To determine which statement is true about Raquel's and Van's gas-price data, let's analyze the given information.
Raquel's data:
- Mean ([tex]$\bar{x}$[/tex]): \[tex]$3.42
- Standard deviation ($[/tex]\sigma[tex]$): 0.07
Van's data:
- Mean ($[/tex]\bar{x}[tex]$): \$[/tex]3.78
- Standard deviation ([tex]$\sigma$[/tex]): 0.23
The standard deviation is a measure of the amount of variation or dispersion in a set of values. A smaller standard deviation indicates that the values tend to be closer to the mean, while a larger standard deviation indicates that the values are spread out over a larger range.
For Raquel:
- Mean is \[tex]$3.42, and the standard deviation is 0.07.
- This small standard deviation suggests that Raquel's gas prices are closely clustered around \$[/tex]3.42.
For Van:
- Mean is \[tex]$3.78, and the standard deviation is 0.23.
- This larger standard deviation suggests that Van's gas prices vary widely around \$[/tex]3.78.
The key to determining which dataset's values are more tightly clustered around their mean is comparing their standard deviations. Raquel's smaller standard deviation (0.07) indicates that her gas prices are more consistently close to \[tex]$3.42 compared to Van's gas prices being close to \$[/tex]3.78, which have a larger spread (0.23).
Therefore, the proper statement based on their data is:
Raquel's data are most likely closer to \[tex]$3.42 than Van's data are to \$[/tex]3.78.
This is the most accurate reflection of the tight clustering of Raquel's data around her mean price compared to Van's.
