Question

the times (in seconds) of 8 runners for the 100 - yard dash are recorded and used to determine the following calculations. mean: 11.8 median: 12.2 mode: 12.5 range: 3.2 iqr: 2.4 the timekeeper had difficulties with the stopwatch and did not start the time until the runners had been running for 1.5 seconds. what would these calculations be if this 1.5 seconds were added to each runners 100 - yard dash time? press each hotspot. label the corresponding number below with the requested value. press to hear a reminder or hint for this problem. adding or subtracting the same value from every data value in a set does not change the measures of dispersion. mean: ① median: ② mode: ③ range: ④ iqr: ⑤

Explanation:

Step1: Effect on Mean

When we add \(1.5\) to each data - point, the mean will increase by \(1.5\). The original mean is \(\bar{x}=11.8\). The new mean \(\bar{y}\) is given by the formula \(\bar{y}=\frac{\sum_{i = 1}^{n}(x_{i}+1.5)}{n}=\frac{\sum_{i = 1}^{n}x_{i}+1.5n}{n}=\bar{x}+1.5\). So, \(\bar{y}=11.8 + 1.5=13.3\).

Step2: Effect on Median

The median is the middle - value (for \(n = 8\), the average of the 4th and 5th ordered values). If we add \(1.5\) to each data - value, the position of the middle - values does not change, but their values increase by \(1.5\). The original median \(M = 12.2\). The new median \(M_{new}=12.2+1.5 = 13.7\).

Step3: Effect on Mode

The mode is the most frequently occurring value. If we add \(1.5\) to each data - value, the most frequently occurring value (in terms of its original value) will now have a value that is \(1.5\) more than the original mode. The original mode \(Mo=12.5\). The new mode \(Mo_{new}=12.5 + 1.5=14.0\).

Step4: Effect on Range

The range is \(R=\text{Max}-\text{Min}\). If we add \(1.5\) to each data - value, the new maximum \(Max_{new}=\text{Max}+1.5\) and the new minimum \(Min_{new}=\text{Min}+1.5\). Then \(R_{new}=(Max + 1.5)-(Min + 1.5)=\text{Max}-\text{Min}\). So the range remains \(3.2\).

Step5: Effect on IQR

The inter - quartile range \(IQR = Q_{3}-Q_{1}\). When we add \(1.5\) to each data - value, the third quartile \(Q_{3new}=Q_{3}+1.5\) and the first quartile \(Q_{1new}=Q_{1}+1.5\). Then \(IQR_{new}=(Q_{3}+1.5)-(Q_{1}+1.5)=Q_{3}-Q_{1}\). So the \(IQR\) remains \(2.4\).

Answer:

Mean: \(13.3\), Median: \(13.7\), Mode: \(14.0\), Range: \(3.2\), IQR: \(2.4\)