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I wanted to ask you for the advice. I am doing some stats exercise and apparently, I cannot get the zero and alternative hypothesis correct. For example, 'At the .01 significance level, can bank management conclude that younger customers use the ATMs more than senior customers?' I thought that the zero hypothesis is that younger customers use the ATMs more or equal to than senior citizens while alternative that less. But the answer sheet says that it's another way around. Do I need to always reverse the question? What would be your advice?

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It seems that the context of this question is traditional frequentist hypothesis testing, and I will answer relative to that. (There are other frameworks for statistical decision-making that might take different approaches.)

To "conclude at the 1% level that young customers use ATMs more than senior customers" $(\mu_y > \mu_s)$ means that we reject the null hypothesis $H_0: \mu_y \le \mu_s$ in favor of the alternative hypothesis $H_a: \mu_y > \mu_s.$

If the study is well-planned, there will have been an earlier power computation to determine the necessary sample sizes of the youth and senior groups (typically the same for both groups). The idea is to take into account the assumed variability in ATM usage within the two groups and the difference in usage between groups that would be of practical importance. Then the sample size would be chosen so that, if the actual difference in usage is some particular number $\Delta = \mu_y - \mu_s$ or greater, then there is reasonable probability, say 90% or 95% or even 99%, that $H_0$ will be rejected.

Sometimes, one distinguishes between the 'research hypothesis' $(H_a)$ and the 'null hypothesis' $(H_0).$ Often the null hypothesis is the unremarkable one of the two and the alternative hypothesis would be more interesting. Perhaps if $H_0$ is rejected in favor of $H_a,$ then more attention will be paid to the needs of young ATM customers in the future (types of transactions available, locations of ATMs, incentives to use ATMs, etc.)

Also, remember that the null hypothesis must always contain an $=$-sign --- whether as $\mu_y \le \mu_s$ (here), or as $\mu_y = \mu_s$ or $\mu_y \ge \mu_s$ (in other situations). This is because the $=$-sign in the null hypothesis determines the 'null distribution' for performing the test (determining the critical value for a test at the 1% significance level or computing the P-value).

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  • $\begingroup$ Thank you, Bruce, but it's not exactly what I am looking for. Let's say there are two exercises, in one assignment it's said: ' At the .01 significance level, can bank management conclude that younger customers use the ATMs more?'. In the answer sheet, I see that H0: µ1 ≤ µ2 H1: µ1 > µ2. Then, there is another example: 'At the 5% level of significance, is it reasonable to conclude that students at West Virginia sleep less than the typical American?' H0: µn = µs H1: µn ≠ µs . For me, the way hypothesises are framed are totally different from the questions asked... $\endgroup$ – Ekaterina Ponkratova Oct 27 '18 at 2:02
  • $\begingroup$ Second example seems strange to me. Maybe something like, 'is it reasonable to conclude that students in WV sleep a different amount than the typical American'. Answer keys are sometimes written by busy graduate students and not carefully vetted by main authors. Best thing may be to ask your instructor to resolve questions with puzzling answers; if key is wrong instructor should be happy to know. $\endgroup$ – BruceET Oct 27 '18 at 3:04
  • $\begingroup$ Thank you Bruce, otherwise, my logic seems to be the same as yours. In the first case, I had H0: µ1>= µ2 H1: µ1 <µ2. Totally confused with some questions asked and what I see in the answer sheet. The worst part is that as a learner I rely on that answer sheet to confirm my understanding! I thought that may be I need to negate the question i.e. i they say 'conclude that younger customers use the ATMs more', I need to reverse the question in order to get H0. This logic works in some cases but don't work in other. That's why I posted here. $\endgroup$ – Ekaterina Ponkratova Oct 27 '18 at 6:30
  • $\begingroup$ Textbook questions need to establish context in a couple of sentences, so rely on 'code' expressions like 'conclude at 5% level that a > b'. That should always mean the alternative hypothesis is a > b. You don't 'conclude' anything by failing to reject $H_0.$ In real applications, you need to go by what you'd like to establish, again that's usually the alternative hyp. // Give up the idea of 'reversing' the question; not always useful. Please read my Answ again carefully. Take situations 1 at a time. Don't try to make stuff up to get a simple gen'l rule--that frequently leads to confusion. $\endgroup$ – BruceET Oct 27 '18 at 7:56

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