To solve the problem, let's apply the process shown in the example image to find the percentile points for each case. Here’s how we proceed for each:

Steps:

1. Express the percentile as a cumulative probability.
2. Use the Z-table to find the closest probability and the corresponding z-score.
3. For probabilities between two z-scores, take the average of the z-scores.

I will now calculate the z-scores for the percentiles $P_{99}, P_{21}, P_{90},$ etc. Let’s begin.

Calculations:

1. $P_{99}$:

• Percentile = $0.99$.
• $P(Z) = 0.99 - 0.5 = 0.49$.
• From the Z-table, $P(2.33) = 0.4901$, which is closest to $0.49$.
• $Z(P_{99}) = 2.33$.

2. $P_{90}$:

• Percentile = $0.90$.
• $P(Z) = 0.90 - 0.5 = 0.40$.
• From the Z-table, $P(1.28) = 0.4000$.
• $Z(P_{90}) = 1.28$.

3. $P_{68}$:

• Percentile = $0.68$.
• $P(Z) = 0.68 - 0.5 = 0.18$.
• From the Z-table, $P(0.47) = 0.1797$, closest to $0.18$.
• $Z(P_{68}) = 0.47$.

4. $P_{40}$:

• Percentile = $0.40$.
• $P(Z) = 0.40 - 0.5 = -0.10$.
• Negative tail: $P(Z) = -0.10$ corresponds to $Z = -0.25$ (symmetry of the normal curve).
• $Z(P_{40}) = -0.25$.

5. $P_{32}$:

• Percentile = $0.32$.
• $P(Z) = 0.32 - 0.5 = -0.18$.
• Negative tail: $P(-0.47) = 0.1797$, closest to $-0.18$.
• $Z(P_{32}) = -0.47$.

6. $P_{21}$:

• Percentile = $0.21$.
• $P(Z) = 0.21 - 0.5 = -0.29$.
• Negative tail: $P(-0.81) = 0.2910$, closest to $-0.29$.
• $Z(P_{21}) = -0.81$.

7. $P_{50}$:

• Percentile = $0.50$.
• $P(Z) = 0.50 - 0.5 = 0.00$.
• $Z(P_{50}) = 0$ (50th percentile is the mean).

8. $P_{14}$:

• Percentile = $0.14$.
• $P(Z) = 0.14 - 0.5 = -0.36$.
• Negative tail: $P(-1.08) = 0.3600$, closest to $-0.36$.
• $Z(P_{14}) = -1.08$.

9. $P_{27}$:

• Percentile = $0.27$.
• $P(Z) = 0.27 - 0.5 = -0.23$.
• Negative tail: $P(-0.62) = 0.2310$, closest to $-0.23$.
• $Z(P_{27}) = -0.62$.

10. $P_{88}$:

• Percentile = $0.88$.
• $P(Z) = 0.88 - 0.5 = 0.38$.
• From the Z-table, $P(1.17) = 0.3800$.
• $Z(P_{88}) = 1.17$.

Summary Table of Results:

Would you like further details or assistance with any specific part?

Relative Questions:

1. How is the Z-score calculated for negative percentiles?
2. What is the significance of the mean in the Z-table?
3. Why does symmetry in the normal curve help with negative tails?
4. How would you calculate probabilities between two Z-scores?
5. What happens when a percentile corresponds exactly to a Z-table value?

Tip: Always check if your percentile is in the upper or lower tail to adjust the probability correctly.