The weight of oranges growing in an orchard is normally distributed with a mean weight of 8 oz. and a standard deviation of 0.5 oz. Using the empirical rule, what percentage of the oranges from the orchard weigh between 7.5 oz. and 8.5 oz.?
According to the empirical rule, approximately 68% of the data falls within one standard deviation from the mean in a normal distribution.
So, to find the percentage of oranges weighing between 7.5 oz. and 8.5 oz., we need to find the z-scores for both weights:
For a weight of 7.5 oz:
z = (7.5 - 8) / 0.5 = -1
For a weight of 8.5 oz:
z = (8.5 - 8) / 0.5 = 1
Next, we find the area under the normal curve between these two z-scores using a standard normal distribution table:
P(-1 < z < 1) = P(z < 1) - P(z < -1)
= 0.8413 - 0.1587
= 0.6826
So, approximately 68.26% of the oranges from the orchard weigh between 7.5 oz. and 8.5 oz.