During the 2011-12 winter season, a homeowner received four deliveries of heating oil, as shown in the following table:
Gallons Purchased Price per Gallon ($)
209 2.60
182 2.40
157 2.78
149 2.74
The homeowner claimed that the mean price he paid for oil during the season was
(2.60 + 2.40 + 2.78 + 2.74)/4 = $2.63 per gallon.
Do you agree with this claim? If not, explain why this method of calculating the mean is not
appropriate in this case and find the correct value of the mean price.
To determine whether the homeowner's claim of a mean price of $2.63 per gallon is correct, we need to analyze the given data.
The homeowner calculated the mean price by simply adding up the prices per gallon for the four deliveries and dividing by 4. However, this method assumes that each delivery is equally important or has the same weight in determining the overall mean price.
In reality, the number of gallons purchased for each delivery varies. This means that some deliveries have a greater impact on the overall price than others. To find the correct mean price, we need to take into account the gallons purchased along with their corresponding prices.
To calculate the weighted mean, we will multiply the gallons purchased by their respective prices, sum up the total cost, and divide it by the total number of gallons purchased.
Let's calculate the weighted mean:
Delivery 1:
Gallons purchased: 209
Price per gallon: $2.60
Total cost: 209 * 2.60 = $543.40
Delivery 2:
Gallons purchased: 182
Price per gallon: $2.40
Total cost: 182 * 2.40 = $436.80
Delivery 3:
Gallons purchased: 157
Price per gallon: $2.78
Total cost: 157 * 2.78 = $436.46
Delivery 4:
Gallons purchased: 149
Price per gallon: $2.74
Total cost: 149 * 2.74 = $408.26
Total gallons purchased: 209 + 182 + 157 + 149 = 697
Total cost: $543.40 + $436.80 + $436.46 + $408.26 = $1824.92
Weighted mean price:
$1824.92 / 697 ≈ $2.62
Therefore, the correct mean price the homeowner paid for oil during the season is approximately $2.62 per gallon, not $2.63 per gallon as claimed.
To determine if the homeowner's claim of a mean price of $2.63 per gallon is correct, we need to evaluate if it is an appropriate method of calculating the mean in this case.
The mean, also known as the average, is obtained by summing up all the values and dividing by the total number of values. However, in this case, the homeowner received different quantities of oil at different prices.
To calculate the correct mean price, we need to consider the total cost of all the oil purchased and divide it by the total number of gallons purchased.
Let's calculate the total cost of oil:
For the first delivery, the homeowner purchased 209 gallons at a price of $2.60 per gallon. Therefore, the total cost for this delivery is 209 * 2.60 = $542.40.
For the second delivery, the homeowner purchased 182 gallons at a price of $2.40 per gallon. So, the total cost for this delivery is 182 * 2.40 = $436.80.
For the third delivery, the homeowner purchased 157 gallons at a price of $2.78 per gallon. Hence, the total cost for this delivery is 157 * 2.78 = $436.46.
Finally, for the fourth delivery, the homeowner purchased 149 gallons at a price of $2.74 per gallon. Thus, the total cost for this delivery is 149 * 2.74 = $407.26.
Now, let's find the total cost of all deliveries combined: $542.40 + $436.80 + $436.46 + $407.26 = $1822.92.
Next, we need to calculate the total number of gallons purchased: 209 + 182 + 157 + 149 = 697 gallons.
To find the correct mean price, we divide the total cost by the total number of gallons: $1822.92 / 697 = $2.61 per gallon.
Therefore, the correct mean price of oil purchased during the season is $2.61 per gallon, which is different from the homeowner's claim of $2.63 per gallon.