April 26th, 2017
Inference when Sigma is Unknown
The t Distributions
When the sampling distribution of (x-bar) is close to Normal, we can ﬁnd probabilities involving
(x-bar) by standardizing:
z = (x-bar) — m ÷ s / √n
When we don’t know ç, we can estimate it using the sample standard deviation sx. What
happens when we standardize? This new statistic does not have a Normal distribution!
Standard Error When ç is Unknown
When we do not know the population standard deviation ç (which is usually the case), we
must estimate it with the sample standard deviation s.
When the standard deviation of a statistic is estimated from data, the result is called the
standard error of the statistic.
The standard error of the sample mean is: s ÷ √n
The t Distributions
When we standardize based on the sample standard deviation sx, our statistic has a new
distribution called a t distribution.
It has a different shape than the standard Normal curve:
It is symmetric with a single peak at 0, however it has much more area in the tails. Like any standardized statistic, t tells us how far (x-bar) is from its mean µ in standard deviation
However, there is a different t distribution for each sample size, speciﬁed by its degrees of
The t density curve is similar in shape to the standard Normal curve. They are both symmetric
about 0 and bell-shaped.
The spread of the t distributions is a bit greater than that of the standard Normal curve (i.e.,
the t curve is slightly “fatter”).
As the degrees of freedom increase, the t density curve approaches the N(0,1) curve more
closely. This is because s estimates ç more accurately as the sample size increases.
When we perform inference about a population mean µ using a t distribution, the appropriate
degrees of freedom are found by subtracting 1 from the sample size n, making df = n — 1.
Draw an SRS of size n from a large population that has a Normal distribution with mean µ and
standard deviation ç. The one-sample t statistic:
t = (x-bar) - m ÷ sx ÷ √n
has the t distribution with degrees of freedom df = n — 1.
When comparing the density curves of the standard Normal distribution and t distributions,
several facts are apparent:
The density curves of the t distributions are similar in shape to the standard Normal curve.
The spread of the t distributions is a bit greater than that of the standard Norm