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Step-by-Step Solution
Step 1: Understand the given information
We have 5 students with an average (mean) height of 150 cm and a variance of 18 cm2. A new student with a height of 156 cm joined them. We need to find the new variance for all 6 students.
Step 2: Express the known quantities mathematically
Let the heights of the original 5 students be $x_1, x_2, x_3, x_4, x_5$. Then:
Mean of the 5 students, $ \overline{x} = 150$ cm.
Variance of the 5 students, $ \sigma^2 = 18$ cm2.
From the mean, we get:
$ \sum_{i=1}^{5} x_i = 5 \cdot \overline{x} = 5 \times 150 = 750.
$
From the variance formula for 5 students:
$ \sigma^2 = \frac{\sum_{i=1}^{5} x_i^2}{5} - \left(\overline{x}\right)^2 = 18.
$
Substitute $\overline{x} = 150$ to find the sum of squares of the original 5 heights:
$ \frac{\sum_{i=1}^{5} x_i^2}{5} - 150^2 = 18 \\
\Rightarrow \frac{\sum_{i=1}^{5} x_i^2}{5} = 18 + 150^2 = 18 + 22500 = 22518 \\
\Rightarrow \sum_{i=1}^{5} x_i^2 = 22518 \times 5 = 112590.
$
Step 3: Incorporate the new student's height
Let the sixth student's height be $x_6 = 156$ cm. Then the total height of all 6 students is:
$ \sum_{i=1}^{6} x_i = 750 + 156 = 906.
$
The new mean, $ \overline{x}_{\mathrm{new}} $, is:
$ \overline{x}_{\mathrm{new}} = \frac{906}{6} = 151.
$
Step 4: Calculate the new sum of squares of heights
The sum of squares of all 6 heights is:
$ \sum_{i=1}^{6} x_i^2 = \sum_{i=1}^{5} x_i^2 + x_6^2 = 112590 + (156)^2. \\
(156)^2 = 24336, \\
\text{so} \quad \sum_{i=1}^{6} x_i^2 = 112590 + 24336 = 136926.
$
Step 5: Apply the variance formula for 6 students
Variance for 6 students is:
$ \sigma_{\mathrm{new}}^2 = \frac{\sum_{i=1}^{6} x_i^2}{6} - \left(\overline{x}_{\mathrm{new}}\right)^2. \\
= \frac{136926}{6} - (151)^2.
$
Compute each term:
$ \frac{136926}{6} = 22821, \quad (151)^2 = 22801.
$
Hence,
$ \sigma_{\mathrm{new}}^2 = 22821 - 22801 = 20.
$
Step 6: Final answer
The variance of the heights of the six students is 20 cm2.
