To solve this, we need to determine the probability that a test score is below 2, given that test scores follow a normal distribution with a mean $\mu = 3.4$ and a standard deviation $\sigma = 0.7$.

The probability can be found by calculating the $Z$-score for $X = 2$ and then finding the corresponding probability from the standard normal distribution.

Step 1: Calculate the $Z$-score

The $Z$-score formula is: $Z = \frac{X - \mu}{\sigma}$ where:

• $X = 2$,
• $\mu = 3.4$, and
• $\sigma = 0.7$.

Substitute the values: $Z = \frac{2 - 3.4}{0.7} = \frac{-1.4}{0.7} = -2$

Step 2: Find the probability corresponding to $Z = -2$

Using a standard normal distribution table or calculator, we find the probability that $Z$ is less than $-2$: $P(Z < -2) \approx 0.0228$

Conclusion

The probability that a test score is below 2 is approximately 0.0228.

Would you like further explanation on any steps?

1. What is a Z-score, and how is it used in probability?
2. How does the normal distribution relate to real-world examples?
3. How can we use the cumulative distribution function (CDF) in normal distribution problems?
4. What is the significance of standard deviation in the normal distribution?
5. Why is it important to round probabilities in statistics?

Tip: When using Z-scores, remember that the Z-score tells you how many standard deviations a point is from the mean. This is very useful for understanding probabilities in normal distributions.