Lecture 4Ttests

STAT 3120Statistical Methods I

STAT3120 - Ttests

The term “Ttest” comes from the application of the t-distribution to evaluate a hypothesis. The t-distribution is used when the sample size is too small (less than 30) to use s/SQRT(n) as a substitute for the population std. Refer to pages 228/229 in your book for a detailed explanation of the t-distribution.

In practice, even hypothesis tests with sample sizes greater than 30, which utilize the normal distribution, are commonly referred to as “ttests”.

Note: a “t-statistic” and a “z-score” are conceptually similar

STAT3120 - Ttests

The t-table (see Appendix 2) is effectively the inverse of the z-score table – the “inside” of the table includes the t-statistics while the “outside” of the table (the designation of the rows and columns) includes the degrees of freedom and the alpha values.

To determine an appropriate t-statistic for a test, you must know the alpha value and the degrees of freedom (n-1).

Note that as the number of observation increase, the t-distribution is assumed to approach normality.

Ttests take three forms:1.One Sample Ttest - compares the mean of the sample to a given number.

• e.g. Is average monthly revenue per customer who switches >$50 ?

Formal Hypothesis Statement examples:

H0: $50

H1: > $50

H0: = $50

H1: $50

STAT3120 - Ttests

STAT3120 - Ttests

Example from page 232:

After a massive outbreak of salmonella, the CDC determined that the source was from a particular manufacturer of ice cream. The CDC sampled 9 production runs if the manufacturer, with the following results (all in MPN/g):

.593 .142 .329 .691 .231 .793 .519 .392 .418

Use this data to determine if the avg level of salmonella is greater than .3 MPN/g, which is considered to be dangerous.

STAT3120 - Ttests

First, Identify the Hypothesis Statements, including the Type I and Type II errors…and your assignment of alpha.

Then, do the computation by hand…

Here is the SAS code that you need to execute a one sample ttest:

Proc ttest data=<dataname> H0= <the null statement> alpha=<alpha value>;Var <quant var> ;Run;

STAT3120 - Ttests

2. Two Sample Ttest - compares the mean of the first sample minus the mean of the second sample to a given number.

• e.g. Is there a difference in the production output of two facilities?

Formal Hypothesis Statement examples:

H0: a - b =0

H1: a - b 0

STAT3120 - Ttests

STAT3120 - Ttests

When dealing with two sample or paired ttests, it is important to check the following assumptions:

1. The samples are independent2. The samples have approximately equal variance3. The distribution of each sample is approximately

normal

Note – if the assumptions are violated and/or if the sample sizes are very small, we first try a transformation (e.g., take the log or the square root). If this does not work, then we engage in non-parametric analysis: Wilcoxan Rank Sum or Wilcoxan Signed Rank tests.

STAT3120 - Ttests

Example: Problem 6.9 from the exercises:

Proc TTEST data=;

Class <the two samples>;

Var <the quant variable to be tested>;

run;

3. Paired Sample Ttest - compares the mean of the differences in the observations to a given number.

e.g. Is there a difference in the production output of a facility after the implementation of new procedures?

Formal Hypothesis Statement example:

H0: diff=0

H1: diff 0

STAT3120 - Ttests

Example: Problem 6.28 from the exercises:

PROC TTEST DATA= H0=0 ALPHA=<>; Var Diff; RUN;

STAT3120 - Ttests