Student's T Distribution Notes

When constructing a confidence interval for the mean, if the population standard deviation $(\sigma)$ is known, the normal distribution is used to find the critical value $\Z<em>{\alpha/2}$ . The confidence interval is calculated as: $\bar{X} \pm Z</em>{\alpha/2} * \frac{\sigma}{\sqrt{n}}$
It is rare to know $(\sigma)$ when estimating $(\mu)$ . In practice, $(\sigma)$ is usually unknown.

When $(\sigma)$ is unknown, it is replaced with the sample standard deviation $(s)$ .
In this case, the critical values are obtained from the Student's t-distribution instead of the normal distribution.
There are multiple Student's t-distributions, each distinguished by its degrees of freedom, which is related to the sample size.

When using the Student's t-distribution for constructing a confidence interval for a population mean, the degrees of freedom are calculated as one less than the sample size:
$\text{degrees of freedom} = n - 1$

Student's t-distributions are symmetric and unimodal, similar to the normal distribution.
However, they are more spread out than the normal distribution because $(s)$ is, on average, a bit smaller than $(\sigma)$ .
Replacing $(\sigma)$ with $(s)$ increases the spread due to the randomness of $(s)$ .
The smaller the number of degrees of freedom, the more pronounced the spread.
When the number of degrees of freedom is large, $(s)$ tends to be close to $(\sigma)$ , making the t-distribution very close to the normal distribution.

The critical value is denoted as $t_{\alpha/2}$ .
The confidence interval for a population mean when $(\sigma)$ is unknown is:
$\bar{X} \pm t_{\alpha/2} * \frac{s}{\sqrt{n}}$
To find the critical value, let $1 - \alpha$ be the confidence level. The critical value is $t<em>{\alpha/2}$ because the area under the t-distribution between $-t</em>{\alpha/2}$ and $t_{\alpha/2}$ is $1 - \alpha$ .
The critical value $t_{\alpha/2}$ can be found in a t-table (Table A3) or using technology.

Example: Find the critical value $t_{\alpha/2}$ needed for a 90% confidence interval when the sample size is 63.
Degrees of freedom: $n - 1 = 63 - 1 = 62$
For a 90% confidence interval, the area in each tail is 0.05.
- TI 84 Plus Calculator:
  - Use the invT command with area to the left as 0.05 and degrees of freedom as 62.
  - invT(0.05, 62) returns -1.6698, so $t_{\alpha/2} = 1.6698$
- Excel:
  - Use the T.INV command.
- Table A3:
  - If the degrees of freedom (62) is not in the table, use the next smaller number (60).
  - The value from the table for a 90% confidence interval with 60 degrees of freedom is 1.671.

Simple random sample.
Large sample size $(n > 30)$ or the population is approximately normal.
If the sample size is small, check if the sample comes from a population that is approximately normal.
- Create a dot plot or box plot of the sample.
- If there are no outliers and the sample is not strongly skewed, it is reasonable to assume approximate normality.

A food chemist analyzes the calorie content of a sample of 8 cookies.
Objective: Find a 98% confidence interval for the mean number of calories.
Assumptions check:
- Simple random sample.
- Small sample size requires the population to be approximately normal. A dot plot shows no strong skewness or outliers.
Using Excel:
- Sample mean: $\bar{x} = 115.375$
- Sample standard deviation: $s = 2.8253$
- Margin of error (using CONFIDENCE.T): CONFIDENCE.T(0.02, 2.8253, 8) = 2.9946
- Confidence interval: $115.375 \pm 2.9946$
Conclusion: We are 98% confident that the mean number of calories per cookie is between 112.4 and 118.4.

Increasing the confidence level increases the critical value and the margin of error.
There is a trade-off between confidence level and margin of error.
Higher confidence is desirable, but so is a smaller margin of error.
Examples:
- 70% confidence intervals have smaller margins of error but cover the population mean only 70% of the time.
- 99.7% confidence intervals almost always cover the population mean, but have a large margin of error.
- 95% confidence level is a good compromise between reliability and margin of error for many purposes.

A 95% confidence interval is constructed using a method that succeeds 95% of the time.
The confidence level describes the success rate of the method, not the success of any particular interval.

When the margin of error is too large, the confidence interval may be too wide to be useful.
Ways to reduce the margin of error:
- Reduce the level of confidence (undesirable).
- Increase the sample size.
Margin of error formula (when $(\sigma)$ is known):
$M = Z_{\alpha/2} * \frac{\sigma}{\sqrt{n}}$
Rewriting the equation to solve for $n$ :
$n = (\frac{Z_{\alpha/2} * \sigma}{M})^2$
This formula gives the minimum value of $n$ needed. Round up to the next whole number.
If $(\sigma)$ is unknown, use $(s)$ from a sample of size greater than 30:
$n = (\frac{Z_{\alpha/2} * s}{M})^2$
Round the result up to the nearest whole number.

Scientists want to estimate the mean weight of mice after a special diet.
A sample of 60 mice has $s = 3$ grams.
Objective: Estimate the number of mice needed for a 95% confidence interval to have a margin of error of 0.5 grams.
Given:
- $Z_{\alpha/2} = 1.96$
- $s = 3$
- $M = 0.5$
Calculation:
$n = (\frac{1.96 * 3}{0.5})^2 = 138.2976$
Round up: $n = 139$
Conclusion: Weigh 139 mice to obtain a 95% confidence interval with a margin of error of 0.5 grams.