Because this is a two-sided alternative hypothesis, the p-value is the combined area to the right of 2.47 and the left of −2.47 in a t-distribution with 35 – 1 = 34 degrees of freedom.
The p-value is p = 0.236. This is not below the .05 standard, so we do not reject the null hypothesis. Thus it is possible that the true value of the population mean is 72. The 95% confidence interval suggests the mean could be anywhere between 67.78 and 73.06.
6th gradersWhen the alternative hypothesis simply states that there is a differencebetween groups, like the alternative hypothesis in this example, it iscalled a nondirectional alternative hypothesis, and a two-tailed significancetest (with critical values in both tails of the sampling distribution)is used.
This is known as a one-tailed hypothesis test because only scoresin one tail of the sampling distribution will lead the researcher to concludesupport for his/her directional prediction.
ANOVA is a test that provides a global assessment of a statistical difference in more than two independent means. In this example, we find that there is a statistically significant difference in mean weight loss among the four diets considered. In addition to reporting the results of the statistical test of hypothesis (i.e., that there is a statistically significant difference in mean weight losses at α=0.05), investigators should also report the observed sample means to facilitate interpretation of the results. In this example, participants in the low calorie diet lost an average of 6.6 pounds over 8 weeks, as compared to 3.0 and 3.4 pounds in the low fat and low carbohydrate groups, respectively. Participants in the control group lost an average of 1.2 pounds which could be called the placebo effect because these participants were not participating in an active arm of the trial specifically targeted for weight loss. Are the observed weight losses clinically meaningful?
However, it is not possible to come up with an example for each cell of these matrices because it is not possible to propose a non-directional and causal hypothesis.
The alternative hypothesis can bedirectional or non-directional.“Eating oatmeal lowers cholesterol” is a directional hypothesis; “Amountof sleep affects test scores” is non-directional.
In general, a p-value is the probability that the test statistic would "lean" as much (or more) toward the alternative hypothesis as it does if the real truth is the null hypothesis.
When testing hypotheses about a mean or mean difference, a t-distribution is used to find the p-value. This is a close cousin to the normal curve. T-Distributions are indexed by a quantity called degrees of freedom, calculated as df = n – 1 for the situation involving a test of one mean or test of mean difference.
This is why, for example, we can be more confident of research results that are consistent with a causal-directional hypothesis, than is the case of findings that are consistent with a non-directional hypothesis.
Now that we have reviewed the critical value and P-value approach procedures for each of three possible hypotheses, let's look at three new examples — one of a right-tailed test, one of a left-tailed test, and one of a two-tailed test.
Notice that the top part of the statistic is the difference between the sample mean and the null hypothesis. The bottom part of the calculation is the standard error of the mean.
The good news is that, whenever possible, we will take advantage of the test statistics and P-values reported in statistical software, such as Minitab, to conduct our hypothesis tests in this course.
where the observed sample mean difference, μ0 = value specified in null hypothesis, sd = standard deviation of the differences in the sample measurements and n = sample size. For instance, if we wanted to test for a difference in mean SAT Math and mean SAT Verbal scores, we would random sample subjects, record their SATM and SATV scores in two separate columns, then create a third column that contained the differences between these scores. Then the sample mean and sample standard deviation would be those that were calculated on this column of differences.
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.