Question

In the following, state whether: the probability is greater than 0.5, lower than 0.5, equal to 0.5, or if it is not possible to tell just by common sense, without computing the actual probabilities. (a) P(X < 612) when X ~N(725, 100) (b) P(X > 11.44) when X~N(15.05, 3.45) (c) P(252 < X 3 277) when XN(245,25)

Answer 1

Since Normal distribution is a symmetrical distribution, so if $X \sim Normal(\mu, \sigma^2)$

Then, $P(X \leq \mu) = P(X \geq \mu) = 0.50$

a) $P(X\leq 612) < 0.5$ since 612 < mean = 725

b) $P(X>11.44) = P(11.44 \leq X 15.05) &plus;P(X>15.05)$

$\Rightarrow P(X>11.44) > P(X>15.05) = 0.50$

$\Rightarrow P(X>11.44) > 0.50$

c) $P(252\leq X \leq 277)$ can not told without calculating probability.