The problem provides a frequency table showing the distribution of heights of seedlings in a nursery. We are asked to calculate the mean and variance of the distribution, correct to one decimal place.

1. Problem Description

The problem provides a frequency table showing the distribution of heights of seedlings in a nursery. We are asked to calculate the mean and variance of the distribution, correct to one decimal place.

2. Solution Steps

First, we need to find the midpoints of each height interval. Let's call them $x_i$.

$x_1 = (60 + 64)/2 = 62$

$x_2 = (65 + 69)/2 = 67$

$x_3 = (70 + 74)/2 = 72$

$x_4 = (75 + 79)/2 = 77$

$x_5 = (80 + 84)/2 = 82$

The frequencies are $f_1 = 7$, $f_2 = 6$, $f_3 = 5$, $f_4 = 8$, $f_5 = 4$.

The total frequency is $N = 7 + 6 + 5 + 8 + 4 = 30$.

(i) Mean:

The formula for the mean of a grouped distribution is:

$\bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{\sum f_i x_i}{N}$

$\sum f_i x_i = (7 \times 62) + (6 \times 67) + (5 \times 72) + (8 \times 77) + (4 \times 82) = 434 + 402 + 360 + 616 + 328 = 2140$

$\bar{x} = \frac{2140}{30} = 71.333... \approx 71.3$

(ii) Variance:

The formula for the variance of a grouped distribution is:

$s^2 = \frac{\sum f_i (x_i - \bar{x})^2}{N-1}$ or $s^2 = \frac{\sum f_i x_i^2}{N-1} - \frac{(\sum f_i x_i)^2}{N(N-1)}$. However, since the mean we calculated is rounded, let's use

$s^2 = \frac{\sum f_i (x_i - \bar{x})^2}{N-1}$

First we must find $\sum f_i(x_i - \bar{x})^2$

$(x_1 - \bar{x})^2 = (62 - 71.3)^2 = (-9.3)^2 = 86.49$

$(x_2 - \bar{x})^2 = (67 - 71.3)^2 = (-4.3)^2 = 18.49$

$(x_3 - \bar{x})^2 = (72 - 71.3)^2 = (0.7)^2 = 0.49$

$(x_4 - \bar{x})^2 = (77 - 71.3)^2 = (5.7)^2 = 32.49$

$(x_5 - \bar{x})^2 = (82 - 71.3)^2 = (10.7)^2 = 114.49$

$f_1(x_1 - \bar{x})^2 = 7 \times 86.49 = 605.43$

$f_2(x_2 - \bar{x})^2 = 6 \times 18.49 = 110.94$

$f_3(x_3 - \bar{x})^2 = 5 \times 0.49 = 2.45$

$f_4(x_4 - \bar{x})^2 = 8 \times 32.49 = 259.92$

$f_5(x_5 - \bar{x})^2 = 4 \times 114.49 = 457.96$

$\sum f_i (x_i - \bar{x})^2 = 605.43 + 110.94 + 2.45 + 259.92 + 457.96 = 1436.7$

$s^2 = \frac{1436.7}{30-1} = \frac{1436.7}{29} = 49.5413... \approx 49.5$

3. Final Answer

Mean = 71.3 cm

Variance = 49.5 $cm^2$