1. Inference for the Mean of a Population

2. Inference for The goal is to make inference about
is a point estimate of
If we sample from a normal distribution, the sampling distribution of is standard normal.
If we sample from a non-normal distribution, the sampling distribution above is approximately standard normal for large n.

3. Standard Error is seldom known Estimate by S
When the standard deviation of a statistic is estimated from the data, the result is called the standard error of a statistic. The standard error of the sample mean is

4. The t Distributions Suppose that an SRS of size n is drawn from a
population. Then the one-sample t statistic
has the t distribution with n – 1 degrees of freedom.
The assumption that we sample from a normal population is important for small n, but not for large n.

5. Properties of the t Distribution
continuous and symmetric about 0
more variable and slightly different shape than standard normal (see Figure page 494)
There are an infinite number of t distributions. They are indexed by the number of degrees of freedom.
As the number of degrees of freedom increase, the t distribution approaches the standard normal.

6. The One-Sample t Confidence Interval
Suppose that an SRS of size n is drawn from a population having unknown mean A level C confidence interval for is
Where t* is the value for t(n – 1) density curve with area C between –t* and t*. This interval is exact when the population distribution is normal and is approximately correct for large n in other cases.

7. Margin of Error
The margin of error for the population mean when we use the data to estimate is

8. The One-Sample t Test
Suppose that an SRS of size n is drawn from a population having unknown mean .
To test the hypothesis Ho: based on an SRS of size n, compute the one-sample t statistic