Questions: Fill in the blanks. (Enter an exact positive number as an integer, fraction, or decimal.) In a normal distribution, x=3 and z=-1.21. This tells you that x=3 is standard deviations to the of the mean.

$Fill in the blanks. (Enter an exact positive number as an integer, fraction, or decimal.) In a normal distribution, x=3 and z=-1.21. This tells you that x=3 is standard deviations to the of the mean.$

Transcript text: Fill in the blanks. (Enter an exact positive number as an integer, fraction, or decimal.) In a normal distribution, $x=3$ and $z=-1.21$. This tells you that $x=3$ is $\square$ standard deviations to the $\square$ of the mean.

Solution

Solution Steps

Step 1: Understanding the Z-Score

In a normal distribution, the z-score is calculated using the formula:

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

where:

$ z $ is the z-score,
$ x $ is the value in question,
$ \mu $ is the mean of the distribution,
$ \sigma $ is the standard deviation.

Step 2: Rearranging the Formula

Given that $ x = 3 $ and $ z = -1.21 $, we can rearrange the formula to express the relationship between $ x $, the mean $ \mu $, and the standard deviation $ \sigma $:

\[ 3 = \mu - 1.21 \sigma \]

This indicates that $ x = 3 $ is located $ 1.21 $ standard deviations to the left of the mean.

Step 3: Conclusion

Since the z-score is negative, it confirms that $ x = 3 $ is indeed $ 1.21 $ standard deviations below the mean.