Question

Suppose in a population, that is normally distributed, µ = 4 and σ = 0.23. Find an interval that contains 95% of the population data. For what value do we have 92% of the data below it?

a) Interval containing 95% of the data: (3.54 < x < 4.46); Value with 92% of data below it: (3.08)
b) Interval containing 95% of the data: (3.77 < x < 4.23); Value with 92% of data below it: (3.38)
c) Interval containing 95% of the data: (3.81 < x < 4.19); Value with 92% of data below it: (3.14)
d) Interval containing 95% of the data: (3.68 < x < 4.32); Value with 92% of data below it: (3.07)

Answer 1

The interval containing 95% of the data in the given population is (3.54, 4.46), and the value below which 92% of the data lies is approximately 4.3243. Option (a) is the closest to these calculated values.

Step-by-step explanation:

The question asks to find an interval containing 95% of the data in a normally distributed population with a mean (µ) of 4 and a standard deviation (σ) of 0.23. Additionally, it inquires about the value below which 92% of the data lies. According to the empirical rule, 95% of values in a normal distribution are within two standard deviations of the mean. Therefore, the interval is μ ± 2σ, which in this case is 4 ± (2 * 0.23), yielding an interval of (3.54, 4.46). To find the value below which 92% of the data lies, we would typically look up the z-score corresponding to 92% in a standard normal distribution table, which is approximately 1.41. Therefore, the value is µ + (z * σ), which calculates to 4 + (1.41 * 0.23) = 4.3243. None of the provided options exactly match these calculations, but option (a) is closest to the accurate values.