The above represents the battery life (in hours) of two brands of cellphones. We will find the shape of each distribution. Note that for both brands the right whisker is longer than the left one, and most of the data are on the left side of the plot. Therefore, the distribution for both brands is skewed right.
b In a box-and-whisker plot, the upper 75 % of the data is given from the first quartile to the rightmost point (the greatest value of the data set). With this in mind, we need to find these two values for each data set to find the range of the upper 75 % of each brand. Let's first do it for Brand A, then for Brand B.
Brand A
The greatest value of the data set is about 7. The first quartile is given by the right point of the left whisker, which is about 3.5. Therefore, by subtracting these values we will find the range of the upper 75 %.
7- 3.5≈ 3.5hours
Brand B
In this case, the greatest value of the data set is about 5.75 and the first quartile is about 3.25. Let's find the range of the upper 75 %.
5.75- 3.25≈ 2.5hours
c The interquartile range (IQR) is given by the difference of the third quartile, Q_3, and the first quartile, Q_1. In a box-and-whisker plot, the left point of the right whisker represents Q_3. For Brand A, Q_3 is about 4.75 and Q_1 is about 3.5. For Brand B, Q_3 is about 4.25 and Q_1 is about 3.25. Let's calculate the interquartile range for each data set.
Brand A's IQR:& 4.75- 3.5=1.25hrs
Brand B's IQR:& 4.25- 3.25=1hour
Therefore, the interquartile range for Brand A is greater.
d To find which brand could have a greater standard deviation, let's compare the range of each data set. We have found that the greatest value for Brand A is 7. The least value is given by the leftmost point of the box-and-whisker plot, which is 2. Let's calculate the range for Brand A.
Brand A's Range: 7- 2=5
In a similar way we can calculate the range for Brand B. The greatest value is about 5.75 and the least value is about 2.25.
Brand B's Range: 5.75- 2.25≈ 3.5
Since Brand A's range is greater, Brand A probably has a greater standard deviation.
e We are asked which brand should we buy if wee need a cell phone that has a battery life of more than 3.5 hours. Since each quartile has 25 % of the data, we can see that 75 % of battery life for Brand A is greater than 3.5. Conversely, for Brand B, only 50 % is greater than 3.5. Therefore, we should buy Brand A.
