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Significance Tests / Hypothesis Testing
In the figure above, I used the to calculate the probability of getting each possible number of males, from 0 to 48, under the null hypothesis that 0.5 are male. As you can see, the probability of getting 17 males out of 48 total chickens is about 0.015. That seems like a pretty small probability, doesn't it? However, that's the probability of getting exactly 17 males. What you want to know is the probability of getting 17 or fewer males. If you were going to accept 17 males as evidence that the sex ratio was biased, you would also have accepted 16, or 15, or 14,… males as evidence for a biased sex ratio. You therefore need to add together the probabilities of all these outcomes. The probability of getting 17 or fewer males out of 48, under the null hypothesis, is 0.030. That means that if you had an infinite number of chickens, half males and half females, and you took a bunch of random samples of 48 chickens, 3.0% of the samples would have 17 or fewer males.
Next, you’ll need to state the null hypothesis (See: ). That’s what will happen if the researcher is wrong. In the above example, if the researcher is wrong then the recovery time is less than or equal to 8.2 weeks. In math, that’s:
H_{0} μ ≤ 8.2
In this example, the hypotheses are:
Ten or so years ago, we believed that there were 9 planets in the solar system. Pluto was demoted as a planet in 2006. The null hypothesis of “Pluto is a planet” was replaced by “Pluto is not a planet.” Of course, rejecting the null hypothesis isn’t always that easy — the hard part is usually figuring out what your null hypothesis is in the first place.
The hypothesis statement in this question is that the researcher believes the average recovery time is more than 8.2 weeks. It can be written in mathematical terms as:
H_{1}: μ > 8.2
Andreas Cellarius hypothesis, showing the planetary motions.
If you prefer an online interactive environment to learn R and statistics, this is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try .
If you prefer an online interactive environment to learn R and statistics, this is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try .
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A good hypothesis statement should:
It’s good science to let people know if your study results are solid, or if they could have happened by chance. The usual way of doing this is to test your results with a . A p value is a number that you get by running a hypothesis test on your data. A P value of 0.05 (5%) or less is usually enough to claim that your results are repeatable. However, there’s another way to test the validity of your results: Bayesian Hypothesis testing. This type of testing gives you another way to test the strength of your results.
Next section: to Inferential statistics (testing hypotheses)
Traditional testing (the type you probably came across in elementary stats or AP stats) is called NonBayesian. It is how often an outcome happens over repeated runs of the experiment. It’s an objective view of whether an experiment is repeatable.
Bayesian hypothesis testing is a subjective view of the same thing. It takes into account how much faith you have in your results. In other words, would you wager money on the outcome of your experiment?
I’m stuck on how to value the null or alternative hypotheses
In the second experiment, you are going to put human volunteers with high blood pressure on a strict lowsalt diet and see how much their blood pressure goes down. Everyone will be confined to a hospital for a month and fed either a normal diet, or the same foods with half as much salt. For this experiment, you wouldn't be very interested in the P value, as based on prior research in animals and humans, you are already quite certain that reducing salt intake will lower blood pressure; you're pretty sure that the null hypothesis that "Salt intake has no effect on blood pressure" is false. Instead, you are very interested to know how much the blood pressure goes down. Reducing salt intake in half is a big deal, and if it only reduces blood pressure by 1 mm Hg, the tiny gain in life expectancy wouldn't be worth a lifetime of bland food and obsessive labelreading. If it reduces blood pressure by 20 mm with a confidence interval of ±5 mm, it might be worth it. So you should estimate the effect size (the difference in blood pressure between the diets) and the confidence interval on the difference.
What descriptive and inferrential statistics to use
Here are three experiments to illustrate when the different approaches to statistics are appropriate. In the first experiment, you are testing a plant extract on rabbits to see if it will lower their blood pressure. You already know that the plant extract is a diuretic (makes the rabbits pee more) and you already know that diuretics tend to lower blood pressure, so you think there's a good chance it will work. If it does work, you'll do more lowcost animal tests on it before you do expensive, potentially risky human trials. Your prior expectation is that the null hypothesis (that the plant extract has no effect) has a good chance of being false, and the cost of a false positive is fairly low. So you should do frequentist hypothesis testing, with a significance level of 0.05.
the null hypothesis is rejected when it is true b.
Bayesian hypothesis testing helps to answer the question: Can the results from a test or survey be repeated?
Why do we care if a test can be repeated? Let’s say twenty people in the same village came down with leukemia. A group of researchers find that cellphone towers are to blame. However, a second study found that cellphone towers had nothing to do with the cancer cluster in the village. In fact, they found that the cancers were completely random. If that sounds impossible, it actually can happen! Clusters of cancer can happen . There could be many reasons why the first study was faulty. One of the main reasons could be that they just didn’t take into account that sometimes things happen randomly and we just don’t know why.
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