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 $3.42 and a standard deviation of 0.07 .
Van's data has a mean of $3.78 and a standard deviation of 0.23 .
Since Raquel's standard deviation is smaller, her data is more likely closer to her mean.
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. Raquel's data has a mean of $3.42 and a standard deviation of $0.07. Van's data has a mean of $3.78 and a standard deviation of $0.23. The standard deviation tells us how spread out the data is around the mean. A smaller standard deviation means the data points are clustered more closely around the mean, while a larger standard deviation means the data points are more spread out.
Compare standard deviations We need to determine which statement is true about their gas-price data. Since Raquel's standard deviation ($0.07) is smaller than Van's standard deviation ($0.23), Raquel's data is more likely to be closer to her mean ($3.42) than Van's data is to his mean ($3.78).
Conclusion Therefore, Raquel's data are most likely closer to $3.42 than Van's data are to $3.78 .
Examples
Understanding the spread of data is crucial in many real-world scenarios. For instance, in manufacturing, a smaller standard deviation in the dimensions of produced parts indicates higher consistency and quality. Similarly, in finance, a smaller standard deviation in investment returns suggests lower risk. In our case, comparing the standard deviations of gas prices helps us understand whose data is more tightly clustered around their average, providing insights into the price consistency in their respective cities.
Raquel's gas price data is more closely clustered around her mean of $3.42 due to a smaller standard deviation of $0.07. Therefore, her data is more likely to be closer to $3.42 than Van's, which has a larger standard deviation of $0.23. The correct choice is A: Raquel's data are most likely closer to $3.42 than Van's data are to $3.78.
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