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# Test the means of two populations

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Two boats the prada (Italy) and the oracle (USA) are competing for a spot in the upcoming Americans cup race. They race over a part of the course several times. Below are the sample times in minutes at the .05 significance level. Can we conclude that there is a difference in their mean times?

12.9, 12.5, 11.0, 13.3, 11.2, 11.4, 11.6, 12.3, 4.2, 11.3

Oracle (USA)
14.1, 14.1, 14.2, 17.4, 15.8, 16.7, 16.1, 13.3, 13.4, 13.6

https://brainmass.com/statistics/hypothesis-testing/test-the-means-of-two-populations-159758

#### Solution Preview

18. The management of White Industries is considering a new method of assembling its golf cart. The present method requires 42.3 minutes, on the average, to assemble a cart. The mean assembly time for a random sample of 24 carts, using the new method, was 40.6 minutes, and the standard deviation of the sample was 2.7 minutes. Using the .10 level of significance, can we conclude that the assembly time using the new method is faster?
Solution. Denote by the mean assembly time. So,

H0: =42.3
H1: <42.3.

Use t-test. The test statistic can be calculated as follows.

So,
p-value=P(Z<-3.085)=0.0< alpha=0.10.

So, we should reject the null hypothesis H0. ...

#### Solution Summary

This solution contains a detailed explanation of testing the difference of the means of two populations

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## ANOVA Problem

When we want to test two samples to determine if it is likely that the population means (estimated by the sample means) are different, we typically use a t-test. If the samples are large, we can also use a z-test. (Note that the formulas for computing s, t and/or z in the case of a two-sample test are different than the formulas for computing the same values in a one-sample test. Use Excel data analysis to conduct tests comparing two sample means.)

Using ANOVA (short for Analysis of Variance), however, we can test 3 or more sample means to determine if at least one of the sample means comes from a population with a mean that is significantly different from all of the others in the test. We actually do this by estimating a combined population variance two different ways and comparing the two estimates (the ratio of these two variance estimates follows the so-called "F distribution").

Question:

Why do we need a new test method to compare the means of 3 or more populations? Why can't we just use a series of z-tests or t-tests to compare all of the possible pairs of population means to see if one (or more) is different?

Most of the testing is to determine one or two things:

1. Is there a statistically significant difference between two or more population means? (based on comparison of 2 or more sample means)

2. Is there a statistically significant relationship between two or more variables? We can use regression analysis or chi-square tests to answer this second question.)

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