Question

determining the effect of outliers
fredrick hit 14, 18, 13, 12, 12, 16, 13, 12, 1, and 15 home runs in 10 seasons of play. which statements are correct?
check all that apply.
fredricks data set contains an outlier.
the median value is 12 home runs.
the mean value is about 12.6 home runs.
the median describes fredricks data more accurately than the mean.
the mean value stays the same when the outlier is not included in the data set.

Explanation:

Step1: Calculate the mean with the outlier

The data set is \(14, 18, 13, 12, 12, 16, 13, 12, 1, 15\).
The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n = 10\) and \(\sum_{i=1}^{10}x_{i}=14 + 18+13 + 12+12 + 16+13 + 12+1+15=126\).
So, \(\bar{x}=\frac{126}{10}=12.6\).

Step2: Calculate the median with the outlier

First, order the data set: \(1,12,12,12,13,13,14,15,16,18\).
Since \(n = 10\) (even), the median \(M=\frac{x_{\frac{n}{2}}+x_{\frac{n}{2}+1}}{2}=\frac{13 + 13}{2}=13\).

Step3: Calculate the mean without the outlier (\(1\))

The new data set is \(12,12,12,13,13,14,15,16,18\), \(n = 9\), \(\sum_{i = 1}^{9}x_{i}=12+12+12+13+13+14+15+16+18 = 125\).
The new mean \(\bar{x}=\frac{125}{9}\approx13.9\).

Step4: Analyze the statements

• "Fredrick's data set contains an outlier": The value \(1\) is much smaller than the other values, so this is correct.
• "The median value is 12 home runs": With the outlier, the median is \(13\), so this is incorrect.
• "The mean value is about 12.6 home runs": With the outlier, \(\bar{x}=12.6\), so this is correct.
• "The median describes Fredrick's data more accurately than the mean": Since there is an outlier, the median is less affected by the outlier than the mean. So this is correct.
• "The mean value stays the same when the outlier is not included": The mean changes from \(12.6\) to approximately \(13.9\), so this is incorrect.

Answer:

Fredrick's data set contains an outlier. The mean value is about 12.6 home runs. The median describes Fredrick's data more accurately than the mean.