Question: Number of brown lentils per quadrat
Group Number 0 1 2 3 4 5 6 7 8 9
Group 1 0 0 0 1 7 6 0 4 1 0
Group 2 0 0 0 1 4 6 2 2 0 0
Group 3 0 0 0 0 7 5 5 0 2 0
Group 4 0 0 0 0 6 6 3 1 1 0
Group 5 0 0 0 1 8 6 1 0 1 0
Group 6 0 0 0 1 5 6 3 2 1 0
Group 7 0 0 0 0 7 4 2 0 1 0
Group 8 0 0 0 0 7 6 6 0 1 0
Total 0 0 0 4 51 45 22 9 8 0
Using the data shown in the Table attached to this Tests and Quizzes page, calculate the variance ($s^2$) around your estimate of the average number of brown lentils per quadrat observed by Group 6. Your answer should be given to TWO decimal places. here is the data
1. **State the problem:** Calculate the variance $s^2$ of the number of brown lentils per quadrat observed by Group 6.
2. **Extract Group 6 data:** The counts for Group 6 are:
Number of lentils ($x$): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Frequency ($f$): 0, 0, 0, 1, 5, 6, 3, 2, 1, 0
3. **Calculate the mean ($\bar{x}$):**
$$\bar{x} = \frac{\sum f x}{\sum f}$$
Calculate total frequency:
$$N = 0+0+0+1+5+6+3+2+1+0 = 18$$
Calculate sum of $f x$:
$$\sum f x = 0\times0 + 0\times1 + 0\times2 + 1\times3 + 5\times4 + 6\times5 + 3\times6 + 2\times7 + 1\times8 + 0\times9$$
$$= 0 + 0 + 0 + 3 + 20 + 30 + 18 + 14 + 8 + 0 = 93$$
Mean:
$$\bar{x} = \frac{93}{18} = 5.1667$$
4. **Calculate variance formula:**
$$s^2 = \frac{\sum f (x - \bar{x})^2}{N - 1}$$
Calculate each squared deviation and multiply by frequency:
\begin{align*}
f(x - \bar{x})^2 &= f(x^2 - 2x\bar{x} + \bar{x}^2) \\
\sum f (x - \bar{x})^2 &= \sum f x^2 - 2\bar{x} \sum f x + \bar{x}^2 \sum f
\end{align*}
Calculate $\sum f x^2$:
$$0^2\times0 + 1^2\times0 + 2^2\times0 + 3^2\times1 + 4^2\times5 + 5^2\times6 + 6^2\times3 + 7^2\times2 + 8^2\times1 + 9^2\times0$$
$$= 0 + 0 + 0 + 9 + 80 + 150 + 108 + 98 + 64 + 0 = 509$$
Calculate variance numerator:
$$\sum f (x - \bar{x})^2 = 509 - 2 \times 5.1667 \times 93 + 5.1667^2 \times 18$$
Calculate each term:
$$2 \times 5.1667 \times 93 = 960.0022$$
$$5.1667^2 = 26.7222$$
$$26.7222 \times 18 = 481.000$$
So:
$$\sum f (x - \bar{x})^2 = 509 - 960.0022 + 481 = 29.9978$$
5. **Calculate variance:**
$$s^2 = \frac{29.9978}{18 - 1} = \frac{29.9978}{17} = 1.7646$$
6. **Final answer rounded to two decimals:**
$$\boxed{1.76}$$
This is the variance of the number of brown lentils per quadrat observed by Group 6.