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T Interval

Table of Contents

1. T Statistic a. Definitions and Terminology
2. Confidence Limits
3. Study Design Considerations
4. Example: [Application] a. Sample Size b. Margin of Error c. Standard Deviation d. Significance e. One Sided Intervals
5. Study Design Derivations a. Sample Size b. Standard Deviation c. Significance d. One Sided Intervals
6. References

1. T Statistic

The $t$-statistic is a standardized measure of the magnitude of difference between a sample's mean and some known, non-random constant. It is similar to a $z$-statistic, but differs in that a $t$-statistic may be calculated without knowledge of the population variance.

1a. Definitions and Terminology

Let $\theta$ be a sample parameter from a sample with standard deviation $s$. Let $\theta_0$ be a constant, and $s_\theta = s/\sqrt{n}$ be the standard error of the parameter $\theta$. $t$ is defined: $$t = \frac{\theta - \theta_0}{s_\theta} = \frac{\theta - \theta_0}{\frac{s}{\sqrt{n}}}$$

2. Confidence Limits

The confidence interval for $\theta$ is written: $$\theta \pm t_{1-\alpha/2} \cdot \frac{s}{\sqrt{n}}$$

The value of the expression on the right is often referred to as the margin of error, and we will refer to this value as $$E = t_{1-\alpha/2} \cdot \frac{s}{\sqrt{n}}$$

3. Study Design Considerations

4. Example: [Application]

4a. Sample Size

4b. Margin of Error

4c. Standard Deviation

4d. Significance

4e. One Sided Intervals

5. Study Design Derivations

5a. Sample Size

$$E = t_{1-\alpha/2} \cdot \frac{s}{\sqrt{n}}$$ $$\frac{E}{t_{1-\alpha/2}} = \frac{s}{\sqrt{n}}$$ $$\frac{E}{t_{1-\alpha/2} \cdot s} = \frac{1}{\sqrt{n}}$$ $$\frac{t_{1-\alpha/2} \cdot s}{E} = \sqrt{n}$$ $$\frac{t_{1-\alpha/2}^2 \cdot s^2}{E^2} = n$$

Since $t_{1-\alpha/2}$ depends on the value of $n$, this is not a problem that is easily reduced to a solution. Many texts encourage using $z_{1-alpha/2}$ as a substitute, but we're using computers here, so we can probably do a little better. Instead, if we write the last line as: $$\frac{t_{1-\alpha/2}^2 \cdot s^2}{E^2} - n = 0$$ $$\big(\frac{t_{1-\alpha/2}^2 \cdot s^2}{E^2} - n\big)^2 = 0$$

We now have a quadratic equation. We'll use the optimize function in the stats package to find a best solution for $n$.

Consider when we have $n=25$, $s=4$ and $\alpha=.05$. The value of $E$ here is $$E = t_{1-\alpha/2} \cdot \frac{s}{\sqrt{n}} = 2.063899 \cdot 4/5 = 1.651119$$.

Now let's rewrite the problem to solve for $n$ using optimize.

fn <- function(n) (qt(1-.05/2, n-1)^2 * 4^2 / 1.651119^2 - n)^2
optimize(fn, c(0, 100))

On the other hand, using the $z$ approximation yields

qnorm(1-.05/2)^2 * 4^2 / 1.651119^2

which is two subjects short of what we would actually need. n_t1samp_interval uses the optimize function and searches over the values 0 to 1,000,000,000.

5b. Standard Deviation

$$E = t_{1-\alpha/2} \cdot \frac{s}{\sqrt{n}}$$ $$\frac{E}{t_{1-\alpha/2}} = \frac{s}{\sqrt{n}}$$ $$\frac{E \cdot \sqrt{n}}{t_{1-\alpha/2}} = s$$

5c. Significance

$$E = t_{1-\alpha/2} \cdot \frac{s}{\sqrt{n}}$$ $$\frac{E \cdot \sqrt{n}}{s} = t_{1-\alpha/2}$$ $$\Phi_t\Big(\frac{E \cdot \sqrt{n}}{s}\Big) = \Phi_t(t_{1-\alpha/2})$$ $$\Phi_t\Big(\frac{E \cdot \sqrt{n}}{s}\Big) = 1 - \frac{\alpha}{2}$$ $$1 - \cdot \Phi_t\Big(\frac{E \cdot \sqrt{n}}{s}\Big) = \frac{\alpha}{2}$$ $$2 \cdot \Big[1 - \Phi_t\Big(\frac{E \cdot \sqrt{n}}{s}\Big)\Big] = \alpha$$

6. References

nutterb/junkyard documentation built on May 24, 2019, 10:51 a.m.