Answer:
a) 300 and 340
b) 95%
c) 5%
d) 81.5%
Step-by-step explanation:
Weight of an organ in adult males has a bell shaped(normal distribution).
Mean weight = 320 grams
Standard deviation = 20 grams
Part a) About 68% of organs weight between:
According to the empirical rule:
- 68% of the data values lie within 1 standard deviation of the mean
- 95% of the data values lie within 2 standard deviation of the mean
- 99.7% of the data values lie within 3 standard deviation of the mean
Thus, 68% of the data values lie in the range: Mean - 1 standard deviation to Mean + 1 Standard Deviation.
Using the values of Mean and Standard deviation, we get:
Mean - 1 Standard Deviation = 320 - 20 = 300 grams
Mean + 1 Standard Deviation = 320 + 20 = 340 grams
This means 68% of the organs will weigh between 300 and 340 grams.
Part b) What percentage of organs weighs between 280 grams and 360 grams?
In order to find what percentage of organs weight between the given range, we need to find how much far these values are from the mean.
Since, mean is 320 and 280 is 40 less than mean, we can write:
280 = 320 - 40
280 = 320 - 2(2)
280 = 320 - 2 Standard Deviations
Similarly,
360 = 320 + 40
360 = 320 + 2 Standard Deviations
So, we have to tell what percentage of values lie within 2 standard deviation of the mean. According to the empirical law, this amount is 95%.
So, 95% of the organs weigh between 280 grams and 360 grams.
Part c) What percentage of organs weighs less than 280 grams or more than 360 grams?
From the previous part we know that 95% of the organs weight between 280 grams and 360 grams.
It is given that the distribution is bell shaped. The total percentage under a bell shaped distribution is 100%. So in order to calculate how much percentage of values are below 280 and above 360, we need to subtract the percentage of values that are between 280 and 360 from 100% i.e.
Percentage of Value outside the range = 100% - Percentage of values inside the range
So,
Percentage of organs weighs less than 280 grams or more than 360 grams = 100 - Percentage of organs that weigh between 280 grams and 360 grams
Percentage of organs weighs less than 280 grams or more than 360 grams = 100% - 95%
= 5%
So, 5% of the organs weigh less than 280 grams or more than 360 grams.
Part d) Percentage of organs weighs between 300 grams and 360 grams.
300 is 1 standard deviation below the mean and 360 is 2 standard deviations above the mean.
Previously it has been established that, 68% of the data values lie within 1 standard deviation of the mean i.e
From 1 standard deviation below the mean to 1 standard deviation above the mean, the percentage of values is 68%. Since the distribution is bell shaped and bell shaped distribution is symmetric about the mean, so the percentage of values below the mean and above the mean must be the same.
So, from 68% of the data values that are within 1 standard deviation from the mean, half of them i.e. 34% are 1 standard deviation below the mean and 34% are 1 standard deviation above the mean. Thus, percentage of values from 300 to 320 is 34%
Likewise, data within 2 standard deviations of the mean is 95%. From this half of the data i.e. 47.5% is 2 standard deviations below the mean and 47.5% is 2 standard deviations above the mean. Thus, percentage of values between 320 and 360 grams is 47.5%
So,
The total percentage of values from 300 grams to 360 grams = 34% + 47.5% = 81.5%
Therefore, 81.5% of organs weigh between 300 grams and 360 grams
is the standard deviation of the population.
