The problem you are working on involves normal distribution and requires shading the corresponding region under the standard normal curve, based on the given data.

Given:

• The original normal distribution has a mean ($\mu$) of 4 and a standard deviation ($\sigma$) of 2.
• The problem asks for the probability of obtaining a value between 3 and 6.

Step 1: Convert to Z-scores

You need to convert the values 3 and 6 from the original distribution to the standard normal distribution (with mean 0 and standard deviation 1). The formula for converting a raw score to a Z-score is:

$Z = \frac{X - \mu}{\sigma}$

Where $X$ is the value from the original distribution.

For $X = 3$:

$Z_1 = \frac{3 - 4}{2} = \frac{-1}{2} = -0.5$

For $X = 6$:

$Z_2 = \frac{6 - 4}{2} = \frac{2}{2} = 1$

Step 2: Shade the Corresponding Region

The standard normal curve should have a shaded region between $Z = -0.5$ and $Z = 1$. Based on the graph below, you should mark the area between these Z-values.

• The left endpoint of the shaded region corresponds to $Z = -0.5$.
• The right endpoint of the shaded region corresponds to $Z = 1$.

Correction for Your Attempt:

It seems the original attempt did not correctly shade this area. You need to select the region between $Z = -0.5$ and $Z = 1$ on the standard normal curve, which reflects the transformation from the original normal distribution.

Would you like further details on how to calculate probabilities or interpret the Z-scores?

Here are some additional questions to further explore this topic:

1. How do you calculate probabilities for a normal distribution using Z-scores?
2. What is the importance of the standard normal curve in statistics?
3. How can you use a Z-table or technology to find the area under a normal curve?
4. Why do we use the standard normal distribution to compare different datasets?
5. How does the shape of the normal curve change with different standard deviations?

Tip: Always check the Z-scores and ensure correct interpretation before shading the area on the standard normal curve.