Question

assume the random variable x is normally distributed with mean μ = 173 and standard deviation σ = 20. find the indicated probability. p(153 < x < 182) p(153 < x < 182)= (round to four decimal places as needed.)

Explanation:

Step1: Standardize the lower - bound

We use the formula $z=\frac{x-\mu}{\sigma}$. For $x = 153$, $\mu=173$ and $\sigma = 20$, we have $z_1=\frac{153 - 173}{20}=\frac{- 20}{20}=-1$.

Step2: Standardize the upper - bound

For $x = 182$, using the same formula $z=\frac{x-\mu}{\sigma}$, we get $z_2=\frac{182-173}{20}=\frac{9}{20}=0.45$.

Step3: Find the probabilities

We know that $P(153 < x<182)=P(-1 < z<0.45)$. Since $P(-1 < z<0.45)=P(z < 0.45)-P(z < - 1)$.
From the standard normal distribution table, $P(z < 0.45)=0.6736$ and $P(z < - 1)=0.1587$.

Step4: Calculate the final probability

$P(-1 < z<0.45)=0.6736 - 0.1587=0.5149$.

Answer:

$0.5149$