1. Statistical Inference for the Mean: t-test
Note 7
Statistical Inference for the Mean: t-test
When variance is estimated from a sample, t-distribution applies.
T-Distribution: a distribution related to normal distribution but taking into account the number of degrees of freedom. Symmetrical and roughly bell-shaped.
Probability
K=df
=degrees of freedom
=n-1 t

2. Statistical Inference for the Mean: t-test
T-Distribution:
Variance is estimated from a sample:
SS: the sum of squares of deviations from the sample mean.
df: the number of degrees of freedom, n-1.
The independent variable of t-distribution is
Normal Distribution:

3. Statistical Inference for the Mean: t-test
t-distribution:
Statistical inferences should be made using
the t-distribution rather than the normal distribution
if the variance is estimated from a sample.
Normal Distribution t
df→30, t-distribution is almost the same as Normal Distribution.
df→∞, t-distribution becomes Normal Distribution.

4. Statistical Inference for the Mean: t-test
Tables of the t-distribution often give one-tail probabilities.
Pr [t>t1]
while Normal Distribution:
Pr [Z<z1]
Tables A2 in Appendix of DeCoursey.
e.g. 5% level of significance corresponds 2.5% one-tail probability.
If n=4, df = n-1 = 4-1=3
Then 2.5% and df of 3 give a value of t of 3.182
Or, if n=4, df = n-1 = 4-1=3, then a value of t of gives one-tail probability of 2.5%, and the two-tail probability is 5%.

5. Statistical Inference for the Mean: t-test
Confidence interval using the t-distribution:
Compared with normal distribution:

6. Statistical Inference for the Mean: t-test
Test of Significance: Comparing a sample mean to a population mean
t-test: a test of significance using the t-distribution.
State the null hypothesis in terms of a population parameter, such as mean μ.
State the alternative hypothesis in terms of the same population parameter.
Determine the variance from the sample.
Calculate the test statistic t of the observation using the mean given by the null hypothesis and determine the degrees of freedom.
State the level of significance.
If probability falls outside of the rejection limit, we reject the Null Hypothesis, which means the sample is from a different population.
Example 23