The question:
The gas in bubbles within amber should be a sample of the atmosphere at the time the amber was formed. Measurements on specimens of amber 75 million years ago give these percents of nitrogen: 63.4, 65.0, 64.4, 63.3, 54.8, 64.5, 60.8, 49.1, 51.0. Construct a 95% confidence interval for the true average % nitrogen in the atmosphere at this time.
My attempt:
I used the following formula for confidence intervals from my textbook:
"A 100(1-$\alpha$)% confidence interval for $\mu$ is $(\bar{X} - t_{n-1, 1-\frac{\alpha}{2}} \frac{S}{\sqrt{n}}, \bar{X} + t_{n-1, 1-\frac{\alpha}{2}} \frac{S}{\sqrt{n}})$."
I know $\bar{X}$ is 59.6, and $n$ is 9. But there are 2 things I'm having trouble with:
Firstly, in the answers they say that $S$ is 6.25 when I am calculating it to be 5.89.
Secondly, I'm having trouble finding $t_{n-1, 1-\frac{\alpha}{2}}$. According to the solutions, it is 2.31. But I'm not sure where they got that number from, or how to find $t_{n-1, 1-\frac{\alpha}{2}}$ in general? I thought it was supposed to be 1.96 but that's incorrect. Any help is appreciated.
Values of the $t$-distribution are read from a table, for example here: http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf
Since you have nine samples, the degrees of freedom is equal to $n-1 = 8$.
You are using the two-tails confidence interval with $\alpha=0.05$ and therefore one half of the tail is equal to $1-0.025$.
Therefore, you must read off the value $t_{8,0.025}$, which, according to the table, is approximately equal to $2.306$.