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I am trying to express a hypothesis to test for the following claim:

A biologist is presented with the data that shows an increase in the average number of bacterias, though he suspects there was no actual change.

I came up with the following:

  • $H_0:\; \mu = 40.10^2$ [no change had occurred]
  • $H_A:\; \mu > 40.10^2$ [there is an increase]
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    $\begingroup$ can you provide more information? Why is $\mu = 4000$ the status quo? $\endgroup$ – Macro Jul 25 '11 at 3:35
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    $\begingroup$ A two-sided alternative hypothesis $\mu \neq 40 \times 10^2$ would usually be considered more appropriate than a one-sided alternative, especially if the direction of the one-sided alternative was only chosen because the data went that way. $\endgroup$ – mark999 Jul 25 '11 at 3:37
  • $\begingroup$ Macro - this is how it is presented. $\endgroup$ – newprint Jul 25 '11 at 3:47
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    $\begingroup$ To echo @Macro and clarify a subtle point, there's a difference between the case where $\mu=4000$ has been estimated from data (either yours or an independent experiment) and where it is given as a theoretical value. In the first case it's subject to observational error which has to be accounted for in the testing. In the second case there's no observational error. More generally, you need to stipulate a probability model for how the data depend on $\mu$. For instance, if you are monitoring bacterial count over time, you might need a regression model. $\endgroup$ – whuber Jul 25 '11 at 14:46
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The null hypothesis you have so far is a good start.

As this question has the homework tag, I'll answer with two questions for you about it, that might nudge you to a better null hypothesis & test of it:

1) What would an observed large decline in the number of bacteria have said about the null hypothesis?

2) Is the null hypothesis you have so far, testable or not? Do you know of any statistical tests which could be done with no information at all about the nature of the underling probability distribution, or the likely statistical pattern of the observation?

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