Question

question #12
an art exhibition was held in a city for a week. the data shown represents the number of visitors to the exhibition on each day of the week.
120, 84, 92, 96, 210, 110, 135
which value is the best measure of center for this data, and why?
a the best measure of center is 210 because it represents the maximum number of visitors.
b the best measure of center is 121 because it is the average number of visitors each day.
c the best measure of center is 110 because it is least affected by the outlier in the data.
d the best measure of center is 96 because it lies in the middle of the data.
question #13
the table shows the average number of people visiting a small park each day for several different years. in 2012, the average number of visitors per day was 62.

year	number of visitors per day
1988	65
1990	50
1992	47
1997	61
2000	58
2006	63
2009	51

which is true if the year 2012 is included in the data set?
a the interquartile range is unchanged.
b the mean increases.
c the range increases.
d the median increases.

Explanation:

Step1: Analyze measure of center for first data - set

The data set for the art - exhibition is 120, 84, 92, 96, 210, 110, 135. The value 210 is an outlier. The mean is affected by outliers. The median is the middle - value when the data is arranged in ascending order. Arranging the data: 84, 92, 96, 110, 120, 135, 210. The median is 110. The median is less affected by outliers than the mean. So, 110 is the best measure of center as it is least affected by the outlier in the data.

Step2: Analyze effect of adding data point for second data - set

For the park - visitors data set: 47, 50, 51, 58, 58, 61, 63, 65. The number of data points \(n = 8\). The median is the average of the 4th and 5th ordered values, \(\text{Median}=\frac{58 + 58}{2}=58\). The mean \(\bar{x}=\frac{47+50 + 51+58+58+61+63+65}{8}=\frac{453}{8}=56.625\). The range is \(65 - 47 = 18\). The inter - quartile range: First, find the first quartile \(Q_1\) (median of lower half) and third quartile \(Q_3\) (median of upper half). The lower half is 47, 50, 51, 58 and \(Q_1=\frac{50 + 51}{2}=50.5\), the upper half is 58, 61, 63, 65 and \(Q_3=\frac{61+63}{2}=62\), \(IQR = Q_3 - Q_1=62 - 50.5 = 11.5\). When we add 62 (the 2012 value) to the data set, the new data set is 47, 50, 51, 58, 58, 61, 62, 63, 65. The new median is the 5th ordered value, which is 58 (unchanged). The new mean \(\bar{x}=\frac{453+62}{9}=\frac{515}{9}\approx57.22\) (mean increases). The new range is \(65 - 47 = 18\) (unchanged). The new \(Q_1=\frac{50+51}{2}=50.5\), \(Q_3=\frac{62 + 63}{2}=62.5\), \(IQR=62.5 - 50.5 = 12\) (IQR changes).

Answer:

Question #12: C. The best measure of center is 110 because it is least affected by the outlier in the data.
Question #13: B. The mean increases.