Final answer:
To find the probability that a score does not exceed 130 on a test normally distributed with a mean of 100 and a standard deviation of 15, calculate the z-score and use a z-table or calculator to determine the probability.
Step-by-step explanation:
To find the probability that a score x does not exceed 130, given that x is normally distributed with a mean of 100 and standard deviation of 15, we use the concept of z-scores and the normal distribution. First, we calculate the z-score for x = 130 using the formula z = (x - mean) / standard deviation, which gives us z = (130 - 100) / 15.
We can then use a z-table, statistical software, or a calculator to find the probability that z is less than or equal to this value. This probability, P(X < 130), represents the percentage of scores that fall below 130.
For example, suppose that the z-score we calculated is approximately 2. This would mean we are looking for the area under the normal curve to the left of 2 standard deviations above the mean.
Checking a z-table or using a calculator, we might find that P(X < 130) is approximately 0.9772, indicating a 97.72% probability that a score does not exceed 130.