Respuesta :
Answer:
a) Within 260 grams and 340 grams.
b) 68%
c) 32%
d) 97.35%
Step-by-step explanation:
The empirical rule 68-95-99.7 for bell-shaped distributions tells us that:
- Approximately 68% of the data is within 1 standard deviation from the mean.
- Approximately 95% of the data is within 2 standard deviation from the mean.
- Approximately 99.7% of the data is within 3 standard deviation from the mean.
a) The data that covers 95% of the organs is within 2 standard deviations (z=±2).
Then we can calculate the bounds as:
[tex]X_1=\mu+z_1\cdot\sigma=300+-2\cdot 20=300+-40=260 \\\\X_2=\mu+z_2\cdot\sigma=300+2\cdot 20=300+40=340[/tex]
b) We have to calculate the number of deviations from the mean (z-score) we have for the values X=280 and X=320.
[tex]z_1=\dfrac{X_1-\mu}{\sigma}=\dfrac{280-300}{20}=\dfrac{-20}{20}=-1\\\\\\z_2=\dfrac{X_2-\mu}{\sigma}=\dfrac{320-300}{20}=\dfrac{20}{20}=1\\\\\\[/tex]
As there are the bounds for one standard devaition, it is expected tht 68% of the data will be within 280 grams and 320 grams.
c) This interval is complementary from the interval in point b, so it is expected that (100-68)%=32% of the organs weighs less than 280 grams or more than 320 grams.
d) We apply the same as point b but with X=240 and X=340 as bounds.
[tex]z_1=\dfrac{X_1-\mu}{\sigma}=\dfrac{240-300}{20}=\dfrac{-60}{20}=-3\\\\\\z_2=\dfrac{X_2-\mu}{\sigma}=\dfrac{340-300}{20}=\dfrac{40}{20}=2\\\\\\[/tex]
The lower bound is 3 deviations under the mean, so it is expected that (99.7/2)=49.85% of the data will be within this value and the mean.
The upper bound is 2 deviations above the mean, so it is expected that (95/2)=47.5% of the data will be within the mean and this value.
Then, within 240 grams and 340 grams will be (49.85+47.5)=97.35% of the organs.
