Question

3-124. (from lessons 3.1.1 and 3.1.3) do you remember malcolm from problem 3-101? he is obsessed with improving his vocabulary. he records the number of new words he learns every day in his journal. 20, 7, 0, 1, 11, 0, 1, 15, 25, 6, 0, 9, 9, 1 a. an average 6th grader learns 9.21 words per day. is malcolm learning more words or fewer words per day than an average 6th grader? explain. b. on how many days did malcolm learn more than 10 words? c. what is the median number of words malcolm learned in a day? how close is malcolm’s median to his mean number of new words per day?

Explanation:

Part a

Step1: Calculate the mean

First, find the sum of the data set. The data is \(20, 7, 0, 1, 11, 0, 1, 15, 25, 6, 0, 9, 9, 1\). The sum \(S=20 + 7+0 + 1+11+0 + 1+15+25+6+0+9+9+1\)
\(S=20+7 = 27; 27+0=27; 27 + 1=28; 28+11 = 39; 39+0=39; 39+1=40; 40+15=55; 55+25 = 80; 80+6=86; 86+0=86; 86+9=95; 95+9=104; 104 + 1=105\)
The number of data points \(n = 14\). The mean \(\bar{x}=\frac{S}{n}=\frac{105}{14}=7.5\)

Step2: Compare with 9.21

The mean number of words Malcolm learns per day is \(7.5\), and the average 6th grader learns \(9.21\) words per day. Since \(7.5<9.21\), Malcolm is learning fewer words per day.

Step1: Identify values >10

We check each data point: \(20>10\), \(7\leq10\), \(0\leq10\), \(1\leq10\), \(11 > 10\), \(0\leq10\), \(1\leq10\), \(15>10\), \(25>10\), \(6\leq10\), \(0\leq10\), \(9\leq10\), \(9\leq10\), \(1\leq10\)

Step2: Count the values

The values greater than 10 are \(20\), \(11\), \(15\), \(25\). So there are \(4\) such values.

Step1: Find the median

First, order the data set: \(0, 0, 0, 1, 1, 1, 6, 7, 9, 9, 11, 15, 20, 25\)
Since \(n = 14\) (even), the median is the average of the \(\frac{n}{2}=7\)-th and \(\frac{n}{2}+ 1=8\)-th values.
The 7th value is \(6\) and the 8th value is \(7\). So the median \(M=\frac{6 + 7}{2}=6.5\)

Step2: Compare with the mean

We already found the mean \(\bar{x}=7.5\) in part (a). The difference between the mean and the median is \(|7.5 - 6.5| = 1\). So the median is \(1\) unit away from the mean.

Answer:

Malcolm is learning fewer words per day. The mean number of words he learns per day is \(7.5\), which is less than \(9.21\).

Part b