Question

the table shows the probabilities of certain prizes in a restaurants contest where the first 100 customers are winners. how does the $100 gift card affect the measure of center of the data? it increases the mean value of the prizes. it decreases the mean value of the prizes. it increases the median value of the prizes. it decreases the median value of the prizes.

Explanation:

Step1: Recall the effect of outliers on mean and median

The mean is calculated by summing all values and dividing by the number of values. The median is the middle value when data is ordered. A large outlier (like the $100$ gift - card) has a greater impact on the mean than on the median.

Step2: Analyze the impact on the mean

The formula for the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. When we add a large value (the $100$ gift - card), the sum $\sum_{i = 1}^{n}x_{i}$ increases. Since $n$ (the number of prizes) is fixed ($n = 100$), the mean $\bar{x}$ will increase.

Step3: Analyze the impact on the median

For $n = 100$ (an even number of data points), the median is the average of the $50^{th}$ and $51^{st}$ ordered values. The $100$ gift - card is just one value. When we order the data (e.g., $1,1,\cdots,1$ (44 times), $5,5,\cdots,5$ (25 times), $5,5,\cdots,5$ (15 times), $10,10,\cdots,10$ (10 times), $20,20,\cdots,20$ (5 times), $100$), the $50^{th}$ and $51^{st}$ values are still $5$. So the median is not affected.

Answer:

It increases the mean value of the prizes.