The problem consists of two parts. Part (a) requires drawing a Venn diagram based on two given statements about businessmen, rich people, and salary workers, and determining the validity of two inferences. Part (b) requires calculating the mean and variance of a given frequency distribution of seedling heights.

1. Problem Description

The problem consists of two parts. Part (a) requires drawing a Venn diagram based on two given statements about businessmen, rich people, and salary workers, and determining the validity of two inferences. Part (b) requires calculating the mean and variance of a given frequency distribution of seedling heights.

2. Solution Steps

(a)

(i) Venn Diagram:

Let B be the set of businessmen, and R be the set of rich people. The statement "Most businessmen are rich" means that there is a significant overlap between B and R, but B is not entirely contained within R. Let S be the set of salary workers. The statement "No salary worker is rich" means that there is no overlap between S and R. The Venn diagram will show sets B and R overlapping, and set S completely outside of set R.

(ii) Validity:

(α) Joe is a businessman and therefore is rich.

Since "Most businessmen are rich," it does not guarantee that every businessman is rich. Thus, this statement is Not Valid.

(β) Kipo is not rich because he is a salary worker.

The statement "No salary worker is rich" implies that if Kipo is a salary worker, then Kipo is not rich. This statement is Valid.

(b)

(i) Mean:

To calculate the mean, we first find the midpoint of each height interval.

60-64: midpoint = $(60+64)/2 = 62$

65-69: midpoint = $(65+69)/2 = 67$

70-74: midpoint = $(70+74)/2 = 72$

75-79: midpoint = $(75+79)/2 = 77$

80-84: midpoint = $(80+84)/2 = 82$

The frequencies are 7, 6, 5, 8, 4 respectively.

Total frequency, $n = 7 + 6 + 5 + 8 + 4 = 30$

Sum of (midpoint * frequency):

$(62*7) + (67*6) + (72*5) + (77*8) + (82*4) = 434 + 402 + 360 + 616 + 328 = 2140$

Mean, $\mu = \frac{\sum (midpoint * frequency)}{n} = \frac{2140}{30} = 71.333...$

Rounded to one decimal place, the mean is 71.

3.

(ii) Variance:

Variance is calculated as $Var(X) = \frac{\sum f_i(x_i - \mu)^2}{n}$, where $f_i$ is the frequency, $x_i$ is the midpoint, $\mu$ is the mean, and $n$ is the total frequency.

Using $\mu = 71.333...$

$(62-71.333)^2 = (-9.333)^2 = 87.111$

$(67-71.333)^2 = (-4.333)^2 = 18.777$

$(72-71.333)^2 = (0.667)^2 = 0.444$

$(77-71.333)^2 = (5.667)^2 = 32.111$

$(82-71.333)^2 = (10.667)^2 = 113.777$

Sum of $f_i(x_i - \mu)^2$:

$7(87.111) + 6(18.777) + 5(0.444) + 8(32.111) + 4(113.777) = 609.777 + 112.662 + 2.22 + 256.888 + 455.108 = 1436.655$

Variance = $1436.655/30 = 47.8885$

Rounded to one decimal place, the variance is 47.

9.

3. Final Answer

(a)

(i) A Venn diagram with overlapping circles for Businessmen and Rich people, and a separate circle for Salary Workers outside the Rich people circle.

(ii)

(α) Not Valid

(β) Valid

(b)

(i) Mean = 71.3 cm

(ii) Variance = 47.9 cm$^2$