T-Interval for mean

Mean: T-Interval

Using the sample mean x̄ as the estimate for the population mean μ
Make an interval estimate at a particular confidence level.

n=10    x̄=100    s=10   
SEM =
CL tc E = tc·SEM CI LowerBound [
= x̄ - E 		CI UpperBound ]
= x̄ + E
95%
99%

Sketch the 95% CL on a number line. Annotate with the bounds of the CI and with the sample mean.
"There is a 95% chance that (i.e. x̄)   ± (i.e. E) contains the true population mean μ."
"There is a 95% chance that [,] contains the true population mean μ."
MSL: "We are 95% confident that these limits actually do contain the value of the population mean μ."

In general, to have a higher confidence level (CL), the margin of error E will be
and the confidence interval (CI) will be .

Effect of sample size n

n= 30
n=30    x̄=100    s=10   
SEM =
CL tc E = tc·SEM CI LowerBound [
= x̄ - E 		CI UpperBound ]
= x̄ + E
95%
99%

Larger n → SEM, critical value tc, margin of error E, and the confidence interval.

n= 100
n=100    x̄=100    s=10   
SEM =
CL tc E = tc·SEM CI LowerBound [
= x̄ - E 		CI UpperBound ]
= x̄ + E
95%
99%

The larger the sample size n, the the margin of error E, and the the confidence interval(s).
So, use as large a sample as you possibly can!

Effect of sample standard deviation s.
Unlike the sample size n, we have no control over the variation in the data, it just is what it is.
So here we presumably are sampling from a different population, with less variation than above. (It's unlikely that from a population with σ=10 we would get a sample with s≤5. ~2% chance with n=10, ~0% chance with n=100)

s=5 n=10
n=10    x̄=100    s=5   
SEM =
CL tc E = tc·SEM CI LowerBound [
= x̄ - E 		CI UpperBound ]
= x̄ + E
95%
99%

Comparing against the first case above of n=10 s=10,
Smaller s → critical values tc, margin of error E, and the confidence interval(s).

s=5 n=100
With this comfortable sample size, this s probably reflects the
n=100    x̄=100    s=5   
SEM =
CL tc E = tc·SEM CI LowerBound [
= x̄ - E 		CI UpperBound ]
= x̄ + E
95%
99%

Comparing against the case above of n=100 s=10,
In general, smaller s indicates variation in the sample and presumably also in the population, thus our confidence interval(s) will be .