For a normal distribution, what is the value of k when li±£g contains 75% of the observations?

The question asks for the value of k such that the interval from μ - kσ to μ + kσ contains 75% of the observations for a normal distribution. This is equivalent to finding the z-score 'k' where the area under the standard normal curve between -k and +k is 0.75.

Due to the symmetry of the normal distribution, the area in each tail is (1 - 0.75) / 2 = 0.125. The cumulative probability up to +k is the sum of the central area and the left tail, which is 0.75 + 0.125 = 0.875. We must find the z-score that corresponds to a cumulative probability of 0.875. Consulting a standard normal (Z) table, a cumulative probability of 0.8749 corresponds to a z-score of 1.15.

A. 0.68: A k-value of approximately 0.6745 corresponds to the interquartile range, which contains the central 50% of observations, not 75%.

B. 1.04: A k-value of 1.04 corresponds to a central area of approximately 70.2% of the observations, which is not the required 75%.

D. 1.28: A k-value of approximately 1.28 corresponds to a central area of 80% of the observations, which is a common value for an 80% confidence interval.

1. Montgomery

D. C. (2013). Introduction to Statistical Quality Control (7th ed.). John Wiley & Sons.

Reference: Appendix A

Table II

"Cumulative Standard Normal Distribution." The table shows that for a z-value of 1.15

the cumulative probability Φ(z) is 0.8749. This confirms that the area to the left of z=1.15 is approximately 0.875.

2. De Veaux

R. D.

Velleman

P. F.

& Bock

D. E. (2016). Stats: Data and Models (4th ed.). Pearson.

Reference: Chapter 6

"The Standard Deviation as a Ruler and the Normal Model

" and the associated Standard Normal probability tables (often in an appendix or inside cover). The text explains that to find the z-score for a given percentile

one finds the probability in the body of the table and reads the corresponding z-score. For the 87.5th percentile (0.875)

the closest z-score is 1.15.

3. PennState University

Eberly College of Science. (n.d.). STAT 414: Introduction to Probability Theory.

Reference: Lesson 21: The Normal Distribution

and Appendix: Table V

The Standard Normal Distribution. The table explicitly shows that P(Z ≤ 1.15) = 0.8749

which is the value required to solve for a 75% central interval. Available via PennState's Open Courseware initiatives.