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Nov 2019 p63 q4
3198
The heights of students at the Mainland college are normally distributed with mean 148 cm and standard deviation 8 cm.
The probability that a Mainland student chosen at random has a height less than h cm is 0.67. Find the value of h.
Solution
Given that the heights are normally distributed with mean \(\mu = 148\) cm and standard deviation \(\sigma = 8\) cm, we need to find the height \(h\) such that \(P(X < h) = 0.67\).
Using the standard normal distribution, we find the z-score corresponding to a probability of 0.67. From standard normal distribution tables, \(P(Z < 0.44) = 0.67\), so \(z = 0.44\).
The z-score formula is \(z = \frac{h - \mu}{\sigma}\).
Substitute the known values: \(0.44 = \frac{h - 148}{8}\).
Solve for \(h\):
\(h - 148 = 0.44 \times 8\)
\(h - 148 = 3.52\)
\(h = 148 + 3.52\)
\(h = 151.52\)
Rounding to the nearest whole number, \(h = 152\).
