Question

probability and statistics
1.8 gcw quantitative data
gcw2 boxplots and outliers
read each set of directions very carefully. all work must be shown to receive credit.
1.)
here are the travel - times in minutes for 15 workers in north carolina, chosen at random by the census bureau: 30 20 10 40 25 20 10 60 15 40 5 30 12 10 10
identify the boundaries for outliers.
low outliers < minutes high outliers > minutes
2.)
here is a stemplot of the areas of the 46 counties in south carolina. note that the data have been rounded to the nearest 10 square miles (mi²).
3 | 9999
4 | 0116689
5 | 01115566778
6 | 47899
7 | 01245579
8 | 0011
9 | 13
10 | 8
11 | 233
12 | 2
key: 6|4 represents a county with an area of 635 to 644.99 square miles.
which of the following is not a correct description of the distribution of area for the 46 south carolina counties?
the county with an area of approximately 1,220 square miles is an outlier.
the median area is less than the mean area of the 46 counties in south carolina.
the distribution of areas is right - skewed.
the area of the counties varies from about 390 square miles to about 1,220 square miles.

Explanation:

Step1: Sort the travel - time data

First, sort the travel - time data for 15 workers: 5, 10, 10, 10, 10, 12, 15, 20, 20, 25, 30, 30, 40, 40, 60.

Step2: Calculate quartiles

The number of data points \(n = 15\). The median (second - quartile \(Q_2\)) is the 8th value, so \(Q_2=20\). The lower half of the data is 5, 10, 10, 10, 10, 12, 15. The median of the lower half (first - quartile \(Q_1\)) is the 4th value, so \(Q_1 = 10\). The upper half of the data is 25, 30, 30, 40, 40, 60. The median of the upper half (third - quartile \(Q_3\)) is the 4th value of the upper half, so \(Q_3=30\).

Step3: Calculate the inter - quartile range (IQR)

\(IQR=Q_3 - Q_1=30 - 10 = 20\).

Step4: Calculate the boundaries for outliers

The lower boundary for outliers is \(Q_1-1.5\times IQR=10-1.5\times20=10 - 30=- 20\). Since time cannot be negative, we can consider 0 as a practical lower - bound for non - negative data. The upper boundary for outliers is \(Q_3 + 1.5\times IQR=30+1.5\times20=30 + 30 = 60\).

For the second part:

1. Analyze the stem - plot data:

• The smallest value is around \(39\times10 = 390\) square miles and the largest value is \(122\times10=1220\) square miles.
• To check if the county with an area of approximately 1220 square miles is an outlier, we would need to calculate quartiles and IQR for this data set as well. But we can also use a visual inspection. The data seems to be right - skewed as there are more values on the lower end and a few larger values pulling the tail to the right.
• For a right - skewed distribution, the mean is usually greater than the median.
• The range of the data is from about 390 to 1220 square miles.
• The statement "The county with an area of approximately 1220 square miles is an outlier" is not necessarily true just based on the given information. We need more formal outlier calculations (using \(Q_1\), \(Q_3\), and \(IQR\)). The other statements about the distribution characteristics (right - skewed, median < mean, range of values) are reasonable based on the stem - plot.

Answer:

1. Low Outliers < 0 minutes, High Outliers > 60 minutes
2. The county with an area of approximately 1220 square miles is an outlier.