If the mean height of a group of students equals 170cm with a standard deviation of 10 cm, then the probability that a student is between 160cm and 180cm is .
We have, "\\mu=170,\\space \\sigma=10"
We are required to find the probability
"p(160\\lt x\\lt 180)"
Since we have the values of "\\mu" and "\\sigma" and this probability can be standardized and the desired probability found using the standard normal tables as follows.
"p((160-\\mu)\/\\sigma\\lt (x-\\mu)\/\\sigma\\lt (180-\\mu)\/\\sigma)=p((160-\\mu)\/\\sigma\\lt Z\\lt (180-\\mu)\/\\sigma)" Substituting for "\\mu" and "\\sigma" gives,
"p((160-170)\/10\\lt Z\\lt (180-170)\/10)=p(-1\\lt Z \\lt 1)=\\phi(1)-\\phi(-1)"
From the standard normal tables, "\\phi(1)=0.8413" and "\\phi(-1)=0.1587" . Therefore, "\\phi(1)-\\phi(-1)=0.8413-0.1587=0.6826"
Therefore, the probability that the height of a student is between 160cm and 180cm is 0.6826.
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