Question

Don’t call me Refer to Exercise $4$ Alejandro, who sent $92$ texts, has a standardized score of $z=1.89$

a. Interpret this $z-$score.

b. The standard deviation of the distribution of the number of text messages sent over the past $24$ hours by the students in Mr. Williams’s class is $34.15$ Use this information along with Alejandro’s $z-$score to find the mean of the distribution.

Short answer

Part (a) Alejandro's $92$ texts are $1.89$ standard deviations over the mean number of texts sent by the students.

Part (b) Mean of the distribution, $\mu =27.45$

Step-by-step solution

Part (a) Step 1: Given information

Value,$x=92$

Standard deviation, $\sigma =34.15$

$z-$score,$z=1.89$

Part (a) Step 2: Concept

The formula used: $z=\frac{(x-\mu )}{\sigma }$

Part (a) Step 3: Explanation

The amount of standard deviations a value deviates from the mean is described by the $z-$score.

The value is below the mean when the $z-$score is negative.

Whereas,

Positive $z-$implies that the value is greater than the mean.

Thus,

According to the sample, $92$ texts sent by Alejandro are $1.89$ standard deviations above the mean number of texts sent by the students.

Part (b) Step 1: Calculation

Calculate the $z-$ score:

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

Substitute values,

$1.89=\frac{92-\mu }{34.15}$

Multiply both sides by $34.15$:

$64.55=92-\mu$

That becomes

$\mu =92-64.55=27.45$

Because the data values and the mean's units are the same.

Thus,

The mean is $27.45$ text messages.