Question

Mark recently took a road trip across the country. The number of miles he drove each day was normally distributed with a mean of 450. If he drove 431.8 miles on the last day with a z-score of -0.7, what is the standard deviation?

Answer 1

Answer:

The (population) standard deviation is 26 miles or $\\ \sigma = 26$ miles.

Step-by-step explanation:

We can solve this question using the concept of z-score or standardized value, which is given by the formula:

$\\ z = \frac{x - \mu}{\sigma}$ [1]

Where

$\\ z$ is the z-score.

$\\ x$ is the raw score.

$\\ \mu$ is the population's mean.

$\\ \sigma$ is the population standard deviation.

Analyzing the question, we have the following data to solve this question:

1. The random variable number of miles driven by day is normally distributed.
2. The population's mean is $\\ \mu = 450$ miles.
3. The raw score, that is, the value we want to standardize, is $\\ x = 431.8$ miles.
4. The z-score is $\\ z = -0.7$. It tells us that the raw value (or raw score) is below the population mean because it is negative. It also tells us that this value is 0.7 standard deviations units (below) from $\\ \mu$.

Therefore, using all this information, we can determine the (population) standard deviation using formula [1].

Then, substituting each value in this formula:

$\\ z = \frac{x - \mu}{\sigma}$

Solving it for $\\ \sigma$

Multiplying each side of the formula by $\\ \sigma$

$\\ \sigma*z = (x - \mu) * \frac{\sigma}{\sigma}$

$\\ \sigma*z = (x - \mu) * 1$

$\\ \sigma*z = x - \mu$

Multiplying each side of the formula by $\\ \frac{1}{z}$

$\\ \frac{1}{z}*\sigma*z = \frac{1}{z}*(x - \mu)$

$\\ \frac{z}{z}*\sigma = \frac{x - \mu}{z}$

$\\ 1*\sigma = \frac{x - \mu}{z}$

$\\ \sigma = \frac{x - \mu}{z}$

Then, this formula, solved for $\\ \sigma$, will permit us to find the value for the population standard deviation asked in the question.

$\\ \sigma = \frac{431.8 - 450}{-0.7}$

$\\ \sigma = \frac{-18.2}{-0.7}$

$\\ \sigma = 26$

Thus, the (population) standard deviation is 26 miles or $\\ \sigma = 26$ miles.