Answer :
To solve the problem using the 68-95-99.7 rule, sometimes referred to as the empirical rule, we first need to understand what this rule tells us about the properties of the normal distribution.
The 68-95-99.7 rule states:
68% of the values lie within one standard deviation (σ) of the mean (μ).
95% of the values lie within two standard deviations of the mean.
99.7% of the values lie within three standard deviations of the mean.
The problem asks for the probability that a standardized normal variable, Z, falls within the range of -3 to 3. In terms of standard deviation (σ) and mean (μ) in a normal distribution, this is expressed as:
[tex]P(-3 \leq Z \leq 3)[/tex]
Given that Z follows a standard normal distribution (with a mean of 0 and a standard deviation of 1), the rule tells us that about 99.7% of the data falls within 3 standard deviations from the mean on both sides.
Therefore,
The probability [tex]P(-3 \leq Z \leq 3) \approx 0.997[/tex]
This means there is about a 99.7% chance that a value of Z will fall between -3 and 3 in a standard normal distribution.
