Question

1. gina is examining gas prices across the county. she randomly selects 10 gas stations, each from different states, and records the gas price. the average price per gallon of regular gas is $3.31. the gas station located in california has a gas price of $4.66 per gallon and a z - score of 2.04. find the standard deviation of the gas prices for these 10 gas stations.
2. values a and b are two different data values from the same distribution. value a has a z - score of 1.4 and value b has a z - score of -2.1.

a. which value is farther from the mean? explain.
b. which value is at the larger percentile? explain.

1. as part of a training program, inexperienced pilots take an online reaction speed test. the mean reaction time for the test is 273 milliseconds with a standard deviation of 82 milliseconds.

a. fiona flyer’s reaction time has a z - score of -0.9. what was fiona’s reaction time?
b. tommy takeoff’s reaction time was the same amount of time away from the mean reaction time as fiona’s, but his time was not the same as fiona’s. how is this possible?
c. what was tommy’s reaction time?

Explanation:

Problem 3

Step1: Recall z - score formula

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $z$ is the z - score, $x$ is the data value, $\mu$ is the mean, and $\sigma$ is the standard deviation. We know that $z = 2.04$, $x=4.66$, and $\mu = 3.31$. We need to solve for $\sigma$.

Step2: Rearrange the formula to solve for $\sigma$

Starting with $z=\frac{x - \mu}{\sigma}$, we can multiply both sides by $\sigma$ to get $z\sigma=x - \mu$. Then divide both sides by $z$ (assuming $z
eq0$) to obtain $\sigma=\frac{x-\mu}{z}$.

Step3: Substitute the known values

Substitute $x = 4.66$, $\mu=3.31$, and $z = 2.04$ into the formula: $\sigma=\frac{4.66 - 3.31}{2.04}$.
First, calculate the numerator: $4.66-3.31 = 1.35$.
Then, divide by the denominator: $\sigma=\frac{1.35}{2.04}\approx0.66$.

The distance of a data value from the mean in terms of z - scores is given by the absolute value of the z - score. For Value A, $|z_A|=|1.4| = 1.4$. For Value B, $|z_B|=|- 2.1|=2.1$. Since $2.1>1.4$, Value B is farther from the mean.

A positive z - score means the value is above the mean, and a negative z - score means the value is below the mean. In a normal distribution (or most distributions), values above the mean are at higher percentiles. Value A has a positive z - score ($z = 1.4$), so it is above the mean, while Value B has a negative z - score ($z=-2.1$), so it is below the mean. Thus, Value A is at a larger percentile.

Answer:

The standard deviation is approximately $\$0.66$ per gallon.

Problem 4

Part (a)