Question

question 8 (3 points)
the data represent the number of cans collected by different classes for a service project.
12 14 22 14 18 23 42 13
9 19 22 14

1. find the mean 18.5 cans
2. find the median. 16 cans
3. eliminate the greatest value, 42, from the data set. explain how the measures of center change.

the mean decreases (increases, decreases) to about 16.36 cans and the median (increases, decreases) to 14 cans.
blank 1: 18.5
blank 2: 16
blank 3: decreases
blank 4:
blank 5:
blank 6:

Explanation:

Step1: Calculate original mean

The original data set is \(12,14,22,14,18,23,42,13,9,19,22,14\). The sum of the data is \(12 + 14+22+14+18+23+42+13+9+19+22+14=222\). There are \(n = 12\) data - points. The mean \(\bar{x}=\frac{222}{12}=18.5\).

Step2: Calculate original median

First, order the data set: \(9,12,13,14,14,14,18,19,22,22,23,42\). Since \(n = 12\) (an even - numbered data set), the median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+1)\)th ordered values. \(\frac{n}{2}=6\) and \(\frac{n}{2}+1 = 7\). The median \(M=\frac{14 + 18}{2}=16\).

Step3: Calculate new mean after removing 42

The new sum of the data set (after removing 42) is \(222-42 = 180\). There are \(n=11\) data - points. The new mean \(\bar{x}=\frac{180}{11}\approx16.36\). The mean decreases from \(18.5\) to about \(16.36\).

Step4: Calculate new median after removing 42

The new ordered data set (after removing 42) is \(9,12,13,14,14,14,18,19,22,22,23\). Since \(n = 11\) (an odd - numbered data set), the median is the \(\frac{n + 1}{2}\)th ordered value. \(\frac{n+1}{2}=6\)th value, which is \(14\). The median decreases from \(16\) to \(14\).

Answer:

Blank 4: \(16.36\)
Blank 5: Decreases
Blank 6: \(14\)