Question

how do you see it? the scatter - plot shows the attendance for each member of a gaming club.

a. the mean attendance for the first four meetings is 20. is the number of students who attended greater than, less than, or equal to 20? explain.

equal to 20: the only way for the mean to equal 20 is if the number of students who attended the fourth meeting is also 20. the mean is always equal to one of the values included in the data set.

greater than 20: the attendance at the second meeting is only 1 above 20 and the attendance at the other two is more than one below 20. the attendance at the fourth meeting must be greater than 20 in order to have a mean of 20.

less than 20: it is not possible for the attendance to be greater than or equal to 20 since the mean is only 20. the total number who attended the four meetings must be less than four times the mean.

less than or equal to 20: the attendance at each of the other three meetings was either greater than or less than 20. to have a mean of 20, the mean must be less than or equal to 20.

Explanation:

Step1: Recall the mean formula

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 4$ (number of meetings) and $\bar{x}=20$. So $\sum_{i = 1}^{4}x_{i}=n\times\bar{x}=4\times20 = 80$.

Step2: Analyze given - data

Let the attendances of the first three meetings be $x_1,x_2,x_3$. Suppose $x_1<20,x_2 = 21,x_3<20$. Then $x_1 + x_2+x_3= x_1 + 21+x_3<20 + 21+20=61$.

Step3: Calculate the fourth - meeting attendance

Let the attendance of the fourth meeting be $x_4$. We know $x_1 + x_2+x_3+x_4 = 80$. So $x_4=80-(x_1 + x_2+x_3)>80 - 61=19$. Since $x_1 + x_2+x_3<61$, $x_4>19$. To make the mean 20, $x_4$ must be greater than 20.

Answer:

Greater than 20: The attendance at the second meeting is only 1 above 20 and the attendance at the other two is more than one below 20. The attendance at the fourth meeting must be greater than 20 in order to have a mean of 20.