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Example 11.2. Hypotheses with One Sample of One Categorical Variable

One of the main goals of statistical hypothesis testing is to estimate the P value, which is the probability of obtaining the observed results, or something more extreme, if the null hypothesis were true. If the observed results are unlikely under the null hypothesis, your reject the null hypothesis. Alternatives to this "frequentist" approach to statistics include Bayesian statistics and estimation of effect sizes and confidence intervals.

State Null and Alternative Hypotheses

(Hint: You should reject the null hypothesis when the percentage inthe box in the center of the distribution is less than 2.5%.)The above exercise is very similar to how a researcher uses anindependent samples t-test.

Example 11.3. Hypotheses with One Sample of One Measurement Variable

State Null and Alternative Hypotheses

The alternative hypothesis is that things are different from each other, or different from a theoretical expectation. For example, one alternative hypothesis would be that male chickens have a different average foot size than female chickens; another would be that the sex ratio is different from 1:1.

You'll analyze similar experiments, with similar null and alternative hypotheses, completely differently depending on which of these three variable types are involved. For example, let's say you've measured variable X in a sample of 56 male and 67 female isopods (Armadillidium vulgare, commonly known as pillbugs or roly-polies), and your null hypothesis is "Male and female A. vulgare have the same values of variable X." If variable X is width of the head in millimeters, it's a measurement variable, and you'd compare head width in males and females with a or a If variable X is a genotype (such as AA, Aa, or aa), it's a nominal variable, and you'd compare the genotype frequencies in males and females with a . If you shake the isopods until they roll up into little balls, then record which is the first isopod to unroll, the second to unroll, etc., it's a ranked variable and you'd compare unrolling time in males and females with a

Example 11.4. Hypotheses with Two Samples of One Categorical Variable

State Null and Alternative Hypotheses

Now instead of testing 1000 plant extracts, imagine that you are testing just one. If you are testing it to see if it kills beetle larvae, you know (based on everything you know about plant and beetle biology) there's a pretty good chance it will work, so you can be pretty sure that a P value less than 0.05 is a true positive. But if you are testing that one plant extract to see if it grows hair, which you know is very unlikely (based on everything you know about plants and hair), a P value less than 0.05 is almost certainly a false positive. In other words, if you expect that the null hypothesis is probably true, a statistically significant result is probably a false positive. This is sad; the most exciting, amazing, unexpected results in your experiments are probably just your data trying to make you jump to ridiculous conclusions. You should require a much lower P value to reject a null hypothesis that you think is probably true.

The other factor affecting the critical t-value(s) is whether thealternative hypothesis is one- or two-tailed (see Hypothesis Testingtutorial for review of one-tailed and two-tailed hypotheses).

Example 11.5. Hypotheses with Two Samples of One Measurement Variable
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Null hypothesis: μ = 72 Alternative hypothesis: μ ≠72

Further, the standard deviation (i.e., the standard error) of thesampling distribution of the difference between the means must beestimated because the decision to reject the null hypothesis is basedon how many standard error units the observed difference in the samplemeans is from zero.

State Null and Alternative Hypotheses

The alternative hypothesis predicts that not only do the two groupsdiffer on the independent variable, they also should be treated asdifferent populations with respect to the variable of interest.

State Null and Alternative Hypotheses

The null hypothesis predicts that two groups who differ on theindependent variable can nonetheless be treated as the same populationwhen it comes to the variable of interest (in our example, thevariable of interest is satisfaction with dorm living).

State Null and Alternative Hypotheses

The alternative hypothesis states that the two samples come from differentpopulations for the variable of interest and can be stated as1 - 2 is not equal to zero(for the two-tailed test) or 1- 2 is greater than zero(for the one-tailed test) or 1- 2 is less than zero (for the one-tailed test in the other direction).

State Null and Alternative Hypotheses

As you'll see in the descriptions of particular statistical tests, sometimes it is important to decide which is the independent and which is the dependent variable; it will determine whether you should analyze your data with a or , for example. Other times you don't need to decide whether a variable is independent or dependent. For example, if you measure the nitrogen content of soil and the density of dandelion plants, you might think that nitrogen content is an independent variable and dandelion density is a dependent variable; you'd be thinking that nitrogen content might affect where dandelion plants live. But maybe dandelions use a lot of nitrogen from the soil, so it's dandelion density that should be the independent variable. Or maybe some third variable that you didn't measure, such as moisture content, affects both nitrogen content and dandelion density. For your initial experiment, which you would analyze using , you wouldn't need to classify nitrogen content or dandelion density as independent or dependent. If you found an association between the two variables, you would probably want to follow up with experiments in which you manipulated nitrogen content (making it an independent variable) and observed dandelion density (making it a dependent variable), and other experiments in which you manipulated dandelion density (making it an independent variable) and observed the change in nitrogen content (making it the dependent variable).

I’m stuck on how to value the null or alternative hypotheses

Another way to classify variables is as independent or dependent variables. An independent variable (also known as a predictor, explanatory, or exposure variable) is a variable that you think may cause a change in a dependent variable (also known as an outcome or response variable). For example, if you grow isopods with 10 different mannose concentrations in their food and measure their growth rate, the mannose concentration is an independent variable and the growth rate is a dependent variable, because you think that different mannose concentrations may cause different growth rates. Any of the three variable types (measurement, nominal or ranked) can be either independent or dependent. For example, if you want to know whether sex affects body temperature in mice, sex would be an independent variable and temperature would be a dependent variable. If you wanted to know whether the incubation temperature of eggs affects sex in turtles, temperature would be the independent variable and sex would be the dependent variable.

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