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    IBMediumStatistics10–12

    Normal Distribution

    Distribuição Normal

    Heights of students follow N(170,82)N(170, 8^{2}). For the standard normal distribution, P(Z>1)≈0.1587P(Z > 1) \approx 0.1587. What percentage of students are taller than 178178 cm?

    A symmetric bell-shaped curve over heights in centimetres, with ticks one standard deviation apart at 146, 154, 162, 170, 178, 186 and 194. The peak sits at the mean, 170 cm, marked by a dashed vertical line. A solid vertical line at 178 cm stands one standard deviation to the right of the mean, exactly where the curve changes from bending downward to bending upward. The area under the curve to the right of 178 cm is shaded, and it is a small part of the total area.146154162170178186194170 cm178 cmtaller than 178 cm

    Standardise the height 178178 with z=x−μσz = \frac{x - \mu}{\sigma} and check that the given P(Z>1)P(Z > 1) is the probability for that zz. What is left is to turn that probability into a percentage and say what share of the students it describes.

    Solution

    Step 1 of 3

    1. 1.z=178−1708=1z = \frac{178 - 170}{8} = 1