## Related questions with answers

Environmental scientists are increasingly concerned with the accumulation of toxic elements in marine mammals and the transfer of such elements to the animals' offspring. The striped dolphin (Stenella coeruleoalba), considered to be a top predator in the marine food chain, was the subject of one such study. The mercury concentrations (micrograms/gram) in the livers of 28 mals striped dolphins were as follows: 1.70 183.00 221.00 286.00 1.72 168.00 406.00 315.00 8.80 218.00 252.00 241.00 5.90 180.00 329.00 397.00 101.00 264.00 316.00 209.00 85.40 481.00 445.00 314.00 118.00 485.00 278.00 318.00 a. Calculate the five-number summary for the data. b. Construct a box plot for the data. c. Are there any outliers? d. If you knew that the first four dolphins were less than 3 years old, while all the others were more than 8 years old, would this information help explain the difference in the magnitude of those four observations? Explain.

Solution

VerifiedGiven:

1.7, 1.72, 8.8, 5.9, 101, 85.4, 118, 183, 168, 218, 180, 264, 481, 485, 221, 406, 252, 329, 316, 445, 278, 286, 315, 241, 397, 209, 314, 318

(a) Order the data values from smallest to largest:

1.7, 1.72, 5.9, 8.8, 85.4, 101, 118, 168, 180, 183, 209, 218, 221, 241, 252, 264, 278, 286, 314, 315, 316, 318, 329, 397, 406, 445, 481, 485

The $\textbf{five-number summary}$ consists of the minimum, the first quartile, the median, the third quartile and the maximum.

The minimum (smallest data value) is 1.70.

The $\textbf{median}$ is the middle value of the sorted data set. Since the number of data values is even, the median is the average of the two middle values (14th and 15th data value) of the sorted data set:

$m=\dfrac{241+252}{2}=\dfrac{493}{2}=246.5$

Since $0.25(n+1)=0.25(28+1)=0.25(29)=7.25$ is not an integer, the $\textbf{lower quartile}$ is the 7th data value increased by 0.25 times the difference between the sixth and the seventh data value.

$Q_1=118+0.25(168-118)=118+12.5=130.5$

Since $0.75(n+1)=0.75(28+1)=0.75(29)=21.75$ is not an integer, the $\textbf{upper quartile}$ is the 21st data value increased by 0.75 times the difference between the 21st and the 22nd data value.

$Q_3=316+0.75(318-316)=316+1.5=317.5$

The maximum (largest data value) is 485.

Thus the five-number summary is then: 1.70, 130.5, 246.5, 317.5, 485.

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