Let’s first analyze and complete the details for your scenario.

We are working with a normal distribution $N(2, 4)$, where:

• Mean ($\mu$) = 2
• Variance ($\sigma^2$) = 4, so the standard deviation ($\sigma$) = $\sqrt{4} = 2$.

Now let's go through each part of your statement:

1. Two standard deviations left of the mean:

   • This means $\mu - 2\sigma = 2 - 2 \times 2 = -2$.
   • The Z-score corresponding to this is $Z = -2$.

2. Two standard deviations right of the mean:

   • This means $\mu + 2\sigma = 2 + 2 \times 2 = 6$.
   • The Z-score corresponding to this is $Z = 2$.

Key formulas:

• The Z-score is calculated as: $Z = \frac{X - \mu}{\sigma}$ where $X$ is the value on the normal distribution, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Completing the formulas:

• The value of $\mu$ (mean) gives 2.
• The value of $\sigma$ (standard deviation) gives 2.
• The value of $\sigma^2$ (variance) gives 4.
• The value of $\mu - 2\sigma$ (two standard deviations left of the mean) gives -2.
• The value of $\mu + 2\sigma$ (two standard deviations right of the mean) gives 6.
• The value of $Z$ for two standard deviations left of the mean gives -2.
• The value of $Z$ for two standard deviations right of the mean gives 2.

Would you like further details or explanations? Here are some additional questions that build on this:

1. What is the probability of a Z-score between -2 and 2 in a normal distribution?
2. How does the Z-score relate to percentiles in a normal distribution?
3. How would you calculate the Z-score for an arbitrary value $X$ not two standard deviations from the mean?
4. Can you explain the 68-95-99.7 rule in the context of standard deviations and normal distribution?
5. How does the standard deviation influence the shape of a normal distribution curve?

Tip: The Z-score provides a standardized way to compare values from different normal distributions, translating them into the same scale.