The weight of oranges growing in an orchard is normally distributed with a mean weight of 6.5 ozand a standard deviation of 0.5 . Using the empirical rule , what percentage of the oranges from the orchard weigh between 5.5 oz. and 7.5 oz.?
Using the empirical rule (68-95-99.7), we know that:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
In this case, we want to find the percentage of oranges that weigh between 5.5 oz and 7.5 oz, which is within one standard deviation above and below the mean.
First, we calculate the z-scores for the weights of 5.5 oz and 7.5 oz:
For 5.5 oz: z = (5.5 - 6.5) / 0.5 = -2
For 7.5 oz: z = (7.5 - 6.5) / 0.5 = 2
Next, we look up the percentage of data falling within -1 and 1 standard deviations from the mean, which is 68%.
Therefore, the percentage of the oranges from the orchard that weigh between 5.5 oz and 7.5 oz is approximately 68%.