We are given that the scores on a standardized test are normally distributed with:

• Mean $\mu = 95$
• Standard deviation $\sigma = 15$

We need to find the test score that is 1.9 standard deviations below the mean.

Step-by-step Solution:

The formula to find a value given the number of standard deviations (denoted as $z$) from the mean is:

$X = \mu + z \times \sigma$

Since we are dealing with a score below the mean, we take $z = -1.9$ (negative because it is below the mean):

$X = 95 + (-1.9) \times 15$ $X = 95 - 28.5$ $X = 66.5$

Thus, the test score 1.9 standard deviations below the mean is 66.5.

Would you like any additional details or clarifications?

Related Questions:

1. What would the score be if it were 2.5 standard deviations above the mean?
2. How would you calculate the probability of getting a score below 66.5 in this distribution?
3. What is the percentile rank for a score of 66.5?
4. If a score is 1 standard deviation below the mean, what is the corresponding score?
5. How does changing the standard deviation affect the range of scores in a normal distribution?

Tip:

In normally distributed data, about 68% of values lie within one standard deviation from the mean, which helps in making quick estimates about spread and probabilities.