Okay.
If you notice the box-and-whisker plot is normally distributed i.e. the mid-point of the interquartile range is the median, and the distribution extends to equal length on either side. Hence, for part (a), the mid-point is the mean, just like in the normal distribution curve, the line of symmetry passes through the mean. i.e. (63+39)/2 = 51.
In the second part, the examiner wants the standard deviation. You have the mean, two values of x and z which can be found by keeping the fact that P(x<=63)=P(x>=39)=0.75. Remember, the quartiles divide the distribution into four parts, such that the Q1 is the lower 1/4th of the distribution, while the Q3 is the 1 - upper 3/4th of the distribution. Meaning thereby, that the probability of x being more than or equal to the Q1 is 0.75, and less than or equal to Q3 is also 0.75. So, z = +0.674 or -0.674. Now just insert these values into the formula:
z = (x - mu)/sigma
Remember, z = -0.674 for x=63, and vice versa. You can use either of the two sets of values.
