AP Statistics Curriculum 2007 StudentsT
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General Advance-Placement (AP) Statistics Curriculum - Student's T Distribution
Very frequently in practive we do now know the population variance and therefore need to estimate it using the sample-variance. This requires us to introduce the T-distribution, which is a one-parameter distribution connecting .
Student's T Distribution
The Student's t-distribution arises in the problem of estimating the mean of a normally distributed population when the sample size is small and the population variance is unknown. It is the basis of the popular Student's t-tests for the statistical significance of the difference between two sample means, and for confidence intervals for the difference between two population means.
Suppose X_{1}, ..., X_{n} are independent random variables that are Normally distributed with expected value μ and variance σ^{2}. Let
- be the sample mean, and
- be the sample variance. We already discussed the following statistic:
is normally distributed with mean 0 and variance 1, since the sample mean is normally distributed with mean μ and standard deviation .
Gosset studied a related quantity under the pseudonym Student),
which differs from Z in that the (unknown) population standard deviation is replaced by the sample standard deviation S_{n}. Technically, has a Chi-square distribution distribution. Gosset's work showed that T has a specific probability density function, which approaches Normal(0,1) as the degree of freedom (df=sample-size -1) increases.
Computing with T-distribution
- You can see the discretized T-table or
- Use the interactive SOCR T-distribution or
- Use the high precision T-distribution calculator.
Example
Suppose a researcher wants to examine CD4 counts for HIV(+) patients seen at his clinic. She randomly selects a sample of n = 25 HIV(+) patients and measures their CD4 levels (cells/uL). Suppose she obtains the following results and we are interested in calculating a 95% (α = 0.025) confidence interval for μ:
Variable | N | N* | Mean | SE of Mean | StDev | Minimum | Q1 | Median | Q3 | Maximum |
CD4 | 25 | 0 | 321.4 | 14.8 | 73.8 | 208.0 | 261.5 | 325.0 | 394.0 | 449.0 |
What do we know from the background information?
- s = 73.8
- SE = 14.8
- n = 25
- [290.85,351.95]
References
- SOCR Home page: http://www.socr.ucla.edu
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