Given the data set 12, 7, 4, 9, 0, 7, 3, compute the mean, median, mode, and Karl Pearson's coefficient of skewness, and describe the shape of the data.

The sum of the data points is $\sum x = 0 + 3 + 4 + 7 + 7 + 9 + 12 = 42$.

$$\text{Mean} (\bar{x}) = \frac{\sum x}{n} = \frac{42}{7} = 6$$

The sorted data is $0, 3, 4, 7, 7, 9, 12$.
Since $n=7$ is an odd number, the median is the $\frac{n+1}{2}$-th value, which is the $\frac{7+1}{2} = 4$-th value.
The 4th value in the sorted data is 7.

$$\text{Median} = 7$$

The mode is the value that appears most frequently in the data set. In the given data, the value 7 appears twice, which is more than any other value.

$$\text{Mode} = 7$$

a) Mean = $\boxed{\text{6}}$, Median = $\boxed{\text{7}}$, Mode = $\boxed{\text{7}}$

b) Calculate Karl Pearson's coefficient of skewness of the data set and also describe its shape.

We found $\bar{x} = 6$.

$$\sum (x - \bar{x})^2 = (0-6)^2 + (3-6)^2 + (4-6)^2 + (7-6)^2 + (7-6)^2 + (9-6)^2 + (12-6)^2$$ $$= (-6)^2 + (-3)^2 + (-2)^2 + (1)^2 + (1)^2 + (3)^2 + (6)^2$$ $$= 36 + 9 + 4 + 1 + 1 + 9 + 36 = 96$$

Using the formula $SK_p = \frac{\text{Mean} - \text{Mode}}{\text{Standard Deviation}}$:

$$SK_p = \frac{6 - 7}{4} = \frac{-1}{4} = -0.25$$

Since Karl Pearson's coefficient of skewness is $-0.25$, which is negative, the distribution is negatively skewed (or left-skewed).

b) Karl Pearson's coefficient of skewness = $\boxed{\text{-0.25}}$. The shape of the distribution is negatively skewed.

c) List the five-number summary.

The sorted data is $0, 3, 4, 7, 7, 9, 12$.
The five-number summary consists of:

• Minimum value: 0
• First quartile ($Q_1$): The median of the lower half ($0, 3, 4$) is 3.
• Median ($Q_2$): The middle value is 7.
• Third quartile ($Q_3$): The median of the upper half ($7, 9, 12$) is 9.
• Maximum value: 12

c) The five-number summary is: Minimum = $\boxed{\text{0}}$, $Q_1$ = $\boxed{\text{3}}$, Median = $\boxed{\text{7}}$, $Q_3$ = $\boxed{\text{9}}$, Maximum = $\boxed{\text{12}}$

d) Construct the box-and-whisker plot and describe the shape.

A box-and-whisker plot is constructed using the five-number summary:

• Minimum: 0
• $Q_1$: 3
• Median: 7
• $Q_3$: 9
• Maximum: 12

To describe the shape from the box plot:

• The median (7) is closer to $Q_3$ (9) than to $Q_1$ (3). The distance from $Q_1$ to Median is $7-3=4$. The distance from Median to $Q_3$ is $9-7=2$. This indicates that the left part of the box is longer than the right part.
• The left whisker extends from 3 to 0 (length 3).
• The right whisker extends from 9 to 12 (length 3). Since the median is closer to the third quartile and the left side of the box is longer, the distribution is negatively skewed.

d) The box-and-whisker plot would show a box from 3 to 9, with a line at 7. Whiskers would extend from 0 to 3 and from 9 to 12. The shape of the distribution is negatively skewed.

e) Compare your answer in (b) and (d).

In part (b), Karl Pearson's coefficient of skewness was calculated as $-0.25$, which indicates negative skewness.
In part (d), the analysis of the box-and-whisker plot also indicated negative skewness because the median is closer to the third quartile and the left portion of the box is longer.
Both methods consistently show that the data distribution is negatively skewed.

e) Both the Karl Pearson's coefficient of skewness (b) and the box-and-whisker plot analysis (d) indicate that the data distribution is negatively skewed.