Question 1166922
```python?code_reference&code_event_index=2
import numpy as np
from scipy.stats import norm

# Given parameters
mu = 520
sigma = 95
percentile = 0.85 # Corresponds to the top 15% (100% - 15% = 85%)

# 1. Find the z-score corresponding to the 85th percentile
# Use the Percent Point Function (inverse of CDF)
z_score = norm.ppf(percentile)

# 2. Calculate the minimum score (X)
# X = mu + Z * sigma
X = mu + z_score * sigma

# 3. Print the intermediate z-score and the final score X
print(f"Mean (mu): {mu}")
print(f"Standard Deviation (sigma): {sigma}")
print(f"Cumulative Probability: {percentile}")
print(f"Z-score (intermediate): {z_score:.4f}")
print(f"Raw Score (X): {X}")

# Round the final answer to the nearest integer
final_score = round(X)
print(f"Final score (rounded to nearest integer): {final_score}")

```

```text?code_stdout&code_event_index=2
Mean (mu): 520
Standard Deviation (sigma): 95
Cumulative Probability: 0.85
Z-score (intermediate): 1.0364
Raw Score (X): 618.46117200191
Final score (rounded to nearest integer): 618

```

The minimum score needed to be in the top $15\%$ on this test is **618**.

This is calculated by finding the score that corresponds to the **$85^{th}$ percentile** of the normal distribution, since the top $15\%$ of scores are those above $100\% - 15\% = 85\%$.

-----

## Calculation Steps

### 1\. Find the Z-score

The first step is to find the **Z-score** ($Z$) that has $85\%$ of the area under the standard normal curve to its left.

Using a Z-table or statistical software (carrying intermediate computations to at least four decimal places):
$$Z \approx \mathbf{1.0364}$$

### 2\. Calculate the Raw Score

We use the Z-score formula rearranged to solve for the raw score ($X$):

$$X = \mu + Z \cdot \sigma$$

Where:

  * $\mu$ (Mean) $= 520$
  * $\sigma$ (Standard Deviation) $= 95$
  * $Z$ (Z-score) $\approx 1.0364$

Substituting the values:
$$X = 520 + (1.0364) \cdot 95$$
$$X = 520 + 98.458$$
$$X \approx 618.458$$

### 3\. Round to the Nearest Integer

Rounding the result to the nearest integer:
$$618.458 \approx \mathbf{618}$$

A score of **618** is the minimum score required to be in the top $15\%$ of test takers.