Math / Data & Statistics

QuestionA population has mean $\mu=29^{\circ}$ and standard deviation $\sigma=6$ . Round the answers to two decimal places as needed.
Part 1 of 3 (a) Find the $z$ -score for a population value of 6 .
The $z$ -score for a population value of 6 is -3.83 .
Part 2 of 3 (b) Find the $z$ -score for a population value of 40 .
The $z$ -score for a population value of 40 is 1.83 .
Part: $2 / 3$
Part 3 of 3 (c) What number has a $z$ -score of -1.1 ?

Studdy Solution

STEP 1

1. The mean of the population, $\mu$ , is $29$ degrees.
2. The standard deviation of the population, $\sigma$ , is $6$ .
3. The $z$ -score formula is $z = \frac{X - \mu}{\sigma}$ , where $X$ is the population value.
4. We are given a $z$ -score of $-1.1$ and need to find the corresponding population value $X$ .

STEP 2

1. Rearrange the $z$ -score formula to solve for the population value $X$ .
2. Substitute the given $z$ -score, mean, and standard deviation into the rearranged formula.
3. Calculate the population value $X$ .

STEP 3

Start with the $z$ -score formula:
$z = \frac{X - \mu}{\sigma}$
Rearrange the formula to solve for $X$ :
$X = z \cdot \sigma + \mu$

STEP 4

Substitute the given $z$ -score ( $-1.1$ ), mean ( $29$ ), and standard deviation ( $6$ ) into the rearranged formula:
$X = (-1.1) \cdot 6 + 29$

STEP 5

Calculate the value of $X$ :
$X = (-1.1) \cdot 6 + 29 = -6.6 + 29 = 22.4$
The population value corresponding to a $z$ -score of $-1.1$ is $22.4$ degrees.

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