Answer:
The empirical rule (also known as the 68-95-99.7 rule) is a statistical guideline that applies to data with a bell-shaped distribution, such as the normal distribution. It states 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.
Given the mean (μ) of 310 grams and the standard deviation (σ) of 25 grams for the organ weight distribution, we can use the empirical rule to answer the questions:
(a) About 95% of organs will be between what weights?
To find the range within which about 95% of the organs fall, we go two standard deviations above and below the mean:
Lower limit: μ - 2σ = 310 - 2(25) = 260 grams
Upper limit: μ + 2σ = 310 + 2(25) = 360 grams
So, about 95% of the organs will weigh between 260 grams and 360 grams.
(b) What percentage of organs weighs between 235 grams and 385 grams?
To find the percentage of organs between 235 grams and 385 grams, we need to consider the range within two standard deviations above and below the mean, as calculated in part (a). That range is from 260 grams to 360 grams.
(c) What percentage of organs weighs less than 235 grams or more than 385 grams?
To find the percentage of organs outside the range of 260 grams to 360 grams, we consider the tails of the distribution.
Percentage less than 260 grams:
Since 260 grams is one standard deviation below the mean, we can use the 68% rule. About 68% of the organs weigh less than 260 grams.
Percentage more than 360 grams:
Since 360 grams is one standard deviation above the mean, we can also use the 68% rule. About 68% of the organs weigh more than 360 grams.
Now, add the percentages for both tails:
Percentage less than 260 grams + Percentage more than 360 grams = 68% + 68% = 136%
So, about 136% of organs weigh less than 235 grams or more than 385 grams. However, this result doesn't make sense in a practical context because percentages should add up to 100%. It suggests that the distribution extends beyond three standard deviations, which is unusual for a normal distribution. This discrepancy may be due to rounding in the percentage values, and in practice, we would expect less than 100% outside the range.
(d) What percentage of organs weighs between 260 grams and 335 grams?
To find the percentage of organs between 260 grams and 335 grams, we are considering the range within one standard deviation below and above the mean:
Lower limit: μ - σ = 310 - 25 = 285 grams
Upper limit: μ + σ = 310 + 25 = 335 grams
So, about 68% of the organs will weigh between 285 grams and 335 grams.
Explanation: