good evening every body

I have a question about t-statistics in first place I need to use t-statistics to estimate the population mean from a sample set, as you see I have a data set the mean and standard deviation are known for sample but not for the population.

how can I use t-statistics to determine the population mean I have X bar and S and n values for sample n=45
how can I compute the degree of freedom and how can I determine the confidence level? thanks for your time huge regards

$df=n-1$

You need to set the confidence level yourself. $\alpha =.05$ is standard.

Confidence interval: $$Upper limit=Xbar+\frac{tquantile(1-\alpha/2,df)*S}{\sqrt{df}}$$ $$Lower limit=Xbar-\frac{tquantile(1-\alpha/2,df)*S}{\sqrt{df}}$$

Where tquantile is the quantile from a t-distribution chart or obtained from statistical software. It is 2.015368 so you have:

$$Upper limit=Xbar+\frac{2.015368*S}{\sqrt{df}}$$ $$Lower limit=Xbar-\frac{2.015368*S}{\sqrt{df}}$$

In R:

> qt(.975,44)
[1] 2.015368


Note: I'm not sure if you need to divide by $\sqrt df$, depends on whether S is the standard dev of the sample or the individual observations...