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I just used the standard formula to determine the sample size of a sample to match the mean of a pupulation with an error margin of 3 percentage points and with a 90% probability. I know would like to check that the mean of my sample is actually within the 3% margin of error of the mean in the population. To do so I am taking the population and appending the sample, identifying observations in the sample with a dummy called sample. Then I run the following regresion: reg var1 sample And I am testing wheter the coefficient for the dummy variable sample is less than 3 percentage points. To do so I am using a one sided t test.

But this is only ok when the coefficient of the dummy variable "sample" has the "right" sign. I would like to test the joing hypoithesis that the coefficient of the dummy variable sample is greater than -3 and lower than 3. Does anyone have any insight on how to do this test? Thanks in advance.

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    $\begingroup$ Your question admits a direct mathematical implementation: you know the population mean $\mu$, you know the sample mean $m$, and the margin of error is $0.03|\mu|$. Why aren't you just checking that $|m-\mu|\le 0.03|\mu|$? That's a calculation with two numbers, nothing more. $\endgroup$
    – whuber ♦
    Commented Jul 16, 2014 at 21:57
  • $\begingroup$ Thanks, I might be missing something, but don't I need to check wheter |m−μ| is STATISTICALLY significantly less than 0.03|μ|? $\endgroup$
    – Laurie
    Commented Jul 16, 2014 at 22:02
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    $\begingroup$ That would make no sense. A test of hypothesis concerns the population mean $\mu$, not the sample mean $m$. It uses the sample mean to draw conclusions about $\mu$, that's all. Your question asks whether two numbers--the mean of this particular sample and the mean of the population--are sufficiently close. If that's what you really mean to ask, then the answer is obtained with a few keystrokes on a calculator, not any statistical test. $\endgroup$
    – whuber ♦
    Commented Jul 16, 2014 at 22:11
  • $\begingroup$ What I mean to say is that: with this particular sample, when making inference about the mean of the population, y get that the mean of the population with 90% probability in a confidence interval. Is the true mean + - 3% in that confidence interval? If this is the case, then the sample is representative with the desired precision... $\endgroup$
    – Laurie
    Commented Jul 16, 2014 at 22:20
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    $\begingroup$ I don't follow at all, because you have no need to make inferences about the population mean: you have stated you know it! $\endgroup$
    – whuber ♦
    Commented Jul 16, 2014 at 22:22

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1) Why not just use confidence intervals?

2) For reference, the test you want here (asking whether a constraint on beta would hold or not) is a straight up Lagrange multiplier test, where your constraint function is

$$ \beta_{\mathrm{sample}}^2 - 9 = 0 $$

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  • $\begingroup$ Shouldn't there be an inequality in the LM test? $\endgroup$
    – Laurie
    Commented Jul 16, 2014 at 20:55
  • $\begingroup$ Yes. And no. Technically the LM test is for whether or not the constraint $h(\theta) = 0$ binds, and the direction of integration over $\lambda$ determines the direction of the inequality. Edit: Or, I should say, the inequality determines the direction of integration. $\endgroup$
    – cycle_cycle_cycle
    Commented Jul 16, 2014 at 21:02

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