Question

Salaries for teachers in a particular elementary school district are normally distributed with a mean of$\backslash (44,000$and a standard deviation of $\backslash )6,500$. We randomly survey ten teachers from that district.

a. Find the$90th$percentile for an individual teacher’s salary.

b. Find the $90th$percentile for the average teacher’s salary.

Short answer

(a)$90th$percentile for individual teacher's salary is .

(b)$90th$percentile for average teacher's salary is $\$46634.20$.

Step-by-step solution

Part (a) Step 1: Given information 

Given in the question that, Salaries for teachers in a particular elementary school district are normally distributed with a mean of $\$44,000$ and a standard deviation of $\$6,500$. We randomly survey ten teachers from that district.

Part (a) Step 2: Explanation

According to the information provided, teacher salaries in a certain elementary school are normally distributed, with a mean of $\$44000$and a standard deviation of $\$6500$, and a random poll of $10$instructors from that district is conducted.

Let $X$represent the teacher's salary, which is normally distributed.

$X~N(\$44000,\$6500)$

Use a Ti-83 calculator to get thelocalid="1648396959689" ${90}^{\text{th}}$percentile for individual teacher salaries. To do so, go tolocalid="1648396972586" $2^{\text{nd}}$, then DISTR, then scroll down to the invNorm option and fill in the required information. After that, press the calculator's ENTER key to get the desired result. The following is a screenshot:

Part (b) Step 1: Given information

Given in the question that, Salaries for teachers in a particular elementary school district are normally distributed with a mean of $\$44,000$ and a standard deviation of $\$6,500$. We randomly survey ten teachers from that district.

Part (b) Step 2: Explanation

According to the supplied details, salaries for teachers in a particular elementary school are normally distributed with mean $\$44000$and standard deviation $\$6500$, and a random survey of $10$teachers is done from that district.

Let denotes the average salary of $10$randomly chosen teachers from the district.

$\overline{X}~N\left(\mu ,\frac{\sigma }{\sqrt{n}}\right)$

$~N\left(44000,\frac{6500}{\sqrt{10}}\right)$

$~N(44000,2055.48)$

To calculate${90}^{\text{th}}$percentile for average teacher salary, use $Ti-83$calculator. For this, click on $2^{\text{nd}}$, then DISTR and then scroll down to the invNorm option and enter the provided details. After this, click on ENTER button of calculator to have the desired result. The screenshot is given as below: