Suppose that scores on a particular test are normally distributed with a mean of 110 and a standard deviation of 19. What is the minimum score needed to be in the top 5% of the scores on the test? Carry your intermediate computations to at least four decimal places, and round your answer to one decimal place.

To determine the minimum score needed to be in the top 5% of the scores on the test, we'll find the score corresponding to the 95th percentile of the normal distribution with the given mean and standard deviation.

### Steps:

1. **Determine the Z-score corresponding to the top 5%**:
   Since we're looking for the top 5%, we need the Z-score that corresponds to the cumulative probability of 95% (or 0.95) in the standard normal distribution.

   From a Z-table or using a calculator, the Z-score for 0.95 is approximately **1.645**.

2. **Convert the Z-score to the corresponding raw score**:
   The Z-score formula is:
   \[
   Z = \frac{X - \mu}{\sigma}
   \]
   Solving for \( X \) (the raw score):
   \[
   X = Z \cdot \sigma + \mu
   \]
   Substituting the values:
   \[
   X = 1.645 \cdot 19 + 110
   \]
   \[
   X \approx 31.255 + 110 = 141.255
   \]

3. **Round the answer to one decimal place**:
   The minimum score needed to be in the top 5% of the scores on the test is approximately **141.3**.

Would you like more details or have any questions about this solution?

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Here are 5 related questions:

1. What score corresponds to the top 1% of test-takers?
2. What is the probability of scoring between 100 and 130 on this test?
3. What score corresponds to the bottom 10% of test-takers?
4. How would the required score change if the standard deviation were 25 instead of 19?
5. What is the probability of scoring exactly the mean score of 110 on this test?

**Tip:** When calculating percentiles in a normal distribution, always ensure accuracy by using Z-tables or statistical software to determine the appropriate Z-scores.

Discover how to find the minimum score needed to be in the top 5% of scores on a test with a normal distribution. Step-by-step calculation using Z-scores and the normal distribution formula. Suitable for advanced high school or college-level students.

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