AP Statistics Curriculum 2007 StudentsT

General Advance-Placement (AP) Statistics Curriculum - Student's T Distribution

Very frequently in practice 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 \(Cauchy=T_{(df=1)} \longrightarrow T_{(df)}\longrightarrow N(0,1)=T_{(df=\infty)}\).

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 X₁, ..., X_n are independent random variables that are Normally distributed with expected value μ and variance σ². Let \[ \overline{X}_n = {X_1+X_2+\cdots+X_n \over n}\] be the sample mean, and

\[{S_n}^2=\frac{1}{n-1}\sum_{i=1}^n\left(X_i-\overline{X}_n\right)^2\] be the sample variance. We already discussed the following statistic: \[Z=\frac{\overline{X}_n-\mu}{\sigma/\sqrt{n}}\]

is normally distributed with mean 0 and variance 1, since the sample mean \(\scriptstyle \overline{X}_n \) is normally distributed with mean \( \mu\) and standard deviation \(\scriptstyle\sigma/\sqrt{n}\).

Gosset studied a related quantity under the pseudonym Student), \[T=\frac{\overline{X}_n-\mu}{S_n / \sqrt{n}},\] which differs from Z in that the (unknown) population standard deviation \(\scriptstyle \sigma\) is replaced by the sample standard deviation \(S_n\). Technically, \(\scriptstyle(n-1)S_n^2/\sigma^2\) has a Chi-square distribution \(\scriptstyle\chi_{n-1}^2\) 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% (\(\alpha=0.025\)) confidence interval for \(\mu\):

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? \[\overline{y}= 321.4\] \[s = 73.8\] \[SE = 14.8\] \[n = 25\]

\[CI(\alpha)=CI(0.025): \overline{y} \pm t_{\alpha\over 2} {1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(x_i-\overline{x})^2\over n-1}}.\]

\[321.4 \pm t_{(24, 0.025)}{73.8\over \sqrt{25}}\]

\[321.4 \pm 2.064\times 14.8\]

\[[290.85, 351.95]\]

CI Interpretation

Still, does this CI (290.85, 351.95) mean anything to us? Consider the following information: The U.S. Government classification of AIDS has three official categories of CD4 counts – asymptomatic = greater than or equal to 500 cells/uL

• AIDS related complex (ARC) = 200-499 cells/uL
• AIDS = less than 200 cells/uL
• Now how can we interpret our CI?

SOCR CI Experiments

The SOCR Confidence Interval Experiment provides empirical evidence that the definition and the construction protocol for Confidence intervals are consistent.

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References

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• SOCR Home page: http://www.socr.ucla.edu

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