For a normal distribution, the value of kkk that corresponds to the central 75% of the observations can be found using standard normal distribution properties.
* Normal Distribution:
* The total area under the normal curve is 1 (or 100%).
* Central 75%:
* This means the middle 75% of the distribution is within a range of ±k\pm k±k standard deviations from the mean.
* The remaining 25% is split equally on both tails, so each tail has 12.5%.
* Z-Scores:
* The Z-score for 87.5% cumulative probability (100% - 12.5%) corresponds to the value of kkk.
* Looking up the Z-score in standard normal distribution tables or using statistical software, we find Z1.28Z \approx 1.28Z1.28.
References: Statistical tables for normal distribution and probability theory texts.