Answer to Question #297899 in Statistics and Probability for clemclem

Question

Oranges chosen at random have a mean mass of 210 grams and a standard deviation of 35 grams. Assuming a normal distribution, what's the probability of choosing an orange with a mass between 170 grams and 250 grams?

Answer 1

Given: $\mu=210,\sigma=35$

$P(170 < X < 250)=P(X<250)-P(X<170)$

Now,

$\begin{aligned} & P[x <250] \\ =& P\left[z <\frac{X-\mu}{\sigma}\right] \\ =& P\left[z <\frac{250-210}{35}\right] \\ =& P[z <1.1428] \\ =& 0.87344 \\ \end{aligned}$

and

$\begin{aligned} & P[x <170] \\ =& P\left[z <\frac{X-\mu}{\sigma}\right] \\ =& P\left[z <\frac{170-210}{35}\right] \\ =& P[z <-1.1428] \\ =& 0.12656 \\ \end{aligned}$

So,

$P(170<X<250)=P(X<250)−P(X<170)=0.87344-0.12656=0.74688$