Final answer:
To calculate the probability of a score between 62 and 76, convert the scores to Z-scores and use a Z-table or software to find the probability associated with each Z-score. Subtracting the smaller from the larger probability gives the required probability.
Step-by-step explanation:
To find the probability that a randomly chosen Junior will score between 62 and 76 on a test when the scores are normally distributed with a mean of 72 and a standard deviation of 8, we will use the properties of the normal distribution. To do this, we convert the scores into Z-scores, which tell us how many standard deviations away from the mean our scores are.
The formula for a Z-score is:
Z = (X - μ) / σ
Where X is the score, μ (mu) is the mean, and σ (sigma) is the standard deviation. So for the scores of 62 and 76, we calculate their Z-scores:
Z62 = (62 - 72) / 8 = -1.25
Z76 = (76 - 72) / 8 = 0.5
You would then use a Z-table or statistical software to find the probabilities for these Z-scores and subtract the smaller probability from the larger one to get the probability of a score falling between 62 and 76.