Final answer:
(a) About 99.7% of organs will weigh between 270 and 330 grams.
(b) Approximately 99.7% of organs weigh between 270 and 330 grams.
(c) About 0.3% of organs weigh less than 270 grams or more than 330 grams.
(d) Around 97.5% of organs weigh between 240 and 330 grams.
Step-by-step explanation:
The empirical rule, also known as the 68-95-99.7 rule, is a statistical rule stating that for a normal distribution:
Approximately 68% of the data falls within one standard deviation of the mean.
Approximately 95% falls within two standard deviations.
Approximately 99.7% falls within three standard deviations.
Given that the mean (μ) is 300 grams and the standard deviation (σ) is 30 grams, we can apply the empirical rule to answer the questions:
(a) About 99.7% of organs will be between what weights?
For a normal distribution, about 99.7% of the data falls within three standard deviations of the mean.
So, the weights will be between 300−3×30 and 300+3×30:
270grams ≤ weight ≤ 330grams
(b) What percentage of organs weigh between 270 grams and 330 grams?
This is the same as the answer to part (a) because the interval [270,330] covers three standard deviations from the mean, and according to the empirical rule, approximately 99.7% of the data falls within this range.
(c) What percentage of organs weigh less than 270 grams or more than 330 grams?
Since about 99.7% of the data falls within three standard deviations of the mean, the percentage of organs weighing less than 270 grams or more than 330 grams is approximately
100 % − 99.7 % = 0.3 %
(d) What percentage of organs weigh between 240 grams and 330 grams?
To find this, we need to determine how many standard deviations below the mean 240 grams is:
z=240-300/30
=2
According to the empirical rule, about 2.5% of the data falls below 2 standard deviations from the mean.
So, the percentage of organs weighing between 240 grams and 330 grams is approximately 95%+2.5%=97.5%.