Question

To qualify for a police academy, candidates must score in the top 10% on a general abilities test. The test has a mean of 200 and a standard deviation of 20. Find the lowest possible score to qualify. Assume the test scores are normally distributed.

Answer 1

To find the lowest possible score to qualify for the police academy by being in the top 10% of the general abilities test, we’ll need to determine the score that corresponds to the 90th percentile of a normal distribution with a mean of 200 and a standard deviation of 20.

1. **Identify the z-score for the 90th percentile**:
The 90th percentile corresponds to a z-score which we can find using a z-table or standard normal distribution calculator. From the z-table, the z-score associated with the 90th percentile is approximately 1.2816.

2. **Calculate the qualifying score using the z-score formula**:
The formula to convert a z-score to a raw score (X) in a normal distribution is:
\[
X = \mu + z\sigma
\]
where:
- \( \mu \) (mean) = 200
- \( z \) = 1.2816
- \( \sigma \) (standard deviation) = 20

3. **Substituting in the values**:
\[
X = 200 + (1.2816)(20)
\]

4. **Calculating**:
\[
X = 200 + 25.632 = 225.632
\]

Since scores are typically whole numbers, we round this up to the nearest whole number:

**The lowest possible score to qualify is 226.**