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][tex]$\bar{x}=3.78$[/tex][/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 [tex][tex]$\$[/tex]3.42$[/tex] than Van's data are to [tex]$\[tex]$3.78$[/tex][/tex].
B. Van's data are most likely closer to [tex]$\$3.42$[/tex] than Raquel's data are to [tex]$\[tex]$3.78$[/tex][/tex].
C. Raquel's data are most likely closer to [tex]$\$3.78$[/tex] than Van's data are to [tex]$\[tex]$3.42$[/tex][/tex].
D. Van's data are most likely closer to [tex]$\$3.78$[/tex] than Raquel's data are to [tex]$\[tex]$3.42$[/tex][/tex].



Answer :

Let's analyze the statements provided regarding Raquel's and Van's gas-price data.

1. Given Data:
- Raquel's average gas price ([tex]\(\bar{x}_R\)[/tex]) is \[tex]$3.42 with a standard deviation (\(\sigma_R\)) of 0.07. - Van's average gas price (\(\bar{x}_V\)) is \$[/tex]3.78 with a standard deviation ([tex]\(\sigma_V\)[/tex]) of 0.23.

2. Understanding the Mean ([tex]\(\bar{x}\)[/tex]) and Standard Deviation ([tex]\(\sigma\)[/tex]):
- The mean is the average value of all gas prices recorded.
- The standard deviation indicates how much the gas prices vary from the mean. A smaller standard deviation signifies that the data points are closer to the mean, whereas a larger standard deviation signifies more variability in the data points around the mean.

3. Comparison of Standard Deviations:
- Raquel's data have a standard deviation of 0.07, which is quite small.
- Van's data have a standard deviation of 0.23, which is larger compared to Raquel's.

4. Interpreting the Standard Deviation in Context:
- Since Raquel's standard deviation (0.07) is smaller than Van's (0.23), Raquel's gas prices are more tightly clustered around her mean price of \[tex]$3.42. - On the other hand, Van's higher standard deviation means his gas prices are more spread out around his mean price of \$[/tex]3.78.

5. Conclusion:
- Given that Raquel has a smaller standard deviation, it means her recorded gas prices are more consistently close to the mean value of \[tex]$3.42. - Therefore, we can conclude that Raquel's data are most likely closer to \$[/tex]3.42 than Van's data are to \[tex]$3.78. Hence, the correct statement is: Raquel's data are most likely closer to \$[/tex]3.42 than Van's data are to \[tex]$3.78. So, the true statement is: Raquel's data are most likely closer to \$[/tex]3.42 than Van's data are to \$3.78.