How to deduce a value is in CI I am trying to learn inferential stats. I am facing few issues while solving a problem. 
I have Confidence Interval for a population mean. I am getting it's vlaue as "
    One Sample t-test

data:  lead
t = 6.1023, df = 36, p-value = 5.074e-07
alternative hypothesis: true mean is not equal to 0
80 percent confidence interval:
 29.27551 45.21098
sample estimates:
mean of x 
 37.24324 

Does this lower confidence bound suggest that the population mean is larger than 30 mg/kg at the $\alpha = 0.10$ significance level?
Can I deduce that the population mean is larger than 30mg/kg at alpha = 0.1?
The lower bound is below 30, so I think we can say that population mean is larger than 30 
but then alpha = 0.1 is the value which is confusing me.
 A: I think you can say:

I am 80% confident that my true population mean is between (29.27551, 45.21098).

This doesn't mean the true mean value is really inside the interval or > 30. What if the population mean is 29.99999? The true value is always unknown and we never know. 
When you report CI, you should simply report your model, the interval and confidence level.
A: All values outside the CI would be rejected as the null value for the mean in a two-sided test at level $\alpha$. All values below the lower confidence limit would be rejected at $\alpha/2$, as would all values above the upper confidence limit (assuming you set up the direction of the test correctly). All values inside the CI would not be rejected at the aforementioned significance levels. 
For example, based on the $80\%$ CI provided, 


*

*a test of $H_0: \mu = 29.2$ would be rejected in favor of $H_1: \mu < 29.2$ at level $\alpha = 0.10$

*similarly, $H_0 : \mu = 45.3$ would be rejected in favor of $H_1 : \mu > 45.3$ at level $\alpha = 0.10$.  


To answer your direct question, $30$ is in the confidence interval, so: No, you could not reject it in favor of $H_1: \mu > 30$ at level $\alpha = 0.10$; $30$ is a plausible value of $\mu$ and the lower bound being below $30$ does not mean you have evidence that $\mu > 30$. 
