Answer:
![P(X<16)=P(\\frac{X-\mu}{\sigma}<(16-\mu)/(\sigma))=P(Z<(16-16.4)/(0.3))=P(Z<-1.33)](https://img.qammunity.org/2021/formulas/mathematics/college/99ip8hwsvvavg9zkvi17jg6p0ecag5p3qv.png)
And we can find this probability on this way using the z table or excel:
![P(Z<-1.33)=0.0917](https://img.qammunity.org/2021/formulas/mathematics/college/lri4vv9jrn9h8hfjzoj409ahnhmgboj9ah.png)
And that represent approximately 9.2% of the data.
Explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the weigths of the cans of a population, and for this case we know the distribution for X is given by:
Where
and
![\sigma=0.3](https://img.qammunity.org/2021/formulas/mathematics/college/kdb1wwx26zzwjdv49n980l6y2jsvinbsd9.png)
We are interested on this probability
![P(X<16)](https://img.qammunity.org/2021/formulas/mathematics/college/aqkkbqwv4juj3sr9chq8418fonyh8a9hc8.png)
And the best way to solve this problem is using the normal standard distribution and the z score given by:
![z=(x-\mu)/(\sigma)](https://img.qammunity.org/2021/formulas/mathematics/high-school/24k01r9qa0a6ibv4tds8q1jpbjh932http.png)
If we apply this formula to our probability we got this:
![P(X<16)=P((X-\mu)/(\sigma)<(16-\mu)/(\sigma))=P(Z<(16-16.4)/(0.3))=P(Z<-1.33)](https://img.qammunity.org/2021/formulas/mathematics/college/jmnujgvllg93wfkc0oqarf07bbxahp3a6i.png)
And we can find this probability on this way using the z table or excel:
![P(Z<-1.33)=0.0917](https://img.qammunity.org/2021/formulas/mathematics/college/lri4vv9jrn9h8hfjzoj409ahnhmgboj9ah.png)
And that represent approximately 9.2% of the data.