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.
Example 4: Suppose your alternative hypothesis H1 is that a newheadache remedy PainX helps a greater proportion of people thanaspirin.
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Can you do a hypothesis test to show that more thanhalf of Lansing households in the proposed district were against thesewer project? (You’re trying to show a majority against, socombine “supporting” and “neutral” since those arenot against.)
Hypothesis testing is the subject of this chapter.
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.
In this example, the significance (p value) of Levene's test is .383.
You can think of a test statistic as a measure ofunbelievability, of disagreement between H0 and your sample. A sample hardly ever matches your null hypothesis perfectly, but thecloser the test statistic is to zero the better the agreement, and thefurther the test statistic is from 0 the worse the sample and the nullhypothesis disagree with each other.
The null hypothesis usually is a statement
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.