example 10: the average monthly mortgage payment including principal and interest is $982 in the united…

example 10: the average monthly mortgage payment including principal and interest is $982 in the united states. if the standard deviation is approximately $180 and the mortgage payments are approximately normally distributed, find the probability that a randomly selected monthly payment is between $800 and $1150.

example 10: the average monthly mortgage payment including principal and interest is $982 in the united states. if the standard deviation is approximately $180 and the mortgage payments are approximately normally distributed, find the probability that a randomly selected monthly payment is between $800 and $1150.

Answer

Explanation:

Step1: Calculate z - scores

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $\mu = 982$ (mean), $\sigma=180$ (standard deviation). For $x = 800$, $z_1=\frac{800 - 982}{180}=\frac{-182}{180}\approx - 1.01$. For $x = 1150$, $z_2=\frac{1150 - 982}{180}=\frac{168}{180}\approx0.93$.

Step2: Find probabilities from z - table

We use the standard normal distribution table. $P(Z < - 1.01)=0.1562$ and $P(Z < 0.93)=0.8238$.

Step3: Calculate the probability between the two values

$P(-1.01<Z<0.93)=P(Z < 0.93)-P(Z < - 1.01)$. $P(-1.01<Z<0.93)=0.8238 - 0.1562=0.6676$.

Answer:

$0.6676$