Let's analyze the given statistical information to determine which statement is true about Raquel's and Van's gas-price data.
Raquel's data:
- Mean (average) gas price: [tex]\(\bar{x} = 3.42\)[/tex]
- Standard deviation: [tex]\(\sigma = 0.07\)[/tex]
Van's data:
- Mean (average) gas price: [tex]\(\bar{x} = 3.78\)[/tex]
- Standard deviation: [tex]\(\sigma = 0.23\)[/tex]
The mean (average) gives us the central value around which the data points are clustered. The standard deviation gives us an indication of how spread out the data points are around the mean. A smaller standard deviation means that the data points tend to be closer to the mean, while a larger standard deviation means the data points are more spread out.
### Comparing the Data
1. Raquel's Data:
- Mean gas price: [tex]$3.42
- Small standard deviation: $[/tex]0.07
Since the standard deviation is small, Raquel's gas prices are tightly clustered around the mean value of [tex]$3.42. This means her data points are generally closer to $[/tex]3.42.
2. Van's Data:
- Mean gas price: [tex]$3.78
- Larger standard deviation: $[/tex]0.23
The larger standard deviation indicates that Van's gas prices are more spread out around the mean value of [tex]$3.78. This means his data points are not as tightly clustered around the mean compared to Raquel's data.
### Conclusion
Given the smaller standard deviation in Raquel's data (\(\sigma = 0.07\)), Raquel's gas-price data are most likely closer to her mean of $[/tex]3.42 than Van's gas-price data are to his mean of [tex]$3.78. This translates to the statement:
"Raquel's data are most likely closer to $[/tex]3.42 than Van's data are to [tex]$3.78."
Thus, the correct statement is:
Raquel's data are most likely closer to $[/tex]3.42 than Van's data are to $3.78.
