INCM 9102 Quantitative Methods

1. INCM 9102Quantitative Methods Ttests

2. Ttests The term “Ttest” comes from the application of the t-distribution to evaluate a hypothesis. Note: a “t-statistic” and a “z-score” are conceptually similar

3. Ttests A side note of interest from Wikipedia: The t-statistic was introduced in 1908 by William Sealy Gosset, a chemist working for the Guiness Brewery in Dublin, Ireland. Gosset had been hired due to Claude Guinness's innovative policy of recruiting the best graduates from Oxford and Cambridge to apply biochemistry and statistics to Guinness' industrial processes. Gosset devised the t-test as a way to cheaply monitor the quality of beer. He published the test in Biometrika in 1908, but was forced to use a pen name by his employer, who regarded the fact that they were using statistics as a trade secret.

4. Ttests The t-table 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.

5. Ttests Ttests take three forms: • 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

6. Ttests Fun and exciting example: 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.

7. 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…

8. Ttests • 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

9. Ttests • When dealing with two sample or paired ttests, it is important to check the following assumptions: • The samples are independent • The samples have approximately equal variance • 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.

10. Ttests • 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