t-test for one sample
Lecture Outline
- How parameters of an unknown population are estimated
- The t-distribution
- Conducting the t-test
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In the previous situations of hypothesis testing (e.g. the z test), we knew what the population parameters were (mean and standard deviation).
However, in most research situations we do not know in advance the characteristics of the population. In those cases, we cannot use z-scores to make quantitative inferences about the population, because we cannot compute the standard error of the sampling distribution of the mean.
So what do we do?
We need to obtain unbiased estimates of the population parameters.
A sample statistic is unbiased if the average
value of the sample statistic (in other words, the mean of the sampling
distribution), obtained over many different samples, is equal to the population
parameter.
Previously (see Table 4.1 of Stats book), we discussed that the sample mean (M) and variance (s2) are unbiased estimates of their corresponding population parameters.
NOTICE: The standard deviation (s) is NOT.
This means that we can use the sample variance as an
unbiased estimate of the population variance and, consequently, estimate
the standard error.
Then we can perform hypothesis testing with a new test, the
t test. For this we need to refer to the t distribution (Table B2 in
Appendix B), which is part of a family of distributions collectively referred to
as Student’s t distributions.
There is a different sampling distribution of t for each possible number of degrees of freedom.
Why?
In z-tests, the standard error does not vary from sample to
sample because it is derived from population variance. IMPLICATION: Samples with
the same mean should also have the same z-score.
However, in the t formula, the standard error is NOT
constant, because it is estimated. I.e. it depends on sample variance, which
will vary from sample to sample, especially when the sample size is small, less
than 30. IMPLICATION: Samples can have the same value for the mean, yet
different values for t, because the estimated error will vary from sample to
sample. Therefore, a t distribution will have more variability than the normal z
distribution.
t = sample mean (from data) – population mean/estimated standard error
Hypothesis testing procedure:
1.State hypotheses and set alpha level.
2.Locate critical region (determine d.f.)
3.Collect data and compute tobt
4.Evaluate H0
If the tcr has no d.f. values listed in the Table, check both surrounding values and choose the larger one.
Assumptions:
1.The values of the sample consist of independent observations.
2.The population sampled must be normal.