Final answer:
The probability that a student scores between 69 and 79 on the Stats final exam is calculated using Z-scores and the standard normal distribution, which requires finding the cumulative probabilities for the Z-scores of the two bounds and subtracting them.
Step-by-step explanation:
The question pertains to finding the probability that a randomly chosen student scores between 69 and 79 on Professor Combs' Stats final exam, given that the scores are normally distributed with a mean of 74 and a standard deviation of 6.8. To calculate this, we would use the standard normal (Z-score) distribution.
First, we must convert the scores 69 and 79 to their corresponding Z-scores using the formula: Z = (X - µ) / σ, where X is the score, µ is the mean, and σ is the standard deviation.
- For X=69: Z = (69 - 74) / 6.8 ≈ -0.735
- For X=79: Z = (79 - 74) / 6.8 ≈ 0.735
Next, we would look these Z-scores up in the standard normal distribution table or use a calculator or software designed for statistical analysis to find the probabilities associated with each Z-score and then find the probability that lies between them.
For example, if the Z-score of -0.735 corresponds to a cumulative probability of 0.231 and the Z-score of 0.735 corresponds to a cumulative probability of 0.769, the probability that a student scores between 69 and 79 is the difference between these probabilities, which is 0.769 - 0.231 = 0.538 or 53.8%.