The mean for Eric's class is $148. The mean for Natalia's class is $152. Natalia's sales total $150. If Natalia moves to Eric's class, the mean for each class increases. Explain why.
Natalia's is below the average of her class but above the average of Eric's class.
To explain why the mean for each class increases when Natalia moves to Eric's class, we need to consider how the mean is calculated and the impact of adding Natalia's sales to Eric's class.
The mean, also known as the average, is calculated by dividing the sum of all the values in a dataset by the number of values. In this case, the mean for Eric's class is $148, and the mean for Natalia's class is $152.
Let's say there are a total of 'n' students in each class. The sum of sales in Eric's class can be represented as 'Sum(Eric)' and the sum of sales in Natalia's class can be represented as 'Sum(Natalia)'. So, Eric's total sales divided by 'n' gives us a mean of $148, and Natalia's total sales divided by 'n' gives us a mean of $152.
Now, when Natalia moves to Eric's class, we need to calculate the new mean for each class. We will have a total of 'n+1' students in Eric's class, as Natalia will join. Hence, the sum of sales in Eric's class will be 'Sum(Eric) + 150' (since Natalia's sales total is $150).
To calculate the new mean for Eric's class, we divide the new total sales in Eric's class by 'n+1'. So, the new mean for Eric's class will be (Sum(Eric) + 150) / (n+1).
On the other hand, in Natalia's class, she is no longer present, so the sum of sales will be 'Sum(Natalia) - 150'. Since there are still 'n' students in Natalia's class, the new mean for Natalia's class will be (Sum(Natalia) - 150) / n.
Comparing the two mean calculations, Eric's mean (with Natalia) is (Sum(Eric) + 150) / (n+1), and Natalia's mean (without Natalia) is (Sum(Natalia) - 150) / n.
Since 'Sum(Eric)' is less than 'Sum(Natalia)', adding a positive value (150) to 'Sum(Eric)' in the numerator of Eric's mean calculation will result in a larger value compared to Natalia's mean calculation. Similarly, subtracting 150 from 'Sum(Natalia)' in the numerator of Natalia's mean calculation will yield a smaller value.
Therefore, when Natalia moves to Eric's class, the numerator in Eric's mean calculation increases more than the numerator in Natalia's mean calculation decreases. As a result, the mean for each class increases.