Question

a researcher for the epa measured the amount of arsenic in the water near a sewage treatment plant. over 5 days, they took n = 20 measurements (in ppb). their sample data is summarized in the ogive graph shown below. what is the median amount of arsenic? ppb what is the iqr of arsenic? ppb question help: video message instructor

Explanation:

Step1: Recall median formula for $n = 20$

For $n = 20$ (even number of data - points), the median is the average of the $\frac{n}{2}$ - th and $(\frac{n}{2}+1)$ - th ordered data - values. Here, $\frac{n}{2}=10$ and $\frac{n}{2}+1 = 11$. We use the ogive graph to find the values corresponding to the 10 - th and 11 - th ordered values. The ogive gives the cumulative frequency. We look for the value on the x - axis when the cumulative frequency is 10 and 11. Since the ogive is a smooth curve representing cumulative frequency, we estimate. The cumulative frequency of 10 and 11 occurs at approximately the same x - value. Looking at the ogive, when the cumulative frequency is around 10 and 11, the amount of arsenic is 45 ppb. So the median is 45 ppb.

Step2: Recall IQR formula

The inter - quartile range $IQR = Q_3−Q_1$. The first quartile $Q_1$ is the value such that 25% of the data lies below it. For $n = 20$, $25\%\text{ of }n=0.25\times20 = 5$. Looking at the ogive, when the cumulative frequency is 5, the value of arsenic is approximately 35 ppb. The third quartile $Q_3$ is the value such that 75% of the data lies below it. For $n = 20$, $75\%\text{ of }n=0.75\times20 = 15$. Looking at the ogive, when the cumulative frequency is 15, the value of arsenic is approximately 55 ppb. Then $IQR=Q_3 - Q_1=55 - 35=20$ ppb.

Answer:

Median: 45 ppb
IQR: 20 ppb