The weighted mean, denoted as \(\bar{x}_{w}\) is another measure of central tendency that can be used to assign weights (or measures of influence) to each of the individual observations in a sample. The formula for the weighted mean is as follows:

\[
\bar{x}_{w}=\frac{\sum_{i=1}^{n} w_{i} \cdot x_{i}}{\sum_{i=1}^{n} w_{i}}
\]

where \(w_{i}\) is the weight associated with each observation \(x_{i}\). Suppose you want to find the mean amount that you spent on gasoline (per gallon) over the course of a 12-week period. The data provided in Table 3.19 gives the number of gallons and the price paid per gallon over the 12-week period.

a. Find \(\bar{x}\), the mean amount paid per gallon for gas over the 12-week period.

b. Find \(\bar{x}_{w}\), the (weighted) mean amount paid per gallon for gas over the 12 -week period.

c. Describe why you think there is a difference between the mean and the weighted mean.

Table 3.19

Transcribed Image Text:

Number of Gallons of Gasoline, Price Per Gallon, and Total Cost Over a 12-Week Period Number of Gallons 13.2 Price Per Gallon (in Dollars) Total (in Dollars) 2.35 31.02 9.8 12.4 2.41 23.62 2.28 28.27 8.1 2.30 18.63 15.2 2.41 36.63 6.8 2.50 17.00 7.5 2.40 18.00 9.3 2.29 21.30 8.4 2.38 19.99 6.2 2.93 18.17 13.3 2.20 29.26 12.8 2.25 28.80