2

How do I do a statistics test that my population mean is less than some value (rather than equal to some value)? I want to do this in two flavors:

  1. sample 30 oranges and test whether the average weight of an orange from my population is less than 1 pound.
  2. sample 30 oranges and 30 apples test whether the the average weight of an apple from my population is more than 2x the the average weight of an orange from my population.

For the first case, I see a Student's t-test for one sample which looks related: could I verify that the sample is different from 1 pound with that test, then actually compute the sample mean and compare to 1 pound to check if I'm below or above.

For the second case, could I do a one-way anova? I would double the weight of each sampled apple and use these new values as a sample to compare vs the sampled oranges. Again, I would use the test to verify that they were different, then actually compute the respective sample means and compare to see if I'm below or above.

Is this the proper way to do such tests that a population mean is less/greater than some value, or is there some better way?

Gaussian0617
  • 704
Alex Reece
  • 123

1 Answers1

1

In order test that your population is less than some value (as opposed to equal to some value), you perform a one-tailed test of significance. You simply compute your test statistic and see if it is less than the critical value corresponding to your desired significance level $\alpha$.

You can use a 2-sample $t$-test in order to compare the two population means in part 2. Your $t$-statistic would be computed as follows:

$$ t = \frac{(\overline{x}_1 - \overline{x}_2) - (\mu_1 - \mu_2)}{\sqrt{s_p^2/n_1 + s_p^2/n_2}} $$

Where the the pooled variance $s_p^2$ is computed as follows:

$$ s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} $$

In order to test the hypothesis that the mean weight of an apple is more than two times the mean weight of an orange, you will need to conduct a test of $H_0: \mu_{apples} > 2\mu_{oranges}$. Thus, your $\overline{x}_2$ and your $s_2^2$ in the above formulae would correspond the the double-weighted sample of oranges.

Gaussian0617
  • 704
  • I believe that a 2-sample t-test is equivalent to 1-way anova: see note on http://www.biostathandbook.com/testchoice.html. Also, this doesn't answer my fundamental question about comparing that my mean is less than some value during a statistics test. – Alex Reece Oct 26 '15 at 21:53
  • That's correct; the 2-sample t-test is a special case of 1-way anova. I've edited my answer and hopefully I've addressed your question now. – Gaussian0617 Oct 26 '15 at 22:04