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do we have to assume normal distribution in order for two tailed tests?

Eg: According to the Australian Bureau of Statistics, the quantity of waste ending up in municipal landfills amounts to 0.80 kg per person per day. Some claim that because of recycling and greater emphasis on the environment, this figure is now lower. Others contend that constant increases in packaging and other life-style developments have pushed this figure higher in spite of recycling efforts. To test whether the amount of garbage per person has changed, a random sample of 54 Australians was taken and they were asked to keep a log of their garbage for a day. The sample mean was 0.92 kg.

Population standard deviation is 0.5 kg, and that the amount of garbage per person per day is normally distributed, use the sample data to determine whether there is sufficient evidence at the 5% level of significance to assert that the amount of garbage per person per day has changed.

Q)Is the assumption that the amount of garbage is normally distributed necessary in order to perform this test?

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Because you're looking at the mean, you can rely on the central limit theorem that says that the distribution of the mean is Normally distributed. The strength of the central limit theorem is that it works for most underlying distributions; thye don't have to be Normal. It is, however, an approximation that gets better as the sample size increases. How large your sample size needs to be depends on the kind of underlying distribution you have. As the comment says, there are some underlying distributions for which the central limit theorem doesn't work.

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    $\begingroup$ Well, if the distribution is Cauchy, then everything is hopeless. The mean is a lie. $\endgroup$ – Clement C. May 7 '15 at 23:11

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