Question

1. the dot plot represents the distribution of wages earned during a one - week period by 12 college students. a. what is the mean? interpret this value based on the situation. b. what is the median? interpret this value based on the situation. c. would a box plot of the same data have allowed you to find both the mean and the median?

Explanation:

Step1: Count data - point frequencies

From the dot - plot:
112 has 1 dot, 114 has 1 dot, 116 has 3 dots, 118 has 2 dots, 120 has 2 dots, 124 has 3 dots.

Step2: Calculate the sum of data values

Sum \(S=(112\times1)+(114\times1)+(116\times3)+(118\times2)+(120\times2)+(124\times3)\)
\(S = 112+114 + 348+236+240+372\)
\(S=1422\)

Step3: Calculate the mean

The number of data points \(n = 12\). Mean \(\bar{x}=\frac{S}{n}=\frac{1422}{12}=118.5\)
Interpretation: On average, the weekly wage of these 12 college students is $118.5.

Step4: Find the median

Since \(n = 12\) (an even number), the median is the average of the \(\frac{n}{2}=6\)th and \((\frac{n}{2}+1)=7\)th ordered data - points.
Ordering the data: 112, 114, 116, 116, 116, 118, 118, 120, 120, 124, 124, 124.
The 6th value is 118 and the 7th value is 118. Median \(M=\frac{118 + 118}{2}=118\)
Interpretation: Half of the 12 college students earn less than $118 per week and half earn more than $118 per week.

Step5: Analyze box - plot

A box - plot shows the minimum, first quartile (\(Q_1\)), median (\(Q_2\)), third quartile (\(Q_3\)), and maximum. It does not directly show the mean. We can find the median from a box - plot, but not the mean without additional calculations.

Answer:

a. Mean is 118.5. On average, the weekly wage of these 12 college students is $118.5.
b. Median is 118. Half of the 12 college students earn less than $118 per week and half earn more than $118 per week.
c. No, a box - plot allows us to find the median but not the mean without additional calculations.