Question

the diagram below is an illustration of the empirical rule. select the missing label for the diagram from the list of choices.

(diagram of a normal distribution curve with a label 0.0013 and a ? to the left of it)

Explanation:

Step1: Recall the Empirical Rule

The empirical rule (68 - 95 - 99.7 rule) for a normal distribution states that about 99.7% of the data lies within \( \mu \pm 3\sigma \), leaving about \( 1 - 0.997 = 0.003 \) of the data in the two tails combined. So each tail (above \( \mu + 3\sigma \) or below \( \mu - 3\sigma \)) has about \( \frac{0.003}{2}=0.0015 \) (approx 0.0013 due to more precise calculation).

Step2: Identify the Label

The diagram shows a normal curve, and the area in the right tail (marked 0.0013) corresponds to the region beyond \( \mu + 3\sigma \). So the missing label "?" should be \( \mu + 3\sigma \) (or the value representing three standard deviations above the mean in the context of the normal distribution's empirical rule).

Answer:

The missing label is \( \boldsymbol{\mu + 3\sigma} \) (or the value representing three standard deviations above the mean, corresponding to the tail area of approximately 0.0013 in the normal distribution's empirical rule).