To solve these problems, we will use the properties of the standard Normal distribution, which is a normal distribution with a mean of 0 and a standard deviation of 1. The proportion of observations corresponds to the area under the curve for the given z-values.

A. Proportion where $z > -1.66$

• To find this proportion, we need to determine the area to the right of $z = -1.66$.

1. Find the area to the left of $z = -1.66$ using the standard Normal distribution table or a calculator. This area is approximately 0.0485.

2. Calculate the area to the right of $z = -1.66$:
   $P(z > -1.66) = 1 - P(z < -1.66)$

   $P(z > -1.66) = 1 - 0.0485 = 0.9515$

   So, the proportion of observations where $z > -1.66$ is approximately 0.9515 or 95.15%.

B. Proportion where $-1.66 < z < 2.85$

• To find this proportion, we need to determine the area between $z = -1.66$ and $z = 2.85$.

1. Find the area to the left of $z = 2.85$ using the standard Normal distribution table or a calculator. This area is approximately 0.9978.

2. Find the area to the left of $z = -1.66$ (already calculated as 0.0485).

3. Calculate the area between $z = -1.66$ and $z = 2.85$:
   $P(-1.66 < z < 2.85) = P(z < 2.85) - P(z < -1.66)$

   $P(-1.66 < z < 2.85) = 0.9978 - 0.0485 = 0.9493$

   So, the proportion of observations where $-1.66 < z < 2.85$ is approximately 0.9493 or 94.93%.

Would you like more details on these calculations or have any other questions?

Here are some related questions you might find useful:

1. What is the probability for $z$ values outside the range of $-1.66 < z < 2.85$?
2. How do you find the z-score given a specific proportion of observations?
3. What is the significance of the z-score in a normal distribution?
4. How do you interpret z-scores in terms of standard deviations?
5. What are the properties of a standard normal distribution?

Tip: Remember that the total area under the standard normal curve is always equal to 1, representing the total probability.