The weight of an organ in adult males has a bell-shaped distribution with a mean of 330 grams and a standard deviation of 15 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 285 grams and 375 grams? (c) What percentage of organs weighs less than 285 grams or more than 375 grams? (d) What percentage of organs weighs between 300 grams and 375 grams?
Accepted Solution
A:
Answer: a) (315,345) b) 99.7%, c) 0.14%, d) 97.36%Step-by-step explanation:Since we have given that Mean = 330 gramsStandard 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%.