Question

1.3 displaying numerical data: dotplots

1. magic words here are data on the lengths of the first 25 words on a randomly selected page from harry potter and the prisoner of azkaban.

2 3 4 10 2 11 2 8 4 3 7 2 7
5 3 6 4 4 2 5 8 2 3 4 4
a. make a dotplot of these data.
b. explain what the dot above 6 represents.
c. what percent of the words have more than 4 letters?

Explanation:

Step1: Count the frequency of each number

First, count how many times each number (word - length) appears in the data set.
For example, for the number \(2\):
Count the number of \(2\)s in the data: \(2,2,2,2,2\). So the frequency of \(2\) is \(5\).
For the number \(3\):
Count the \(3\)s: \(3,3,3\). Frequency of \(3\) is \(3\).
For the number \(4\):
Count the \(4\)s: \(4,4,4,4,4,4\). Frequency of \(4\) is \(6\).
For the number \(5\):
Count the \(5\)s: \(5,5\). Frequency of \(5\) is \(2\).
For the number \(6\):
Count the \(6\)s: \(6\). Frequency of \(6\) is \(1\).
For the number \(7\):
Count the \(7\)s: \(7,7\). Frequency of \(7\) is \(2\).
For the number \(8\):
Count the \(8\)s: \(8,8\). Frequency of \(8\) is \(2\).
For the number \(10\):
Count the \(10\)s: \(10\). Frequency of \(10\) is \(1\).
For the number \(11\):
Count the \(11\)s: \(11\). Frequency of \(11\) is \(1\).

Step2: Draw the dot - plot

Draw a horizontal axis with the values \(2,3,4,5,6,7,8,10,11\).
For each value \(x\), place a dot above \(x\) on the horizontal axis. The number of dots above \(x\) is equal to its frequency.

Step3: Explain the dot above \(6\)

In a dot - plot, each dot represents one data point. So the dot above \(6\) represents a word that has a length of \(6\) letters.

Step4: Calculate the percentage of words with more than \(4\) letters

First, count the number of words with more than \(4\) letters.
Words with more than \(4\) letters: \(5,6,7,8,10,11\)
Number of words with \(5\) letters: \(2\)
Number of words with \(6\) letters: \(1\)
Number of words with \(7\) letters: \(2\)
Number of words with \(8\) letters: \(2\)
Number of words with \(10\) letters: \(1\)
Number of words with \(11\) letters: \(1\)
Total number of words with more than \(4\) letters \(n=2 + 1+2+2+1+1=9\)
Total number of words \(N = 25\)
The percentage \(P=\frac{n}{N}\times100=\frac{9}{25}\times100 = 36\%\)

Answer:

a. The dot - plot has dots placed above the values \(2,3,4,5,6,7,8,10,11\) on the horizontal axis with the number of dots corresponding to their frequencies (\(5\) dots above \(2\), \(3\) dots above \(3\), \(6\) dots above \(4\), \(2\) dots above \(5\), \(1\) dot above \(6\), \(2\) dots above \(7\), \(2\) dots above \(8\), \(1\) dot above \(10\), \(1\) dot above \(11\)).
b. The dot above \(6\) represents a word with a length of \(6\) letters.
c. \(36\%\)