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Null and Alternative Hypotheses for a Mean

Why the Null Hypothesis (H)?

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

The null hypothesis in ANOVA is always that there is no difference in means. The research or alternative hypothesis is always that the means are not all equal and is usually written in words rather than in mathematical symbols. The research hypothesis captures any difference in means and includes, for example, the situation where all four means are unequal, where one is different from the other three, where two are different, and so on. The alternative hypothesis, as shown above, capture all possible situations other than equality of all means specified in the null hypothesis.

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

Suppose, for instance, that he/she formulates the following hypothesis: "Students taught by Method A will be better readers." How can we really 'test' this hypothesis if we don't know for sure what 'better' means here?!

How to Set Up a Hypothesis Test: Null versus Alternative

This leads us into the concept of "null" hypothesis!

The p-value= .004 indicates that we should decide in favor of the alternative hypothesis. Thus we decide that less than 40% of college women think they are overweight.

Step 4: We compare the p-value to alpha, which we will let alpha be 0.05. Since 0.0044 is less than 0.05 we will reject the null hypothesis and decide in favor of the alternative, Ha.

Null and Alternative Hypotheses for a Mean

The method for finding the p-value is based on the alternative hypothesis:

The null hypothesis is that any change in mean levels of pain from time 1 to time 2 is simply random (explained by chance error) and the true score does not vary from time 1 to time 2.

This is called the null hypothesis. Null meaning nothing. And the hypothesis is that nothing is there in our data, no differences from what we expect except chance variation or chance error.

Null hypothesis: μ = 72  Alternative hypothesis: μ ≠72
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  • the null hypothesis is rejected when it is true b.

    The two competing statements about a population are called the null hypothesis and the alternative hypothesis.

  • the null hypothesis is not rejected when it is false c.

    In general, the smaller the p-value the stronger the evidence is in favor of the alternative hypothesis.

  • the null hypothesis is probably wrong b.

    Power = the probability of correctly rejecting a false null hypothesis = 1 - .

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the result would be unexpected if the null hypothesis were true c.

How do you know which hypothesis to put in H0 and which one to put in Ha? Typically, the null hypothesis says that nothing new is happening; the previous result is the same now as it was before, or the groups have the same average (their difference is equal to zero). In general, you assume that people’s claims are true until proven otherwise. So the question becomes: Can you prove otherwise? In other words, can you show sufficient evidence to reject H0?

the null hypothesis is probably true d.

A related criticism is that a significant rejection of a null hypothesis might not be biologically meaningful, if the difference is too small to matter. For example, in the chicken-sex experiment, having a treatment that produced 49.9% male chicks might be significantly different from 50%, but it wouldn't be enough to make farmers want to buy your treatment. These critics say you should estimate the effect size and put a on it, not estimate a P value. So the goal of your chicken-sex experiment should not be to say "Chocolate gives a proportion of males that is significantly less than 50% (P=0.015)" but to say "Chocolate produced 36.1% males with a 95% confidence interval of 25.9 to 47.4%." For the chicken-feet experiment, you would say something like "The difference between males and females in mean foot size is 2.45 mm, with a confidence interval on the difference of ±1.98 mm."

One can never prove the truth of a statistical (null) hypothesis.

Finally, say you work for the company marketing the pie, and you think the pie can be made in less than five minutes (and could be marketed by the company as such). The less-than alternative is the one you want, and your two hypotheses would be

failing to reject the null hypothesis when it is false.

This criticism only applies to two-tailed tests, where the null hypothesis is "Things are exactly the same" and the alternative is "Things are different." Presumably these critics think it would be okay to do a one-tailed test with a null hypothesis like "Foot length of male chickens is the same as, or less than, that of females," because the null hypothesis that male chickens have smaller feet than females could be true. So if you're worried about this issue, you could think of a two-tailed test, where the null hypothesis is that things are the same, as shorthand for doing two one-tailed tests. A significant rejection of the null hypothesis in a two-tailed test would then be the equivalent of rejecting one of the two one-tailed null hypotheses.

failing to reject the null hypothesis when it is true.

where k is the number of comparison groups and N is the total number of observations in the analysis. If the null hypothesis is true, the between treatment variation (numerator) will not exceed the residual or error variation (denominator) and the F statistic will small. If the null hypothesis is false, then the F statistic will be large. The rejection region for the F test is always in the upper (right-hand) tail of the distribution as shown below.

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