Given a normally distributed test with a mean of 50 and a standard deviation of 10, we can calculate the probabilities using the standard normal distribution (Z-distribution).

1. Probability of scoring between 40 and 65:

   • We need to find the Z-scores for 40 and 65.
   • The Z-score formula is: $Z = \frac{X - \mu}{\sigma}$ where $X$ is the score, $\mu$ is the mean, and $\sigma$ is the standard deviation.

   Let's compute the Z-scores for 40 and 65.

2. Probability of scoring lower than 40 or higher than 65:

   • This is the complement of scoring between 40 and 65.
   • We can find it by subtracting the probability of scoring between 40 and 65 from 1.

Let's calculate these probabilities.1. Probability of scoring between 40 and 65: Approximately 77.45%.

2. Probability of scoring lower than 40 or higher than 65: Approximately 22.55%.

Would you like more details or have any questions? Here are some related questions for further exploration:

1. What is the probability of scoring exactly 50 on the test?
2. How does the probability change if the standard deviation increases to 15?
3. What is the Z-score for a score of 30 on this test?
4. What is the probability of scoring between 55 and 70?
5. If the mean changes to 60 while keeping the standard deviation the same, what is the new probability of scoring between 40 and 65?

Tip: The standard normal distribution is a useful tool to quickly find probabilities when dealing with normally distributed data.