Question

You find yourself on planet Coraciidae where the indigenous creatures are smart, logical, and quite conveniently, speak English. However, after asking a few questions, you realize that they have no computers or advanced technology of any kind. You reach into your pocket to show them your cellphone, sure that they will be amazed, but realize, to your horror, that your battery is dead and you have no charger. You’re on your own here with no computer, no phone, no technology whatsoever. You’d better make yourself comfortable and useful, it seems like you might be stuck here for a while. On one of the first days on the planet, you meet Ektuz, a scientist who explains that he is trying to convince his peers that he’s discovered an odor that can be used to attract cockroaches (apparently cockroaches bring good luck on this planet). The apparatus he’s built to test his belief is quite simple: The scientist explains that he randomly chooses two of the six arms of the maze and puts a small drop of the attractant smell in those two arms. The other four arms he leaves empty. He then releases a roach in the center of the maze and records whether the roach walks down one of the legs that has the attractant odor in it, or if it chooses a leg that is empty. He says that he has repeated this process with 110 roaches and that 49 of the 110 roaches chose a leg that did have the odor he was testing. That’s better than chance, so he thinks this evidence supports his claims. Sadly, he explains that his colleagues dismiss his results as just the random behavior of the insect, since most of the roaches still wandered down the legs that did not have the odor in them. Fully explain how you would help your new friend Ektuz run a statistical test to see if his data supports the claim that the odor is attracting roaches. You don’t have access to a computer, phone, or calculator, so you won’t be able to provide an exact probability in your answer, but you need to present a step-by-step process for arriving at the answer and explain why you are performing each step. Your process/description needs to be rigorous enough to convince the skeptics (who are quite bright and will listen to reason). It might take a while to run the test, but Ektuz is willing to work on it for weeks if it will convince his colleagues. Remember, you don't have access to a computer or any other technology, but Ektuz’s lab is right next door to a casino, so you have unlimited access to coins, decks of cards, and dice, all of which are exactly like the stuff we have back on earth, and which may prove helpful for this project.

Answer 1

Answer to the question)

Statistically if out of 6 arms only two have odours, and if the scientist believes than roaches come to that odour the proportion of roaches flowing to that odour must be GREATER than 2/6 = 0.33

The proportion by chance must be 0.33

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Thus we have population parameter proportion P = 0.33 if the odour is ineffective

And the population parameter proportion P > 0.33 if the odour is effective

These two statements are called null and alternate hypothesis respectively

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Now based on the sample we get:

Sample size = 110

Sample proportion (p^) = 49/110 = 0.45

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Now for running the statistical test, we use the one sample proportion Z test

The formula of test statistic Z is as follows:

Z = (p^ - P) / sqrt(p*(1-p)/n)

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On plugging the values we get:

Z = (0.45 -0.33) / sqrt(0.33*0.67/110)

Z = 2.68

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By empirical rule we know that if the Z value > 2, then the probability of the null hypothesis to be true is less than 0.05

And hence we conclude that the conclusion drawn by the scientist is correct

The proportion of roaches attracted towards the odour is more than just by chance