IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. What is the probability of scoring below 55?


To find the probability of scoring below 55 on an IQ test that follows a normal distribution with a mean (μ) of 100 and a standard deviation (σ) of 15, we need to calculate the Z-score first and then use the standard normal distribution to find the probability.

### Step 1: Calculate the Z-score
The Z-score formula is:

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

Where:
- \( X \) is the IQ score (55 in this case),
- \( \mu \) is the mean (100),
- \( \sigma \) is the standard deviation (15).

Substituting the values:

\[
Z = \frac{55 - 100}{15} = \frac{-45}{15} = -3
\]

### Step 2: Use the Z-score to find the probability
Now that we have the Z-score of -3, we look up the cumulative probability for this Z-score in the standard normal distribution table, or we can use a calculator. The cumulative probability for \( Z = -3 \) is approximately 0.00135.

### Conclusion
The probability of scoring below 55 is about **0.135%** or **0.00135**.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the probability of scoring above 145 in the same IQ distribution?
2. What percentage of people score between 85 and 115 on this IQ test?
3. How can we calculate the probability of scoring between two IQ scores, like 90 and 110?
4. What does a Z-score represent in the context of a normal distribution?
5. How would you interpret an IQ score of 70 in terms of standard deviations from the mean?

**Tip:** Always double-check Z-tables or use statistical software to ensure accuracy when working with probabilities.

IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. What is the probability of scoring below 55?

Find out the probability of scoring below 55 on an IQ test with a mean of 100 and standard deviation of 15. This solution explains key statistical concepts such as Z-scores and normal distribution. Perfect for high school students studying statistics or probability.

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