Question 1143990: the average price of new story townhouse is 1,600,000, find the maximum and minimum prices of the townhouse that a contractor will build to include the middle 70% of the market. Assume that the standard deviation of prices is 300,000 and the variable is normally distributed.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the mean is 700,000
the standard deviation is 300,000
use the z-score formula to find the low and high z-score where 70% of the prices are in the middle of the normal distribution curve.
in the normal distribution curve, 70% in the middle means that 30% are outside the limits which means that 15% are to the left of the middle and 15% are to the right of the middle.
since the z-score tables give you the are to the left, you would look for a z-scored that has 15% of the area to the left of it and you would look for a z-score that have 85% of the area to the left of it.
alternatively, you can use a z-score calculator to find the same, since it is much more accurate when the areas in the table are not right on.
one such calculator can be found at https://stattrek.com/online-calculator/normal.aspx
using this calculator, i get the following z-scores for the 15% to the left and the 85% to the left of the z-score.
the low z-scorei is -1.036.
the high z-score is 1.036.
the z-scores are the same except for the sign because the normal distribution curve is symmetric about the mean.
now that you have the z-scores, you find the raw scores by using the z-score formula of z = (x-m)/s
z is the z-score.
x is the raw score.
m is the raw mean.
s is the standard deviation.
for the low z-score, the formula becomes -1.306 = (x-1,600,000)/300,000.
solve for x to get x = -1.036 * 300,000 + 1,600,000 = 1,289,200.
for the high z-score, the formula becomes 1.036 * 300,000 +1,600,000 = 1,910,800.
that's those are your solutions.
this can be seen visually in the following display.
any differences in the numbers is due to differences in rounding between the two calculators used.
the calculator used in the display can be found at http://davidmlane.com/hyperstat/z_table.html
the first pictures below are finding the z-score and the third picture below is the visual display of the results.
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