A kind of Paired T Test, Student’s T distribution is used for finding confidence intervals for the population mean when the sample size is less than 30 and the population standard deviation is unknown. If you need to evaluate something with a population greater than 30, use the Z distribution => t distribution is flatter and wider than the z distribution. The t distribution becomes narrower (taller) as sample sizes increase, and gradually becomes very close to the Normal Distribution. Both z and t distributions are symmetric and bell-shaped, and both have a mean of zero.

*The more degrees of freedom, the better.

Student t-test would be used to compare two population means using samples from each. See this example.

Also used to test hypotheses about population means based on sample data.

## Student’s T Distribution Example Questions

### Confidence of a mean 1

At a soda bottling factory, the normal filling specification (goal) is 16 fluid ounces. A sample of 20 bottles is tested with the following results: x = 16.13 fl oz and s = 0.24 fl oz. What interval would allow you to say, with 95% confidence, that the interval contains the actual mean of the filling process?

### Confidence of a mean 2

An automated computer attendant is supposed to respond to all inquiries within 15 seconds. The company claims with 99 percent confidence that the average response time is less than 17 seconds. A random sample of 25 calls reveals that X = 13.5 seconds and s = 0.95 seconds. Based on this data is the company correct?

### Confidence of a mean 3

A photography developing machine uses ink to produce photographs. A sample of 15 pictures showed an average of 1.5 mL of ink used per picture, with a standard deviation o 0.075 mL. Develop a 99% confidence band for the actual average amount of ink used by the machine to produce each photograph.

Also See:

http://stattrek.com/probability-distributions/t-distribution.aspx

Julioskij says

Hi, I don´t get it. What is the formula t = (xbar – u) / (s/Sqrt(n)) useful for? In the examples all results are drawn with T-values from the table. Can you clarify?

Thanks 🙂

Six Sigma Study Guide says

Julio, in the examples we are trying to solve the confidence interval. We get the critical t value from the chart. We then substitute that value into the t equation to find the confidence interval. If you look closely, you’ll see that the t = equation has been rewritten for u = and the critical t value is substituted in.

Hope that helps!