Final answer:
1. The probability that a student had a score lower than 61.8 is 0.1423, which is not considered an unusual event. 2. The probability that a student had a score between 55.6 and 68 is 0.475, which is also not considered an unusual event.
Step-by-step explanation:
1. To find the probability that a student had a score lower than 61.8, we need to calculate the z-score first. The formula for calculating the z-score is:
z = (x - mean) / standard deviation
Plugging in the values, we get:
z = (61.8 - 68) / 6.2 = -1.06
Using a standard normal distribution table or calculator, we can find the probability associated with a z-score of -1.06, which is approximately 0.1423. Therefore, the probability that a student had a score lower than 61.8 is 0.1423.
This is not considered an unusual event because it falls within two standard deviations from the mean.
2. To find the probability that a student had a score between 55.6 and 68, we need to calculate the z-scores for both scores and find the area between them on the standard normal distribution.
The z-score for 55.6 is:
z = (55.6 - 68) / 6.2 = -1.97
The z-score for 68 is:
z = (68 - 68) / 6.2 = 0
Using a standard normal distribution table or calculator, we can find the probabilities associated with these z-scores, which are approximately 0.025 and 0.5 respectively. Subtracting the probability associated with -1.97 from the probability associated with 0 gives us the probability of the score being between 55.6 and 68, which is approximately 0.475.
This is also not considered an unusual event because it falls within two standard deviations from the mean.