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.?

Answer 1

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.