Respuesta :
Answer:
a) About 95% of organs will be within 230 and 430 grams.
b) 99.7%
c) 0.3%
d) 83.85%
Step-by-step explanation:
The empirical rule 68-95-99.7 tells us that:
- 68% of the data is expected to be within 1 standard deviation from the mean.
- 95% of the data is expected to be within 2 standard deviation from the mean.
- 99.7% of the data is expected to be within 3 standard deviation from the mean.
We have a bell shaped distribution (we can assume as approximately normal) with mean of 330 g. and standard deviation of 50 g.
a) This happens for an interval with ±2 standard deviations from the mean.
That is:
[tex]X_1=\mu+z_1\cdot\sigma=330-2\cdot 50=330-100=230\\\\ X_2=\mu+z_2\cdot\sigma=330+2\cdot 50=330+100=430[/tex]
About 95% of organs will be within 230 and 430 grams.
b) We can calculate the z-scores for each value to know how many standard deviations are from the mean.
[tex]z_1=\dfrac{X_1-\mu}{\sigma}=\dfrac{180-330}{50}=\dfrac{-150}{50}=-3\\\\\\z_2=\dfrac{X_2-\mu}{\sigma}=\dfrac{480-330}{50}=\dfrac{150}{50}=3\\\\\\[/tex]
As the values are 3 standard deviations from the mean each, it is expected that 99.7% of the organs weigh between 180 and 480 grams.
c) This is the complementary of the point b.
Then, it is expected that (100-99.7)%=0.3% of the organs weigh less than 180 grams or more than 480 grams.
d) We can calculate the z-scores for each value to know how many standard deviations are from the mean.
[tex]z_1=\dfrac{X_1-\mu}{\sigma}=\dfrac{280-330}{50}=\dfrac{-50}{50}=-1\\\\\\z_2=\dfrac{X_2-\mu}{\sigma}=\dfrac{480-330}{50}=\dfrac{150}{50}=3\\\\\\[/tex]
Between 280 and 330 there is 68%/2=34% of the data.
Between 330 and 480 there is 99.7%/2=49.85% of the data.
Then, between 280 grams and 480 grams there is (34+49.85)%=83.85% of the data.
