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Significance Tests / Hypothesis Testing

Five Steps in a Hypothesis Test

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The Language of Hypothesis Testing

The theory of statistical hypothesis testing was developed in the early 20th century and has been the mainstay of practical statistics ever since. It was designed to apply the scientific method to situations involving data with random fluctuations (and almost all real-world data has random fluctuations). Following are a few terms commonly used in hypothesis testing.

The test statistic for examining hypotheses about one population mean:

Area under theoretical models of distributions is the method that classical hypothesis testing employs to estimate probabilities. A major part of an intermediate course in mathematical statistics is the theoretical justification of the models that are used in hypothesis testing.

Call the ``null'' hypothesis and the ``alternative''.

Hypothesis tests are procedures for making rational decisions about the reality of effects.

State what will happen if the experiment doesn’t make any difference. That’s the null hypothesis–that nothing will happen. In this experiment, if nothing happens, then the recovery time will stay at 8.2 weeks.

This number, 0.030, is the P value. It is defined as the probability of getting the observed result, or a more extreme result, if the null hypothesis is true. So "P=0.030" is a shorthand way of saying "The probability of getting 17 or fewer male chickens out of 48 total chickens, IF the null hypothesis is true that 50% of chickens are male, is 0.030."

Next section: to Inferential statistics (testing hypotheses)

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.

After you do a statistical test, you are either going to reject or accept the null hypothesis. Rejecting the null hypothesis means that you conclude that the null hypothesis is not true; in our chicken sex example, you would conclude that the true proportion of male chicks, if you gave chocolate to an infinite number of chicken mothers, would be less than 50%.

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  • How to Determine a p-Value When Testing a Null Hypothesis

    Steps for testing hypotheses

  • Null hypothesis: μ = 72 Alternative hypothesis: μ ≠72

    Research Hypothesis (H)

  • Null hypothesis: μ = 72 Alternative hypothesis: μ ≠72

    Why the Null Hypothesis (H)?

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That’s How to State the Null Hypothesis!

The primary goal of a statistical test is to determine whether an observed data set is so different from what you would expect under the null hypothesis that you should reject the null hypothesis. For example, let's say you are studying sex determination in chickens. For breeds of chickens that are bred to lay lots of eggs, female chicks are more valuable than male chicks, so if you could figure out a way to manipulate the sex ratio, you could make a lot of chicken farmers very happy. You've fed chocolate to a bunch of female chickens (in birds, unlike mammals, the female parent determines the sex of the offspring), and you get 25 female chicks and 23 male chicks. Anyone would look at those numbers and see that they could easily result from chance; there would be no reason to reject the null hypothesis of a 1:1 ratio of females to males. If you got 47 females and 1 male, most people would look at those numbers and see that they would be extremely unlikely to happen due to luck, if the null hypothesis were true; you would reject the null hypothesis and conclude that chocolate really changed the sex ratio. However, what if you had 31 females and 17 males? That's definitely more females than males, but is it really so unlikely to occur due to chance that you can reject the null hypothesis? To answer that, you need more than common sense, you need to calculate the probability of getting a deviation that large due to chance.

1. the null hypothesis were true (there really were no effects), and

Note that if the alternative hypothesis is the less-than alternative, you reject H0 only if the test statistic falls in the left tail of the distribution (below –2). Similarly, if Ha is the greater-than alternative, you reject H0 only if the test statistic falls in the right tail (above 2).

Null Hypothesis (1 of 4) - David Lane

It is important to distinguish between biological null and alternative hypotheses and statistical null and alternative hypotheses. "Sexual selection by females has caused male chickens to evolve bigger feet than females" is a biological alternative hypothesis; it says something about biological processes, in this case sexual selection. "Male chickens have a different average foot size than females" is a statistical alternative hypothesis; it says something about the numbers, but nothing about what caused those numbers to be different. The biological null and alternative hypotheses are the first that you should think of, as they describe something interesting about biology; they are two possible answers to the biological question you are interested in ("What affects foot size in chickens?"). The statistical null and alternative hypotheses are statements about the data that should follow from the biological hypotheses: if sexual selection favors bigger feet in male chickens (a biological hypothesis), then the average foot size in male chickens should be larger than the average in females (a statistical hypothesis). If you reject the statistical null hypothesis, you then have to decide whether that's enough evidence that you can reject your biological null hypothesis. For example, if you don't find a significant difference in foot size between male and female chickens, you could conclude "There is no significant evidence that sexual selection has caused male chickens to have bigger feet." If you do find a statistically significant difference in foot size, that might not be enough for you to conclude that sexual selection caused the bigger feet; it might be that males eat more, or that the bigger feet are a developmental byproduct of the roosters' combs, or that males run around more and the exercise makes their feet bigger. When there are multiple biological interpretations of a statistical result, you need to think of additional experiments to test the different possibilities.

The null hypothesis is an hypothesis about a population parameter

Figure out the . The alternate hypothesis is the opposite of the null hypothesis. In other words, what happens if our experiment makes a difference?

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