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Complete Question
The weight of an organ in adult males has a bell shaped distribution with a mean of 320 grams and a standard deviation of 20 grams. Use the empirical rule to determine the following. A.) About 68% of organs will be between what weights? B.) what percentage of organs weighs between 280 grams and 360 grams? C.) what percentage of organs weighs less than 280 grams or more than 360 grams? D.) what percentage of organs weighs between 300 grams and 360 grams?
Answer:
A.) About 68% of organs will be between what weights?
Therefore, 68% of the organs will weigh between 300 and 340 grams.
B.) what percentage of organs weighs between 280 grams and 360 grams?
95%
C.) what percentage of organs weighs less than 280 grams or more than 360 grams?
5%
D.) what percentage of organs weighs between 300 grams and 360 grams?
81.5%
Step-by-step explanation:
Empirical formula states that:
68% of data falls within 1 standard deviation from the mean - that means between μ - σ and μ + σ .
95% of data falls within 2 standard deviations from the mean - between μ – 2σ and μ + 2σ .
Where:
μ = Population mean
σ = Population standard deviation
From the above question, we have the following information
Mean weight = 320 grams
Standard deviation = 20 grams
a) About 68% of organs weight between:
To solve for this, we would use the empirical rule:
68% of the data values lie within 1 standard deviation of the mean
Hence, 68% of the data values lie in the range:
Mean - 1 standard deviation to Mean + 1 Standard Deviation.
Mean - 1 Standard Deviation
μ - σ
= 320grams - 20grams = 300 grams
Mean + 1 Standard Deviation
μ + σ
= 320grams + 20grams = 340 grams
Therefore, 68% of the organs will weigh between 300 and 340 grams.
B.) what percentage of organs weighs between 280 grams and 360 grams?
Mean weight = 320 grams
Standard deviation = 20 grams
Hence,
x - mean = 280 grams - 320 grams = -40 grams
x - mean = 360 grams - 320 grams = 40 grams
Note that both differences = 2 × standard deviation
Hence, from the empirical formula above,
95% of data falls within 2 standard deviations from the mean - between μ – 2σ and μ + 2σ applies
Therefore, 95% of the organs fall between 280 grams and 320 grams
C.) what percentage of organs weighs less than 280 grams or more than 360 grams?
Mean weight = 320 grams
Standard deviation = 20 grams
Hence,
x - mean = 280 grams - 320 grams = -40 grams
x - mean = 360 grams - 320 grams = 40 grams
So, from the empirical formula above,
95% of data falls within 2 standard deviations from the mean - between μ – 2σ and μ + 2σ applies
Therefore, 95% of the organs fall between 280 grams and 320 grams
It is important to note that finding the percentage of data that weighs less than or more than the given range of values, the formula is given as:
Percentage of Value outside( less than or more than) the range = 100% - Percentage of values within the range
100% - 95%
= 5%
Therefore, the percentage of organs weighs less than 280 grams or more than 360 grams is 5%
D.) what percentage of organs weighs between 300 grams and 360 grams?
Mean weight = 320 grams
Standard deviation = 20 grams
Hence,
x - mean = 300 grams - 320 grams = -20 grams
20 grams = 1 × standard deviation
For 300 grams, the empirical formula that states that:
68% of data falls within 1 standard deviation from the mean - that means between μ - σ and μ + σ applies.
It is important to note that the percentage of values below the mean and above the mean must be the same.
But since this is a bell shaped distribution, the percentage of data between 300 and 320 grams = 68%/2
= 34%
x - mean = 360 grams - 320 grams = 40 grams
40 = 2 × standard deviation
For 360 grams , the empirical formula that states that,
95% of data falls within 2 standard deviations from the mean - between μ – 2σ and μ + 2σ applies
It is important to note that the percentage of values below the mean and above the mean must be the same.
Since this is a bell shaped distribution, the percentage between 320 and 360grams = 95%/2
= 47.5%
Therefore, the percentage of organs that weighs between 300 grams - 360 grams = 34% + 47.5%
= 81.5%
