Sure, let's solve the problem step-by-step using the Empirical Rule.
The Empirical Rule, also known as the 68-95-99.7 Rule, gives us a way to estimate the percentage of data that falls within a certain number of standard deviations of the mean in a normal distribution. The key points of the Empirical Rule are:
- Approximately 68% of the data falls within one standard deviation (σ) of the mean (μ), i.e., between -1σ and +1σ.
- Approximately 95% of the data falls within two standard deviations of the mean, i.e., between -2σ and +2σ.
- Approximately 99.7% of the data falls within three standard deviations of the mean, i.e., between -3σ and +3σ.
Given that we need to find the area to the left of [tex]\( z = -2 \)[/tex]:
1. According to the Empirical Rule, 95% of the data falls within ±2 standard deviations of the mean. This means that 95% of the data is within the interval [tex]\([-2σ, +2σ]\)[/tex].
2. Since the normal distribution is symmetric, the remaining 5% of the data is split equally between the two tails of the distribution. Thus, half of this 5% lies in the left tail (to the left of [tex]\(-2σ\)[/tex]) and the other half lies in the right tail (to the right of [tex]\(+2σ\)[/tex]).
3. Half of 5% is 2.5%. Therefore, the area to the left of [tex]\( z = -2 \)[/tex], which corresponds to the left tail beyond [tex]\(-2σ\)[/tex], is 2.5%.
Thus, the area under the standard normal curve to the left of [tex]\( z = -2 \)[/tex] is 0.025.
