Question

d. sum of two rolled dice (data from 40 rolls)
5 - # - summary:
min: __ q1: med: q3: max: __
the mean is __ which is __ than the median.
the shape of this data set is ____
sum of two dice in 40 rolls

Explanation:

Step1: Identify minimum value

The minimum value in the data - set of sums of two dice is 2.

Step2: Find first - quartile (Q1)

Since there are \(n = 40\) data points, the position of Q1 is \(\frac{n+1}{4}=\frac{40 + 1}{4}=10.25\). The 10th and 11th ordered data - points are considered. Counting from the left of the dot - plot, Q1 = 5.

Step3: Calculate the median (Med)

The position of the median for \(n = 40\) (an even number of data points) is \(\frac{n}{2}=20\) and \(\frac{n}{2}+1 = 21\). The median is the average of the 20th and 21st ordered data - points. Counting from the left, Med = 7.

Step4: Determine third - quartile (Q3)

The position of Q3 is \(\frac{3(n + 1)}{4}=\frac{3\times(40+1)}{4}=30.75\). The 30th and 31st ordered data - points are considered. Counting from the left, Q3 = 8.

Step5: Find maximum value

The maximum value in the data - set is 12.

Step6: Calculate the mean

The data set has 1 value of 2, 3 values of 3, 4 values of 4, 5 values of 5, 6 values of 6, 7 values of 7, 6 values of 8, 5 values of 9, 4 values of 10, 3 values of 11, and 1 value of 12.
The sum of the data is \(1\times2+3\times3 + 4\times4+5\times5+6\times6+7\times7+6\times8+5\times9+4\times10+3\times11+1\times12\)
\[=2 + 9+16+25+36+49+48+45+40+33+12\]
\[=315\]
The mean \(\bar{x}=\frac{315}{40}=7.875\).
Since \(7.875>7\), the mean is greater than the median.

Step7: Determine the shape of the data - set

The data is symmetric around the value 7, so the shape of the data - set is approximately symmetric.

Answer:

Min: 2
Q1: 5
Med: 7
Q3: 8
Max: 12
The mean is 7.875 which is greater than the median.
The shape of this data set is approximately symmetric.