Let's analyze the problem step-by-step to determine which statement is true about their gas-price data.
Both Raquel and Van recorded the lowest prices for a gallon of gas at gas stations in their respective cities on the same day.
- For Raquel, we have:
- Mean price ([tex]\(\bar{x}\)[/tex]) = [tex]$3.42
- Standard deviation (\(\sigma\)) = $[/tex]0.07
- For Van, we have:
- Mean price ([tex]\(\bar{x}\)[/tex]) = [tex]$3.78
- Standard deviation (\(\sigma\)) = $[/tex]0.23
To evaluate the closeness of their data points to the mean values, we use the concept of standard deviation. The standard deviation provides a measure of how spread out the data points are from the mean ([tex]\(\bar{x}\)[/tex]).
1. Raquel's data:
- The mean price is [tex]$3.42, with a standard deviation of $[/tex]0.07.
- A smaller standard deviation indicates that Raquel's gas prices are more tightly clustered around the mean value of [tex]$3.42.
2. Van's data:
- The mean price is $[/tex]3.78, with a standard deviation of [tex]$0.23.
- A larger standard deviation suggests that Van's gas prices are more spread out from the mean value of $[/tex]3.78 compared to Raquel's data.
Considering the standard deviations, we see that Raquel's data have a smaller standard deviation ([tex]\(\sigma = 0.07\)[/tex]) than Van's data ([tex]\(\sigma = 0.23\)[/tex]). This indicates that Raquel's recorded gas prices are more likely to be closer to her mean value of [tex]$3.42 than Van's gas prices are to his mean value of $[/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 $[/tex]3.78."
