Download (s/sqrt(n)) - People Server at UNCW

yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

History of statistics wikipedia, lookup

Student's t-test wikipedia, lookup

Taylor's law wikipedia, lookup

Bootstrapping (statistics) wikipedia, lookup

Confidence interval wikipedia, lookup

Resampling (statistics) wikipedia, lookup

Degrees of freedom (statistics) wikipedia, lookup

Foundations of statistics wikipedia, lookup

Statistical inference wikipedia, lookup

• Inference about the mean of a population of
measurements (m) is based on the standardized
value of the sample mean (Xbar).
• The standardization involves subtracting the
mean of Xbar and dividing by the standard
deviation of Xbar – recall that
– Mean of Xbar is m ; and
– Standard deviation of Xbar is s/sqrt(n)
• Thus we have (Xbar - m )/(s/sqrt(n)) which has a
Z distribution if:
– Population is normal and s is known ; or if
– n is large so CLT takes over…
• But what if s is unknown?? Then this
standardized Xbar doesn’t have a Z distribution
anymore, but a so-called t-distribution with n-1
degrees of freedom…
• Since s is unknown, the standard deviation of
Xbar, s/sqrt(n), is unknown. We estimate it by
the so-called standard error of Xbar, s/sqrt(n),
where s=the sample standard deviation.
• There is a t-distribution for every value of the
sample size; we’ll use t(k) to stand for the
particular t-distribution with k degrees of
freedom. There are some properties of these tdistributions that we should note…
• Every t-distribution looks like a N(0,1) distribution; i.e., it
is centered and symmetric around 0 and has the same
characteristic “bell” shape… however, the standard
deviation of t(k) {sqrt(k/(k-2))} is greater than 1, the s.d.
of Z so the t-distribution density curve is more spread out
than Z. Probabilities involving r.v.s that have the t(k)
distributions are given by areas under the t(k) density
curve … the pt function in R gives us the probabilities we
pt(q, k) = Prob(t(k)<= q)
• The good news is that everything we’ve already
learned about constructing confidence intervals
and testing hypotheses about m carries through
under the assumption of unknown s …
• So e.g., a 95% confidence interval for m based
on a SRS from a population with unknown s is
Xbar +/- t*(s.e.(Xbar))
Recall that s.e.(Xbar) = s/sqrt(n). Here t* is the
appropriate quantile from the t-distribution so that
the area between –t* and +t* is .95
• As we did before, if we change the level of
confidence then the value of t* must change
• e.g., for 95% confidence with df=12, qt(.975,12)
gives the correct t* ….
• Similarly, we may test hypotheses using this tdistributed standardized Xbar… e.g., to test the
H0: m =m0 against Ha: m >m0 we use
(Xbar - m0)/(s/sqrt(n)) which has a tdistribution with n-1 df, assuming the null
hypothesis is true. See the last page of these
notes for a summary of hypothesis testing in the
case of “the one-sample t-test” …
• HW: Read the online Chapter 10 on Hypothesis
Testing with Standard Errors (start with the first 3
sections… the third deals with the t-distribution).
Work on the second problem set handout…
• Note: a statistic is robust if it is insensitive to
violations of the assumptions made when the
statistic is used. For example, the t-statistic
requires normality of the population… how
sensitive is the t-statistic to violations of
normality?? Consider these practical guidelines
for inference on a single mean:
– If the sample size is < 15, use the t procedures if the
data are close to normal.
– If the sample size is >= 15 then unless there is strong
non-normality or outliers, t procedures are OK
– If the sample size is large (say n >= 40) then even if
the distribution is skewed, t procedures are OK