**Question:** In a group of 100 children, the mean weight of children is 15 kg. The standard deviation is 1.5 kg. Which one of the following is true?

A. 95% of children weigh between 13.2 kg and 16.2 kg.
B. 90% of children weigh between 12 kg and 16 kg.
C. 99% of children weigh between 13.4 kg and 15.6 kg.
D. 90% of children weigh between 12 kg and 16 kg.

**Correct Answer:** D. 90% of children weigh between 12 kg and 16 kg.

**Core Concept:** In statistics, the normal distribution (or Gaussian distribution) is a continuous probability distribution that describes a data set that has a bell-shaped curve with a specific set of characteristics. The mean (μ) is the central value and the standard deviation (SD) is a measure of the dispersion or spread of the data.

**Why the Correct Answer is Right:**

The correct answer (D) is based on the Central Limit Theorem, which states that the distribution of sample means will be approximately normal regardless of the distribution of the population. In this case, the mean weight of the children is 15 kg, and the standard deviation is 1.5 kg.

**Why Each Wrong Option is Incorrect:**

A. The statement is incorrect because it only covers the range for 95% of the data, which is not the same as the 90% mentioned in option D.
B. The statement is incorrect because it only covers the range for 90%, not 90% as mentioned in option D.
C. The statement is incorrect because it covers only 99% of the data, not 90%.

**Why Option D is Correct:**

The correct answer (D) is based on the following calculation:

For 90% of the data to fall within a range, we need to calculate the two values that are 1.645 standard deviations away from the mean (μ = 15 kg). These values are the 25th percentile (Q1) and the 75th percentile (Q3) of the normal distribution.

**Core Concept:**

The Q1 and Q3 values can be calculated using the following formulas:

Q1 = (μ - 1.645 * SD) and Q3 = (μ + 1.645 * SD)

**Why Option D is Right:**

Using the formulas, we calculate Q1 and Q3 for the given mean (μ = 15 kg) and standard deviation (SD = 1.5 kg):

Q1 = (15 kg - 1.645 * 1.5 kg) = (15 kg - 2.57 kg) = 12.43 kg (rounded to one decimal place)
Q3 = (15 kg + 1.645 * 1.5 kg) = (15 kg