Or a very large set of data the measured mean is found to be 288.6 with a standard deviation of 21.2. assuming the data to be normally distributed, determine the range within which 75% of the data are expected to fall.

The 68-95-99.7 rule tells us 68% of the probability is between -1 standard deviation and +1 standard deviation from the mean. So we expect 75% corresponds to slightly more than 1 standard deviation.

Usually the unit normal tables don't report the area between -σ and σ but instead a cumulative probability, the area between -∞ and σ. 75% corresponds to 37.5% in each half so a cumulative probability of 50%+37.5%=87.5%. We look that up in the normal table and get σ=1.15.

So we expect 75% of normally distributed data to fall within μ-1.15σ and μ+1.15σ

That's 288.6 - 1.15(21.2) to 288.6 + 1.15(21.2)

264.22 to 312.98