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The problem involves a normal distribution of aptitude test scores with a **mean (μ) = 400** and **standard deviation (σ) = 100**. I'll solve each part step by step:

---

### (a) Proportion of candidates with scores higher than 525:
We need to find the proportion of candidates scoring higher than 525. First, calculate the **Z-score** for 525 using the formula:

\[
Z = \frac{X - \mu}{\sigma}
\]
Where:
- \( X = 525 \) (the test score),
- \( \mu = 400 \) (mean),
- \( \sigma = 100 \) (standard deviation).

\[
Z = \frac{525 - 400}{100} = \frac{125}{100} = 1.25
\]

Now, we find the area to the right of \( Z = 1.25 \). Using standard normal distribution tables or a calculator, the cumulative probability for \( Z = 1.25 \) is approximately **0.8944**. Therefore, the proportion of candidates scoring higher than 525 is:

\[
1 - 0.8944 = 0.1056 \quad \text{or} \quad 10.56\%
\]

---

### (b) Cut-off score for the top 10%:
We need to find the score that separates the top 10% from the rest. This means we need the Z-score corresponding to the 90th percentile (since 90% of the data is below this score). From Z-tables, the Z-score for the 90th percentile is approximately **1.28**.

Now, use the Z-score formula to find the corresponding test score \( X \):

\[
Z = \frac{X - \mu}{\sigma}
\]

Substitute the known values:

\[
1.28 = \frac{X - 400}{100}
\]

Solve for \( X \):

\[
X = 1.28 \times 100 + 400 = 128 + 400 = 528
\]

Thus, the cut-off score for the top 10% is approximately **528**.

---

### (c) Percentage of applicants within two standard deviations of the mean:
For a normal distribution, **68.27%** of the data falls within one standard deviation of the mean, and **95.45%** of the data falls within two standard deviations of the mean. Therefore, the percentage of applicants with scores within two standard deviations (i.e., between 200 and 600) is:

\[
95.45\%
\]

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Would you like further details or have any questions?

Here are five related questions for further practice:
1. What is the probability that a randomly chosen applicant scores below 300?
2. How would the results change if the mean score was increased to 450 but the standard deviation remained 100?
3. What score would be needed to fall into the top 5% of applicants?
4. What proportion of applicants scores between 300 and 500?
5. If the standard deviation was reduced to 80, how would the cut-off for the top 10% change?

**Tip:** For normally distributed data, using Z-scores simplifies calculating probabilities and percentiles!

The aptitude test scores of applicants to a university graduate program are normally distributed with a mean of 400 and a standard deviation of 100. An applicant needs a test score higher than 525 to be admitted to the graduate program. What proportion of candidates qualifies? What is the required cut-off score for the top 10%? What percentage of applicants has test scores within two standard deviations of the mean?

Solve questions about normally distributed aptitude test scores for a graduate program. Find out what proportion of candidates score higher than 525, determine the cut-off for the top 10%, and calculate the percentage within two standard deviations of the mean.

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