In this model we take the difference of each pair and

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Finite Mathematics for the Managerial, Life, and Social Sciences: An Applied Approach
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Chapter 8 / Exercise 2
Finite Mathematics for the Managerial, Life, and Social Sciences: An Applied Approach
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In this model we take the difference of each pair and create a new population of differences, so if effect, the hypothesis test is a one population test of mean that we already covered in the prior section. Example - rental cars An independent testing agency is comparing the daily rental cost for renting a compact car from Hertz and Avis. A random sample of 15 cities is obtained and the following rental information obtained. At the .05 significance level can the testing agency conclude that there is a difference in the rental charged? Notice in this example that cities are the single population being sampled and that two measurements (Hertz and Avis) are being taken from each city. Using the matched pair design, we can eliminate the variability due to cities being differently priced (Honolulu is cheap because you can’t drive very far on Oahu!) Design Research Hypotheses: H o : µ 1 = µ 2 (Hertz and Avis have the same mean price for compact cars.) H a : µ 1 µ 2 (Hertz and Avis do not have the same mean price for compact cars.) Model will be matched pair t-test and these hypotheses can be restated as: H o : µ d = 0 H a : µ d 0 The test will be run at a level of significance ( α ) of 5%.
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Finite Mathematics for the Managerial, Life, and Social Sciences: An Applied Approach
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Chapter 8 / Exercise 2
Finite Mathematics for the Managerial, Life, and Social Sciences: An Applied Approach
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P a g e | 170 Model is two-tailed matched pairs t-test with 14 degrees of freedom. Reject Ho if t < -2.145 or t >2.145. Data/Results We take the difference for each pair and find the sample mean and standard deviation. 𝑡 = 1.80 0 2.513 15 = 2.77 Reject Ho under either the critical value or p-value method. Conclusion There is a difference in mean price for compact cars between Hertz and Avis. Avis has lower mean prices. The advantage of the matched pair design is clear in this example. The sample standard deviation for the Hertz prices is $5.23 and for Avis it is $5.62. Much of this variability is due to the cities, and the matched pairs design dramatically reduces the standard deviation to $2.51, meaning the matched pairs t-test has significantly more power in this example. 10.4 Independent sampling – comparing two population variances or standard deviations Sometimes we want to test if two populations have the same spread or variation, as measured by variance or standard deviation. This may be a test on its own or a way of checking assumptions when deciding between two different models (e.g.: pooled variance t-test vs. unequal variance t-test). We will now explore testing for a difference in variance between two independent samples. 15 513 . 2 80 . 1 = = = n s X d d
P a g e | 171 Characteristics of F Distribution It is positively skewed It is non-negative There are 2 different degrees of freedom (df num , df den ) When the degrees of freedom change, a new distribution is created The expected value is 1. 10.4.1 F distribution The F distribution is a family of distributions related to the Normal Distribution. There are two different degrees of freedom, usually represented as numerator (df num ) and denominator (df den ). Also, since the F represents squared data, the inference will be about the variance rather than the about the standard deviation.

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