There is a relationship between confidence interval and standard deviation:
[tex]\theta=\overline{x} \pm \frac{z\sigma}{\sqrt{n}}[/tex]
Where [tex]\overline{x}[/tex] is the mean, [tex]\sigma[/tex] is standard deviation, and n is number of data points.
Every confidence interval has associated z value. This can be found online.
We need to find the standard deviation first:
[tex]\sigma=\sqrt{\frac{\sum(x-\overline{x})^2}{n}[/tex]
When we do all the calculations we find that:
[tex]\overline{x}=123.8\\
\sigma=11.84[/tex]
Now we can find confidence intervals:
[tex]($90\%,z=1.645): \theta=123.8 \pm \frac{1.645\cdot 11.84}{\sqrt{15}}=123.8 \pm5.0\\($95\%,z=1.960): \theta=123.8 \pm \frac{1.960\cdot 11.84}{\sqrt{15}}=123.8 \pm 5.99\\ ($99\%,z=2.576): \theta=123.8 \pm \frac{2.576\cdot 11.84}{\sqrt{15}}=123.8 \pm 7.87\\[/tex]
We can see that as confidence interval increases so does the error margin. Z values accociated with each confidence intreval also get bigger as confidence interval increases.
Here is the link to the spreadsheet with standard deviation calculation:
https://docs.google.com/spreadsheets/d/1pnsJIrM_lmQKAGRJvduiHzjg9mYvLgpsCqCoGYvR5Us/edit?usp=sharing
