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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.

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."

Every hypothesis test contains a set of two opposing statements, or hypotheses, about a population parameter. The first hypothesis is called the denoted H_{0}. The null hypothesis always states that the population parameter is to the claimed value. For example, if the claim is that the average time to make a name-brand ready-mix pie is five minutes, the statistical shorthand notation for the null hypothesis in this case would be as follows:

When you set up a hypothesis test to determine the validity of a statistical claim, you need to define both a null hypothesis and an alternative hypothesis.

In the second experiment, you are going to put human volunteers with high blood pressure on a strict low-salt 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 label-reading. 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.

which we get by inserting the hypothesized value of the population mean difference (0) for the population_quantity. If or (that is, ), we say the data are not consistent with a population mean difference of 0 (because does not have the sort of value we expect to see when the population value is 0) or "we **reject the hypothesis that the population mean difference is 0**". If t were -3.7 or 2.6, we would reject the hypothesis that the population mean difference is 0 because we've observed a value of t that is unusual if the hypothesis were true.

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50 years ago, Milton Friedman articulated the natural rate hypothesis. It was composed of two sub-hypotheses: First, the natural rate of unemployment is independent of monetary policy. Second, there is no long-run trade-off between the deviation of unemployment from the natural rate and inflation. Both propositions have been challenged. The paper reviews the arguments and the macro and micro evidence against each. It concludes that, in each case, the evidence is suggestive, but not conclusive. Policy makers should keep the natural rate hypothesis as their null hypothesis, but keep an open mind and put some weight on the alternatives.

If (that is, ), we say the data are consistent with a population mean difference of 0 (because has the sort of value we expect to see when the population value is 0) or "we **fail to reject the hypothesis that the population mean difference is 0**". For example, if t were 0.76, we would fail reject the hypothesis that the population mean difference is 0 because we've observed a value of t that is unremarkable if the hypothesis were true.

That is, if the null hypothesis were to be rejected at= 0.05, this would be reported as 'p Small p-values suggest that the null hypothesis is unlikely to be true.

It indicates the strength of evidence for say, rejecting the null hypothesis H, rather than simply concluding 'reject H' or 'do not reject H'. The power of a statistical hypothesis test measures the test's ability to reject the null hypothesis when it is actually false - that is, to make a correct decision.In other words, the power of a hypothesis test is the probability of not committing a .

This probability of a type I error can be precisely computed as,**P(type I error)** = significance level = The exact probability of a type II error is generally unknown.If we do not reject the null hypothesis, it may still be false (a type II error) as the sample may not be big enough to identify the falseness of the null hypothesis (especially if the truth is very close to hypothesis).For any given set of data, type I and type II errors are inversely related; the smaller the risk of one, the higher the risk of the other.A type I error can also be referred to as an error of the first kind. In a hypothesis test, a type II error occurs when the null hypothesis H, is not rejected when it is in fact false.

Presumably, we would want to test the null against the first alternative hypothesis since it would be useful to know if there is likely to be less than 50 matches, on average, in a box (no one would complain if they get the correct number of matches in a box or more).Yet another alternative hypothesis could be tested against the same null, leading this time to a two-sided test:That is, nothing specific can be said about the average number of matches in a box; only that, if we could reject the null hypothesis in our test, we would know that the average number of matches in a box is likely to be less than or greater than 50. A one sample t-test is a hypothesis test for answering questions about the mean where the data are a random sample of independent observations from an underlying normal distribution: The null hypothesis for the one sample t-test is:That is, the sample has been drawn from a population of given mean and unknown variance (which therefore has to be estimated from the sample).This null hypothesis, H is tested against one of the following alternative hypotheses, depending on the question posed: A two sample t-test is a hypothesis test for answering questions about the mean where the data are collected from two random samples of independent observations, each from an underlying normal distribution:When carrying out a two sample t-test, it is usual to assume that the variances for the two populations are equal, that is:The null hypothesis for the two sample t-test is:That is, the two samples have both been drawn from the same population.This null hypothesis is tested against one of the following alternative hypotheses, depending on the question posed.

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