10_1_sample_t - Thettest Inferences about Population Means...

The t-test
Inferences about Population Means
when population SD is unknown

Confidence intervals in z (Review)

Want to estimate height of students at USF.
Sampled N=100 students. Found mean =68 in
and SD = 6 in.

Best guess for population mean is 68 inches plus
or minus some.

95%CI =

95%CI=68±(1.96)[6/sqrt(100)]

68 ±1.96(.6) = 68 ±1.18

Interval is 66.82 to 69.18. Such an interval will
contain the mean 95% of the time.
X
z X σ
05 .
±
N
X
X
σ
σ =

Problem with z

Formulas so far use population SD, and they have
been correct, but SD is usually unknown, so we
have to estimate

Estimate will be off a bit; would be nice to account
for this

The statistic called ‘ t ’ adjusts for error in estimate of
SD. Estimate of SD is better as sample size
increases, so t changes with N. The values of t are
basically the same as z , but t spreads out more and
more as the sample size gets small.

The t Distribution
We use t when the population variance is unknown (the
usual case) and sample size is small (N<100, the usual
case). If you use a stat package for testing hypotheses
about means, you will use t .
The t distribution is a short, fat relative of the normal. The shape of t depends on
its df. As N becomes infinitely large, t becomes normal.

Example values from t and z
Area