Respuesta :
Answer: a) (315,345) b) 99.7%, c) 0.14%, d) 97.36%
Step-by-step explanation:
Since we have given that
Mean = 330 grams
Standard deviation = 15 grams
(a) About 68% of organs will be between what weights?
We will use 68-95-99.7 rule, it is empirical.
as we know that
[tex]P(\mu-\sigma\leq X\leq \mu +\sigma)\approx 0.6827[/tex]
So,
[tex]\mu -\sigma=330-15=315\\\\\mu+\sigma=330+15=345[/tex]
So,the 68% of the data falls within three standard deviation or will be between (315 grams and 345 grams)
(b) What percentage of organs weighs between 285 grams and 375 grams?
[tex]\mu+3\sigma=330+3\times 15=330+45=375\\\\\mu-3\sigma=330-3\times 15=330-45=285[/tex]
so, 99.7 % of organs weighs between 285 grams and 315 grams, because the these two values are within one standard deviation of the mean.
(c) What percentage of organs weighs less than 285 grams or more than 375 grams?
Since it is 3 standard deviation below the mean and 3 standard deviations above the mean.
so, it becomes
[tex]1-0.9973=0.0027\\\\\dfrac{0.0027}{2}=0.00135\\\\So,0.00135\times 100=0.135\%=0.14\%[/tex]
Since 285 grams is 3 standard deviation below the mean(=330 grams) and 375 grams is 3 standard deviation above the mean(=330 grams).
So 0.14% data is lies below the 285 grams and 0.14% data is lies above the 375 grams.
(d) What percentage of organs weighs between 300 grams and 375 grams?
[tex]\mu-2\sigma=330-2\times 15=330-30=300\\\\\mu+3\sigma=330+3\times 15=330+45=375[/tex]
So, 300 grams is 2 standard deviation below the mean i.e. 330 grams and 375 grams is 3 standard deviation above the mean i.e. 330 grams.
so, percentage of organs weighs would be
[tex]100-2.5-0.14=97.36\%[/tex]
Hence, a) (315,345) b) 99.7%, c) 0.14%, d) 97.36%.
Using the Empirical Rule, it is found that:
a) Between 315 and 345 grams.
b) 99.7%.
c) 0.3%.
d) 97.35%.
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The Empirical Rule states that for a normal variable:
- The middle 68% of the measures is within 1 standard deviation of the mean.
- The middle 95% is within 2 standard deviations.
- The middle 99.7% is within 3 standard deviations.
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- Mean of 330, standard deviation of 15 grams.
Item a:
Within 1 standard deviation of the mean, thus:
330 - 15 = 315 grams.
330 + 15 = 345 grams.
Between 315 and 345 grams.
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Item b:
330 - 3(15) = 285 grams.
330 + 3(15) = 375 grams.
Within 3 standard deviations of the mean, thus, 99.7%.
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Item c:
More than 3 standard deviations of the mean, thus, 100% - 99.7% = 0.3%.
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Item d:
330 - 2(15) = 300 grams.
330 + 3(15) = 375 grams.
- Between 2 standard deviations below the mean and 3 above.
- The normal distribution is symmetric, which means that 50% of the measures are below the mean and 50% are above.
- Of the 50% below, 95% is within 2 standard deviations, and of the 50% above, 99.7% is within 3 standard deviations, thus:
[tex]P = 0.5(95) + 0.5(99.7) = 97.35[/tex]
The percentage is 97.35%.
A similar problem is given at https://brainly.com/question/21728562
