From a set of 1000 observations known to be normally distributed, the mean is 534 cm and SD is 13.5 cm. How many observations are likely to exceed 561 cm? How many will be between 520.5 cm and 547.5 cm? Between what limits will the middle 50% of the observations lie?

You need to know how your distribution works.

Have a quick sketch of the distribution.

Let X be the length.
Therefore, X ~ N (534, 13.5^2)

Find the z value of 561, using

Then, find the corresponding probability in the z table, using your sketch to make sure you get the correct probability.

Then, find the z values for the other lengths, and use your sketch again, by shading the required part if necessary.

The last part is the reverse of the first two parts. The middle observations lie 25% to the left and 25% to the right of the mean. Find the z value corresponding to the probabilities, using your sketch again to make sure you get the right probabilities, and convert the z value into the length using the formula I mentioned above.