The following data sets represent the highway fuel efficiency in miles per gallon (mpg) of randomly selected cars from two car companies. Which company has cars that are more consistent in highway fuel efficiency?
Car Company A: 35, 28, 35, 30, 31, 36, 35, 30
Car Company B: 29, 33, 40, 27, 34, 34, 34, 25
(1 point) bot
h companies have the same mean highway fuel efficiency of 32.5 mpg. However, Car Company B has a smaller range (15 mpg) compared to Car Company A (8 mpg), indicating that their cars have more consistent highway fuel efficiency. Additionally, the standard deviation of Car Company B's data set (4.29 mpg) is smaller than that of Car Company A (2.48 mpg), further supporting the conclusion that Car Company B has cars that are more consistent in highway fuel efficiency. Therefore, Car Company B has cars that are more consistent in highway fuel efficiency.
To determine which car company has cars that are more consistent in highway fuel efficiency, we can calculate the standard deviation of each data set. The lower the standard deviation, the more consistent the data's values are.
For Car Company A:
Step 1: Calculate the mean (average) of the data set:
Mean = (35 + 28 + 35 + 30 + 31 + 36 + 35 + 30) / 8
= 260 / 8
= 32.5
Step 2: Calculate the differences between each data point and the mean:
Differences = (35 - 32.5), (28 - 32.5), (35 - 32.5), (30 - 32.5), (31 - 32.5),
(36 - 32.5), (35 - 32.5), (30 - 32.5)
Step 3: Square each difference:
Squared Differences = (35 - 32.5)^2, (28 - 32.5)^2, (35 - 32.5)^2, (30 - 32.5)^2,
(31 - 32.5)^2, (36 - 32.5)^2, (35 - 32.5)^2, (30 - 32.5)^2
Step 4: Calculate the average of the squared differences (this is also known as the variance):
Variance = ( (35 - 32.5)^2 + (28 - 32.5)^2 + (35 - 32.5)^2 + (30 - 32.5)^2 +
(31 - 32.5)^2 + (36 - 32.5)^2 + (35 - 32.5)^2 + (30 - 32.5)^2 ) / 8
= (2.25 + 20.25 + 2.25 + 6.25 + 2.25 + 12.25 + 2.25 + 6.25) / 8
= 54.75 / 8
= 6.84375
Step 5: Calculate the square root of the variance to find the standard deviation:
Standard Deviation = √6.84375
≈ 2.61799
For Car Company B:
Step 1: Calculate the mean (average) of the data set:
Mean = (29 + 33 + 40 + 27 + 34 + 34 + 34 + 25) / 8
= 256 / 8
= 32
Step 2: Calculate the differences between each data point and the mean:
Differences = (29 - 32), (33 - 32), (40 - 32), (27 - 32), (34 - 32),
(34 - 32), (34 - 32), (25 - 32)
Step 3: Square each difference:
Squared Differences = (29 - 32)^2, (33 - 32)^2, (40 - 32)^2, (27 - 32)^2,
(34 - 32)^2, (34 - 32)^2, (34 - 32)^2, (25 - 32)^2
Step 4: Calculate the average of the squared differences (this is also known as the variance):
Variance = ( (29 - 32)^2 + (33 - 32)^2 + (40 - 32)^2 + (27 - 32)^2 +
(34 - 32)^2 + (34 - 32)^2 + (34 - 32)^2 + (25 - 32)^2 ) / 8
= (9 + 1 + 64 + 25 + 4 + 4 + 4 + 49) / 8
= 160 / 8
= 20
Step 5: Calculate the square root of the variance to find the standard deviation:
Standard Deviation = √20
≈ 4.47214
Comparing the standard deviations, we see that Car Company A has a standard deviation of approximately 2.61799, while Car Company B has a standard deviation of approximately 4.47214.
Therefore, Car Company A has cars that are more consistent in highway fuel efficiency.