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June 2018 p63 q2
3210
The random variable X has the distribution \(N(-3, \sigma^2)\). The probability that a randomly chosen value of X is positive is 0.25.
Find the value of \(\sigma\). [3]
Find the probability that, of 8 random values of X, fewer than 2 will be positive. [3]
Solution
(i) To find \(\sigma\), we use the standard normal distribution. The probability that \(X > 0\) is 0.25, which corresponds to a \(z\)-value of 0.674 (since the probability to the right of the mean is 0.25).
Using the standardization formula:
\(z = \frac{0 - (-3)}{\sigma} = 0.674\)
Solving for \(\sigma\):
\(\sigma = \frac{3}{0.674} \approx 4.45\)
(ii) The probability that a single value of \(X\) is positive is 0.25. We need to find the probability that fewer than 2 out of 8 values are positive. This is a binomial distribution problem with \(n = 8\) and \(p = 0.25\).
