(a) A random variable X follows a normal distribution with mean 60 and standard deviation 5 - Leaving Cert Mathematics - Question 2 - 2013

To find P(52 ≤ X ≤ 68), we can calculate it as follows:

$P(52 ≤ X ≤ 68) = P(X ≤ 68) - P(X < 52)$

First, let's standardize 52:

$Z = \frac{52 - 60}{5} = \frac{-8}{5} = -1.6$

Using the Z-table again:

$P(Z ≤ -1.6) ≈ 0.0548$

Now, substituting back:

$P(52 ≤ X ≤ 68) = P(X ≤ 68) - P(X < 52)$ $= 0.9452 - 0.0548 = 0.8904$

Sketch a new distribution that is shifted upwards to indicate an increase in height for all plants, while maintaining the same variance.

Sketch a new distribution with a narrower peak to indicate that there are fewer really small and really tall plants, while keeping the mean at µ.

Sketch a new distribution that is flatter than the original to indicate an increase in both small and tall plants, with the mean remaining unchanged.