z chart question.

[hcc221]

The selling price of various homes in a community follows the normal distribution with a mean of $176,000 and a standard deviation of $22,300. What is the probability that the next house will sell for less than $190,000?

i know that you use the conversion formula which would equal to 190,000-176,000 / 22,300 = .63 and that .63 would equal to .2357 in a z chart. my question is whether you would add .5 to the .2357 or minus .5 to .2357. please explain to me why you would add or minus? thank you

 

[chrisr]

The z-chart that you are using maybe displays only positive values of z.

In your case z=0.63 returns a value of 0.2357,
meaning that the probability that z could be between 0 and 0.63 is 0.2357.
For values of house price less than the mean, z is negative.

If you had a z-table showing positive and negative values of z,
you could directly read off the corresponding percentiles or probabilities.

Since the numerator of z is (x-mean), then z=0 corresponds to x = mean,
while x less than the mean gives negative z,
x greater than the mean gives positive z.
The graph is symmetrical about z=0.

Your value, z=0.63 corresponds to 0.7357 on a z-chart with positive and negative z,
meaning that 73.57% of the graph lies to the left of z=0.63.

Your reading, 0.2357, indicates that 23.57% of the graph lies to the right of z=0 and to the left of z=0.63.

As you are looking for the probability that the next house will sell for under $190,000,
which is the probability that the house price is less than the mean or between the mean and $190,000,
then you want to calculate P(z<0.63) which is P(z<0) + P(0<z>0.63),
so yes, you add your result to 0.5.