrow(z_{1}+\cdots+z_{10}+x)=55$

$\Rightarrow 40+x=55$

$\Rightarrow x=15$

When $y$ is added in this list as the twelfth number, the overall mean is 4. So we have,

$\frac{1}{12}(z_{1}+\cdots+z_{10}+x+y)=4$

$\Rightarrow(z_{1}+\cdots+z_{10}+x+y)=48$

$\Rightarrow 40+15+y=48$

$\Rightarrow y=-7$

Answer: Therefore, the values are,

- $x=15$

- $y=-7$

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