It is simple, straightforward, easy to use, and adaptable to a broad range of situations. Its utility is occasioned by the fact that scientific research very often examines the phenomena of nature two variables at a time, with an eye toward answering the basic question: Are these two variables related? If we examine two different levels of one variable, will we find them to be associated with different levels of the other?

The power is about 0. Significance Tests for Unknown Mean and Unknown Standard Deviation In most practical research, the standard deviation for the population of interest is not known.

In this case, the standard deviation is replaced by the estimated standard deviation salso known as the standard error. Since the standard error is an estimate for the true value of the standard deviation, the distribution of the sample mean is no longer normal with mean and standard deviation.

Instead, the sample mean follows the t distribution with mean and standard deviation. The t distribution is also described by its degrees of freedom. For a sample of size n, the t distribution will have n-1 degrees of freedom.

The notation for a t distribution with k degrees of freedom is t k. As the sample size n increases, the t distribution becomes closer to the normal distribution, since the standard error approaches the true standard deviation for large n. The test statistic follows the t distribution with n-1 degrees of freedom.

The test statistic z is used to compute the P-value for the t distribution, the probability that a value at least as extreme as the test statistic would be observed under the null hypothesis. Example The dataset "Normal Body Temperature, Gender, and Heart Rate" contains observations of body temperature, along with the gender of each individual and his or her heart rate.

The t test statistic is equal to This result is significant at the 0. Data presented in Mackowiak, P. Matched Pairs In many experiments, one wishes to compare measurements from two populations. This is common in medical studies involving control groups, for example, as well as in studies requiring before-and-after measurements.

Such studies have a matched pairs design, where the difference between the two measurements in each pair is the parameter of interest. Analysis of data from a matched pairs experiment compares the two measurements by subtracting one from the other and basing test hypotheses upon the differences.

Usually, the null hypothesis H0 assumes that that the mean of these differences is equal to 0, while the alternative hypothesis Ha claims that the mean of the differences is not equal to zero the alternative hypothesis may be one- or two-sided, depending on the experiment.

Using the differences between the paired measurements as single observations, the standard t procedures with n-1 degrees of freedom are followed as above.

Example In the "Helium Football" experiment, a punter was given two footballs to kick, one filled with air and the other filled with helium. The punter was unaware of the difference between the balls, and was asked to kick each ball 39 times.

The balls were alternated for each kick, so each of the 39 trials contains one measurement for the air-filled ball and one measurement for the helium-filled ball.

Given that the conditions leg fatigue, etc. The Sign Test Another method of analysis for matched pairs data is a distribution-free test known as the sign test.See Chapter r-bridal.com for additional guidance on the mentoring and data management plan requirements for collaborative proposals.

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