# Testing hypothesis & $\alpha$

Hey I have a question that I found in a textbook for some practice before a test, but there aren't any solutions for it. I'm pretty sure it's related to testing a hypothesis, but I'm not sure.

If anyone can point me into the right direction of solving this that would be great.

Q1: Five soft drink bottling companies have agreed to implement a time management program in hopes of increasing productivity (measured in cases of soft drinks bottled per hour). The number of cases of soft drinks bottled per hour before and after implementation of the program are listed below.

        Company
#  1   2   3   4   5
Before 500 475 525 490 530
After  510 480 525 495 533


Test at $\alpha$ = .05 if the time management program is efficient in increasing the productivity.

• Which textbook are these questions you're posting from? What have you tried with this question? Dec 11, 2013 at 14:54

The data is obviously paired. Furthermore there are few observations so assessing normality and homoscedasticity may not be reliable. A Wilcoxon signed rank test seems appropriate.

• Welcome to our site, Matt. This does not seem like the best advice. The data are not obviously paired, because there is no basis to associate each hour before the intervention with the hours after. Furthermore, one does not need to assess normality of the data. We must contemplate the possible approximate normality of the sampling distribution of the sample mean. There is no evidence against that, so a t-test would be perfectly appropriate and have greater power than a nonparametric test.
– whuber
Dec 11, 2013 at 14:32
• Thanks for the welcome. The measurements are made before and after a certain change, but in the same companies. It's equivalent to the classical paired data example of weight of the same persons before and after a diet. There may be no evidence against that assumption, however that does not make it correct. With such a small sample size the power for detecting non-normality is small. The power gain of the t-test when compared to the Wilcoxon signed rank test is also quite small (see e.g. Lehmann), so if unsure the nonparametric alternative may be safest (though a little conservative). Dec 11, 2013 at 14:58
• Thank you for the clarification (+1). I see now that the data are for five companies, rather than for a sequence of five hours, so I agree with you: they are paired by companies. The issue is not to detect non-normality, but to assess whether one should defend against frank non-normality of the sampling distribution of the mean. Because the sample size is small, one would hate to lose any power. Also, I believe the tie at $525$ might cause some difficulties for most implementations of the Wilcoxon signed rank test.
– whuber
Dec 11, 2013 at 15:35
• Agree on the tie issues. However, various authors have shown that nonparametric tests are more appropriate for small sample sizes (if the underlying sampling distribution is unknown) and even have higher power for some distributional shapes. Some authors state that one should never do parametric tests with very small sample sizes (e.g. Siegel & Castellan) Anyhow, with only 5 observations, ties and the 5% significance level the Wilcoxon test will probably be not so useful also... Dec 11, 2013 at 16:11
• Thank you for the advice and references. The issue with the tie actually could be important: a few quick calculations suggest the p-values are hovering in the 5-10% range (depending on which test is chosen), right on the threshold of "significance." Of course that raises other issues, such as how reliable would a decision be that depends on the details of how a single tie is treated, but since this is evidently a textbook exercise, pursuing such points here would perhaps be unhelpful.
– whuber
Dec 11, 2013 at 18:03

im pretty sure I its relating to testing a hypothesis, but im not too sure If anyone can point me into the right direction of solving this that would be great !

After and before are samples of given populations. Samples have their own distributions and two samples from the same population won't be necessarily identical. Consequently, you have to test if the differences between the after of before samples can be assigned to the sampling process (both are part of the same population) or to improvements in the productivity (the samples were taken from different populations).

If I'm not clear enough, don't hesitate to ask.