Question

Data for a particular subdivision near downtown Houston indicate that the average price per square foot for a home is $100 with a standard deviation of $5 (normally distributed). What is the probability that the average price per square foot for a home is greater than $110

Answer 1

Answer:

$P(X>110)=P(\frac{X-\mu}{\sigma}>\frac{110-\mu}{\sigma})=P(Z>\frac{110-100}{5})=P(z>2)$

And we can find this probability with the complement rule and we got:

$P(Z>2) = 1-P(Z<2)$

And using the normal standard table or excel we got:

$P(Z>2) = 1-P(Z<2)=1-0.97725=0.02275$

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the prices of a population, and for this case we know the distribution for X is given by:

$X \sim N(100,5)$  

Where $\mu=100$ and $\sigma=5$

We are interested on this probability

$P(X>110)$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

$z=\frac{x-\mu}{\sigma}$

If we apply this formula to our probability we got this:

$P(X>110)=P(\frac{X-\mu}{\sigma}>\frac{110-\mu}{\sigma})=P(Z>\frac{110-100}{5})=P(z>2)$

And we can find this probability with the complement rule and we got:

$P(Z>2) = 1-P(Z<2)$

And using the normal standard table or excel we got:

$P(Z>2) = 1-P(Z<2)=1-0.97725=0.02275$

And we can find this probability on this way:

$P(-0.50<z<0.65)=P(z<0.65)-P(-0.5)$