Question

salary range: proportion within $30,000 and $70,000
the average starting salary is $mu = 49000$ and $sigma = 9000$. what proportion of working recent college graduates earn a salary within the range of $30,000$ to $70,000$
to answer this question, follow the steps below and respond to the questions after each step.
step 1: choose the correct rule for finding the proportion within two values.
? question: what rule do we choose?
step 2: compute the two z - scores and
$z=\frac{x - mu}{sigma}$

1. compute $z_1$ for $30,000$
2. compute $z_2$ for $70,000$

note: round both z - scores to two decim
? question: what is the value of $z_1$?
? question: what is the value of $z_2$?

Explanation:

Step1: Recall normal - distribution rule

To find the proportion of values within a range in a normal distribution, we use the z - score formula $z=\frac{X - \mu}{\sigma}$ and look up the z - scores in the standard normal distribution table. The rule for finding the proportion within two values is to convert the values to z - scores and then find the difference between the cumulative distribution function values of the two z - scores. Here, the correct rule is to use the z - score transformation and standard normal table lookup. We assume Rule 1 is the correct rule for this process (as it is likely the rule related to z - score and normal distribution proportion finding).

Step2: Compute $z_1$

Given $\mu = 49000$, $\sigma=9000$, and $X_1 = 30000$. Using the z - score formula $z_1=\frac{X_1-\mu}{\sigma}=\frac{30000 - 49000}{9000}=\frac{- 19000}{9000}\approx - 2.11$.

Step3: Compute $z_2$

Given $X_2 = 70000$, $\mu = 49000$, and $\sigma = 9000$. Using the z - score formula $z_2=\frac{X_2-\mu}{\sigma}=\frac{70000 - 49000}{9000}=\frac{21000}{9000}\approx2.33$.

Answer:

Rule: Rule 1
$z_1$: - 2.11
$z_2$: 2.33