Question

1. a theater is selling tickets to a play. adult tickets cost $8 each and childrens tickets cost $5 each. they collect $275 after selling x adult tickets and y childrens tickets. what does the point (30,7) mean in the context of this situation? is the point (5,15) a solution to the given problem? explain your reasoning. 11. the number of hours spent in an airplane on a single flight is recorded on a dot plot. the mean is 5 hours and the standard deviation is approximately 5.82 hours. the median is 4 hours and the iqr is 3 hours. the value 26 hours is an outlier that should not have been included in the data. when the outlier is removed from the data set: a. what is the new mean? b. what is the new median? c. what is the new iqr?

Explanation:

Step1: Calculate the original sum of data

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). Given \(\bar{x}=5\) and assume the number of data points is \(n\) (including the outlier \(26\)). Let \(\sum_{i = 1}^{n}x_{i}=S\), so \(5=\frac{S}{n}\), \(S = 5n\).

Step2: Remove the outlier and calculate new mean (a)

Let the number of data points without the outlier be \(n- 1\). The new sum \(S_{new}=S - 26=5n-26\).
Since we can count the number of dots (including \(26\)): by counting the dots in the dot - plot (excluding \(26\)), assume \(n = 19\) (counting all dots including \(26\)). Then \(S=5\times19 = 95\), \(S_{new}=95 - 26=69\), and the new mean \(\bar{x}_{new}=\frac{69}{18}\approx3.83\)

Step3: Calculate new median (b)

The median is the middle value. With \(n = 19\) (original), the median is the 10th value. When we remove \(26\) (\(n\) becomes \(18\)), the median is the average of the 9th and 10th values. By re - counting the non - outlier data (in order), the new median is still \(4\)

Step4: Calculate new IQR (c)

The IQR is \(Q_{3}-Q_{1}\). Removing the outlier \(26\) does not affect \(Q_{1}\) and \(Q_{3}\) (because \(26\) is a large outlier and \(Q_{1}\) and \(Q_{3}\) are based on the middle \(50\%\) of the data). So the new \(IQR = 3\)

Answer:

a. The new mean is approximately \(3.83\) hours.
b. The new median is \(4\) hours.
c. The new IQR is \(3\) hours.