# Null and Alternative Hypotheses for a Mean

In the χ^{2} goodness-of-fit test, we conclude that either the distribution specified in H_{0} is false (when we reject H_{0}) or that we do not have sufficient evidence to show that the distribution specified in H_{0} is false (when we fail to reject H_{0}). Here, we reject H_{0} and concluded that the distribution of responses to the exercise question following the implementation of the health promotion campaign was not the same as the distribution prior. The test itself does not provide details of how the distribution has shifted. A comparison of the observed and expected frequencies will provide some insight into the shift (when the null hypothesis is rejected). Does it appear that the health promotion campaign was effective?

## We then have three null hypotheses (and three alternatives).

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

The p-value is p = 0.019. This is below the .05 standard, so the result is statistically significant. This means we decide in favor of the alternative hypothesis. We're deciding that the population mean is not 72.

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

We must first assess whether the sample size is adequate. Specifically, we need to check min(np_{0}, np_{1,} ..., n p_{k}) __>__ 5. The sample size here is n=470 and the proportions specified in the null hypothesis are 0.60, 0.25 and 0.15. Thus, min( 470(0.65), 470(0.25), 470(0.15))=min(282, 117.5, 70.5)=70.5. The sample size is more than adequate so the formula can be used.

## T-test | Stata Annotated Output

Stata calculates the difference () as , or proportion of non-religious people who answered true minus proportion of religious people who answered true. Thus the hypothesis that religious people are less likely to answer true is and the very low p-value associated with it suggests we should reject the null and accept that alternative hypothesis.

## Stata | FAQ: One-sided tests for coefficients

Alternatively, a two-tailed prediction means that we do not make a choice over the direction that the effect of the experiment takes. Rather, it simply implies that the effect could be negative or positive. If Sarah had made a two-tailed prediction, the alternative hypothesis might have been: