Hypothesis testing for a sample mean

Testing a sample mean is much like testing a proportion except you use the t-distribution instead of the normal curve, and the t-distribution takes sample size and degrees of freedom into account. Like with the confidence intervals for samples.

Confidence Intervals for Sample Means

When we find confidence intervals for samples means we use the student-t distribution.
Two conditions must be met: the sample must be random and it the sample size must be large enough for the central limit theorem. (Around 30)

The general formula is:

upper limit: sample mean estimate + t-value*standard error
lower limit: sample mean estimate - t-value*standard error

In math this can be
upper limit: X + t*(s/sqrt(n))
lower limit: X - t*(s/sqrt(n))

The t value is determined by the point on the x-axis that represents the amount of probability we want and is found much like the z-value for proportion confidence intervals.

The t-value must also generally be found by using a computer or a table. The t-distribution takes the number of the sample size into account, and calls this accounting "degrees of freedom". Degrees of freedom are n-1, or one less than your sample size. Generally the more degrees of freedom, the better your estimates.

Sample Means and Deviations

For any random sample (of size n) taken from a population with mean u and standard deviation H

we can say that the mean is equal to X and that the standard deviation is equal to H/sqrt(n)

That is to say, that as the sample (n) gets larger the estimate of the standard deviation will become smaller and more accurate.

Distributions of sample populations are represented by the "t-distribution" that varies in shape by sample size, and looks much like the normal curve covered earlier. As the sample size approaches the population size the t-distribution looks more and more like the normal curve, but that is tomorrow's lesson.