Module 05 – Hypothesis Tests Using Two Samples

Class Objectives:

· Identify whether two samples are independent or dependent.

· Compare the testing procedures for two sample tests.

· Test hypothesis about two population parameters.

 

Module 05 – Part 1

Last week we took one sample to see if it supported our alternative hypothesis. This week we are going to increase to TWO samples and see if there is a significant difference between them.

 

When would we use this?

 

 

 

 

· Two samples are __________________________________ if the sample values from one population are not related to or somehow naturally paired or matched with the sample values from the other population.

· Example:

 

 

 

 

· Two samples are _____________________________ (or consist of ______________________________________) if the sample values are somehow matched, where the matching is based on some inherent relationship.

· Example:

 

 

 

 

Hint: If the two samples have different sample sizes with no missing data, they must be independent. If the two samples have the same sample size, the samples may or may not be independent.

 

Put the variables in for each population in the table below.

  Population 1 Population 2
Population Mean    
Population Standard Deviation    
Population Proportion    
Sample Size    
Sample Mean    
Sample Standard Deviation    
Sample Proportion    

 

Note: We are going to approach the problem as if are unknown. This is the most common and means that we will be using the t test statistic.

 

· The test statistic is given by the formula below:

 

 

 

where we assume .

 

To calculate the degrees of freedom, pick the _______________________ n value and subtract 1.

 

We will be doing the same steps as before to test the hypothesis (either critical value or p-value test). There are just different formulas.

 

· The null hypothesis is given as _____________________________.

· The alternative hypothesis will be either ____________________________, ___________________________, or _____________________________.

 

Example 1. Data Set 26 “Cola Weights and Volumes” in Appendix B includes weights (lb) of the contents of cans of Diet Coke (n = 36, x = 0.78479 lb, s = 0.00439 lb) and of the contents of cans of regular Coke (n = 36, x = 0.81682 lb, s = 0.00751 lb). Use a 0.05 significance level to test the claim that the contents of cans of Diet Coke have weights with a mean that is less than the mean for regular Coke.

 

 

 

 

 

 

 

 

 

 

 

 

 

Example 2. Researchers from the University of British Columbia conducted trials to investigate the effects of color on creativity. Subjects with a red background were asked to think of creative uses for a brick; other subjects with a blue background were given the same task. Responses were scored by a panel of judges and results from scores of creativity are given below. Higher scores correspond to more creativity. The researchers make the claim that “blue enhances performance on a creative task.” Use a 0.05 significance level to test the claim that blue enhances performance on a creative task.

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Example 3. A study of seat belt use involved children who were hospitalized after motor vehicle crashes. For a group of 123 children who were wearing seat belts, the number of days in intensive care units (ICU) has a mean of 0.83 and a standard deviation of 1.77. For a group of 290 children who were not wearing seat belts, the number of days spent in ICUs has a mean of 1.39 and a standard deviation of 3.06. Use a 0.01 significance level to test the claim that children wearing seat belts have a lower mean length of time in an ICU than the mean for children not wearing seat belts.

 

 

 

 

Module 05 – Part 2

Inferential statistics involves forming conclusions about population parameters.

 

· These population parameters could be:

 

 

 

 

 

 

 

The activities that we could perform on two samples are estimating the value of the population parameters using confidence intervals and testing claims made about the population parameters.

 

Independent Samples Dependent Samples
Samples taken from two different populations, where the selection process for one sample is independent of the selection process for the other sample.

 

Samples taken from two populations where either (1) the element samples is a member of both populations or (2) the element samples in the second population is selected because it is similar on all other characteristics, or “matched,” to the element selected from the first population.

 

Example.

 

 

 

 

 

 

 

Example.

 

 

 

 

The hypothesis Test for two dependent samples is a bit different because we need to use the difference from each matched pair to test the claim.

 

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