Raquel and Van live in two different cities. As part of a project, they each record the lowest prices for a gallon of gas at gas stations around their cities on the same day. Raquel's data show [tex]$\bar{x}=3.42$[/tex] and [tex]$\sigma=0.07$[/tex]. Van's data show [tex]$\bar{x}=3.78$[/tex] and [tex]$\sigma=0.23$[/tex].
Which statement is true about their gas-price data?
A. Raquel's data are most likely closer to $3.42 than Van's data are to $3.78.
B. Van's data are most likely closer to $3.42 than Raquel's data are to $3.78.
C. Raquel's data are most likely closer to $3.78 than Van's data are to $3.42.
D. Van's data are most likely closer to $3.78 than Raquel's data are to $3.42.
Answer (2)
Raquel's data has a mean of x R ˉ = 3.42 and a standard deviation of σ R = 0.07 .
Van's data has a mean of x V ˉ = 3.78 and a standard deviation of σ V = 0.23 .
Since σ R < σ V , Raquel's data is more likely to be closer to its mean.
Therefore, Raquel's data are most likely closer to $3.42 than Van's data are to $3.78 .
Explanation
Understand the problem and provided data We are given the mean and standard deviation of gas prices recorded by Raquel and Van in their respective cities. Raquel's data has a mean of x R ˉ = 3.42 and a standard deviation of σ R = 0.07 . Van's data has a mean of x V ˉ = 3.78 and a standard deviation of σ V = 0.23 . The standard deviation tells us how spread out the data is around the mean. A smaller standard deviation indicates that the data points are clustered more closely around the mean, while a larger standard deviation indicates that the data points are more spread out.
Objective and key idea We need to determine which statement is true about their gas-price data based on the given means and standard deviations. The key idea is that the data with the smaller standard deviation is more likely to be closer to its mean.
Compare standard deviations Comparing the standard deviations, we have σ R = 0.07 and σ V = 0.23 . Since 0.07 < 0.23 , Raquel's data has a smaller standard deviation than Van's data. This means that Raquel's data points are more likely to be closer to her mean of $3.42 than Van's data points are to his mean of $3.78.
Conclusion Therefore, Raquel's data are most likely closer to $3.42 than Van's data are to $3.78 .
Examples
Imagine you're tracking the fuel efficiency of two different car models. If one model has a lower standard deviation in its MPG (miles per gallon) data, it means its fuel efficiency is more consistent and predictable compared to the other model. This information is valuable for consumers looking for reliable fuel economy.
Raquel's gas prices are more consistently clustered around the mean of $3.42 due to a lower standard deviation of $0.07, compared to Van's higher standard deviation of $0.23 around his mean of $3.78. Therefore, the correct statement is A: Raquel's data are most likely closer to $3.42 than Van's data are to $3.78.
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