Given a list of 20 scores, construct a frequency distribution table, find the mode, median, and mean.

Question

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2 (i)
The scores are: 70, 80, 78, 98, 84, 67, 98, 70, 80, 100, 87, 83, 70, 70, 88, 91, 70, 78, 88, 88.
First, arrange the scores in ascending order:
67, 70, 70, 70, 70, 70, 78, 78, 80, 80, 83, 84, 87, 88, 88, 88, 91, 98, 98, 100.
There are 20 scores in total.

a) Represent the above scores on a frequency distribution table.

Create a table with scores and their frequencies.
$\begin{array}{|c|c|} \hline \textbf{Score} & \textbf{Frequency} \\ \hline 67 & 1 \\ 70 & 5 \\ 78 & 2 \\ 80 & 2 \\ 83 & 1 \\ 84 & 1 \\ 87 & 1 \\ 88 & 3 \\ 91 & 1 \\ 98 & 2 \\ 100 & 1 \\ \hline \textbf{Total} & \textbf{20} \\ \hline \end{array}$
b) The mode of the distribution
Identify the score with the highest frequency from the table.
The score 70 has the highest frequency of 5.
The mode is $\boxed{\textbf{70}}$ .

c) The median score
Since there are 20 scores (an even number), the median is the average of the 10th and 11th scores in the ordered list.
The 10th score is 80.
The 11th score is 83.
Calculate the average of these two scores.
$\text{Median} = \frac{80 + 83}{2} = \frac{163}{2} = 81.5$
The median score is $\boxed{\textbf{81.5}}$ .

d) The mean score
Calculate the sum of all scores.
$\sum \text{scores} = 67(1) + 70(5) + 78(2) + 80(2) + 83(1) + 84(1) + 87(1) + 88(3) + 91(1) + 98(2) + 100(1)$ $\sum \text{scores} = 67 + 350 + 156 + 160 + 83 + 84 + 87 + 264 + 91 + 196 + 100 = 1638$
Divide the sum of scores by the total number of scores (20).
$\text{Mean} = \frac{\sum \text{scores}}{\text{Number of scores}} = \frac{1638}{20} = 81.9$
The mean score is $\boxed{\textbf{81.9}}$ .

2 (ii)
A straight line $l_1$ passes through the points P(3,4) and Q(1,-2).

a) Determine the equation of the line $l_1$ .
Calculate the gradient (slope) $m$ of the line using the points P $(x_1, y_1) = (3,4)$ and Q $(x_2, y_2) = (1,-2)$ .
$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 4}{1 - 3} = \frac{-6}{-2} = 3$
Use the point-slope form of a linear equation, $y - y_1 = m(x - x_1)$ , with point P(3,4) and $m=3$ .
$y - 4 = 3(x - 3)$ $y - 4 = 3x - 9$
Rearrange the equation into the slope-intercept form $y = mx + c$ .
$y = 3x - 9 + 4$ $y = 3x - 5$
The equation of the line $l_1$ is $\boxed{\textbf{y = 3x - 5}}$ .

b) The coordinates of the point R.
The line meets the y-axis at point R.
On the y-axis, the x-coordinate is 0.
Substitute $x=0$ into the equation of the line $y = 3x - 5$ .
$y = 3(0) - 5$ $y = -5$
The coordinates of the point R are $\boxed{\textbf{(0, -5)}}$ .

Given a list of 20 scores, construct a frequency distribution table, find the mode, median, and mean.

Step-by-step solution

Create a table with scores and their frequencies.

Identify the score with the highest frequency from the table.

Since there are 20 scores (an even number), the median is the average of the 10th and 11th scores in the ordered list.

Calculate the average of these two scores.

Calculate the sum of all scores.

Divide the sum of scores by the total number of scores (20).

Calculate the gradient (slope) $m$ of the line using the points P $(x_1, y_1) = (3,4)$ and Q $(x_2, y_2) = (1,-2)$ .

Use the point-slope form of a linear equation, $y - y_1 = m(x - x_1)$ , with point P(3,4) and $m=3$ .

Rearrange the equation into the slope-intercept form $y = mx + c$ .

The line meets the y-axis at point R.

Still stuck on this one?

More Mathematics Questions

Given a list of 20 scores, construct a frequency distribution table, find the mode, median, and mean.

Step-by-step solution

Create a table with scores and their frequencies.

Identify the score with the highest frequency from the table.

Since there are 20 scores (an even number), the median is the average of the 10th and 11th scores in the ordered list.

Calculate the average of these two scores.

Calculate the sum of all scores.

Divide the sum of scores by the total number of scores (20).

Calculate the gradient (slope) mmm of the line using the points P(x1,y1)=(3,4)(x_1, y_1) = (3,4)(x1​,y1​)=(3,4) and Q(x2,y2)=(1,−2)(x_2, y_2) = (1,-2)(x2​,y2​)=(1,−2).

Use the point-slope form of a linear equation, y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1​=m(x−x1​), with point P(3,4) and m=3m=3m=3.

Rearrange the equation into the slope-intercept form y=mx+cy = mx + cy=mx+c.

The line meets the y-axis at point R.

Still stuck on this one?

More Mathematics Questions

Calculate the gradient (slope) $m$ of the line using the points P $(x_1, y_1) = (3,4)$ and Q $(x_2, y_2) = (1,-2)$ .

Use the point-slope form of a linear equation, $y - y_1 = m(x - x_1)$ , with point P(3,4) and $m=3$ .

Rearrange the equation into the slope-intercept form $y = mx + c$ .