step 3: apply the rule & report the final proportion\nthe average starting salary is μ = $49,000 and σ =…
step 3: apply the rule & report the final proportion\nthe average starting salary is μ = $49,000 and σ = $9,000. ? question: what is the final proportion of salaries between $30,000 and $70,000?\nproportion within a range\n1. calculate the z - scores for both $30,000 and $70,000\n2. then use the z - table to find the two proportions.\n3. subtract the proportions (z2 - z1) to find the proportion between them.\nselect the correct final proportion between from the options below.\np(z_final)=.0250\np(z_final)=.0273\np(z_final)=.9750\np(z_final)=.9727
Answer
Explanation:
Step1: Calculate z - score for $30,000
The formula for the z - score is $z=\frac{x-\mu}{\sigma}$. Here, $x = 30000$, $\mu=49000$, and $\sigma = 9000$. So, $z_1=\frac{30000 - 49000}{9000}=\frac{- 19000}{9000}\approx - 2.11$.
Step2: Calculate z - score for $70,000
Using the same formula, with $x = 70000$, $\mu=49000$, and $\sigma = 9000$. So, $z_2=\frac{70000 - 49000}{9000}=\frac{21000}{9000}\approx2.33$.
Step3: Find proportions from z - table
From the standard normal distribution table, the proportion corresponding to $z_1=-2.11$ is $P(Z < - 2.11)=0.0174$. The proportion corresponding to $z_2 = 2.33$ is $P(Z < 2.33)=0.9901$.
Step4: Calculate proportion between the two values
The proportion between $z_1$ and $z_2$ is $P(-2.11<Z<2.33)=P(Z < 2.33)-P(Z < - 2.11)=0.9901 - 0.0174=0.9727$.
Answer:
$p(z_{final}) = 0.9727$