2. Two Sample Problems

3.  Compare the responses of two treatments or compare the characteristics of 2 populations  Separate samples from each population  *Different from Matched pairs  May have 2 different sample sizes  No matching of the units That means you could test results from a group of real men like me with a group of geeks like that guy?!!

4.  Conditions  1. Two SRS’s; Independent samples (no matching); measuring same variable  2. Both populations are normally distributed w/ unknown parameters.  Needs  1. Sample Sizes (n 1, n 2 )  2. Sample Statistics (x-bar 1, x-bar 2, s 1, s 2 )  Purpose  Compare 2 means to look for a SIGNIFICANT difference

5.  Conservative Estimate when using the t-table  Calculate degrees of freedom for each sample and use the smaller of the two  n 1 - 1  n 2 – 1  Use the smaller of the two… If you use the calculator or statistical software, they will use a formula to calculate the t-statistic & degrees of freedom….

6.  We are testing the H o :µ 1 =µ 2  Write the Hypotheses (context)  Check the Conditions (show)  Calculate the T Statistic  Find the P-Value for the appropriate df  Make statistical decision and interpret results in context

7.  The hypotheses for this test will be as follows, depending on the situation and context of the problem:  Ho: µ1 = µ2  Ha: µ1 > µ2, µ1 < µ2, µ1 ≠ µ2

8. Standard Error Using the smaller of the n-1 degrees of freedom 2-Sample T- Statistic formula:

9.  Fantastic Fishy Food advertises the best fish growing formula on the market, but so does Nibbles n’ Bits, however. As a research project, Ronnie has decided to study the growth rates of fish given these two foods over a set period of time. After carefully setting up the experiment, Ronnie measured a SRS of 48 fish from the FFF group, finding an average growth of 15 g with a standard deviation of 2.32g. The Nibbles n’ Bits group of fish grew an average of 16.7 g with a standard deviation of 1.87g out of a SRS of 52 measurements. Help Ron decide if there’s a significant difference between the two food types and the growth they produce in the fish at a 5% level.

10.  H o : The average mass increase of fish is the same between Fantastic Fishy Food and Nibbles n’ Bits. µ FFF = µ NnB  H a : The average mass increase of fish is significantly different between Fantastic Fishy Food and Nibbles n’ Bits. µ FFF ≠ µ NnB t = -4.0138 w/ df = 49 p <.0005 <  =.05 so we reject that the 2 foods cause the same growth.

11.  Using the calculator gives even more accurate results using exact degrees of freedom (not our limited chart)  Stat – Tests – 4:2-SampTTest

12.  Fantastic Fishy Food advertises the best fish growing formula on the market, but so does Nibbles n’ Bits, however. As a research project, Ronnie has decided to study the growth rates of fish given these two foods over a set period of time. After carefully setting up the experiment, Ronnie measured a SRS of 48 fish from the FFF group, finding an average growth of 15 g with a standard deviation of 2.32g. The Nibbles n’ Bits group of fish grew an average of 16.7 g with a standard deviation of 1.87g out of a SRS of 52 measurements. Help Ron decide if there’s a significant difference between the two food types and the growth they produce in the fish. Try using the Calculator this time…

13.  There is an option on your calculator to pool your degrees of freedom.  This option can only be used if:  the sample sizes are exactly the same  But also, only if the variances of the two population are known to be the same. This basically means we WON’T be using the Pooled option, Sucka!