Let's solve each part of this question related to finding z-scores.

Problem (a)

We are given two parts in (a) to find z-scores based on probabilities.

Part (i) $P(Z < z) = 0.645$

To find $z$ such that $P(Z < z) = 0.645$, we look up this probability in the standard normal distribution table or use statistical software.

From the z-table or software:

• $z \approx 0.37$ when $P(Z < z) = 0.645$.

Thus, the answer is:

• $z \approx 0.37$ (rounded to 2 decimal places).

Part (ii) $P(Z > z) = 0.8$

For this part, we need to find $z$ such that $P(Z > z) = 0.8$.

Since $P(Z > z) = 0.8$, then $P(Z < z) = 1 - 0.8 = 0.2$.

Using the z-table or statistical software:

• $z \approx -0.84$ when $P(Z < z) = 0.2$.

Therefore, the answer is:

• $z \approx -0.84$ (rounded to 2 decimal places).

Problem (b)

In this part, we are asked to find the z-score that separates the bottom 85% from the top 15%, or the 85th percentile.

To find this, we need to find $z$ such that $P(Z < z) = 0.85$.

From the z-table or using software:

• $z \approx 1.04$ when $P(Z < z) = 0.85$.

So, the answer is:

• $z \approx 1.04$ (rounded to 2 decimal places).

Would you like more detailed explanations on any part of this solution?

Follow-Up Questions

1. What is the process to find z-scores for any given percentile?
2. How does one interpret a z-score in a standard normal distribution?
3. Why is the z-score for $P(Z > z) = 0.8$ negative in part (ii)?
4. How are cumulative probabilities used in z-score calculations?
5. How would these calculations change if the distribution were not standard normal?

Tip

When using a z-table, remember that it typically gives cumulative probabilities from the left up to a z-score. Adjust accordingly for right-tail probabilities by subtracting from 1.