## DNP 830 Discussion Basic Assumptions of T-Test Normality and Sample Size

*DNP 830 Discussion Basic Assumptions of T-Test Normality and Sample Size*

What are the assumptions for a t-test? How should these be run? Do you want to see a p value of <0.05? Why or why not?

Science is based on probability. It is impossible to say with certainty whether the events observed in nature will occur or not. The same rule applies when testing hypotheses through research. Null hypotheses are basically assumed to be true. Fig. 1 presents two normal distribution probability density curves under the respective assumptions that the null hypothesis is true (left) or false (alternative hypothesis is true, right). They show the typical curves that approach but never reach the x-axis on both ends. Whether the assumption is true or not, the probability never becomes zero; this means that any result obtained from research always has a possibility of unreliability. It is always possible for conclusions from research to be wrong, and an appropriate hypothesis is required to conduct research as well as to reduce the risk of false conclusions.

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Fig. 1.

Concept of hypothesis testing in independent t-test. H0: null hypothesis, H1: alternative hypothesis, μ1 and μ2: mean values of two groups.

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If the null hypothesis is concluded to be true when the value is less than a specific point on the x-axis, and false when the value is greater, then the point is called the critical value. The probability curve of the null hypothesis partially exists on the right side of the critical value. This means that the null hypothesis is true, but as it exceeds the critical value, it is mistakenly thought to be false; this is called a Type I error. In contrast, even if the null hypothesis is false (orange probability distribution curve), because it exceeds the critical value, it is mistakenly accepted to be true; this is called a Type II error. The size of the error can be represented with a probability, which is the area under the curve lying outside the critical value. The probability of a Type I error is called the α or level of significance. The probability of a Type II error is called β. Subtracting α from the probability of the null hypothesis becomes the confidence interval, and subtracting β from the probability of the alternative hypothesis becomes the power. Under the assumption that the null hypothesis is true, the P value—the probability of observing the test statistic of the data—can be obtained. In order to determine whether the hypothesis is true or false, it is necessary to confirm whether the observed event is statistically likely to occur under the assumption that the hypothesis is true. The P value is compared to a preset standard that determines the null hypothesis as false, which is the α. The α is the standard which has been agreed upon by researchers seeking the unknown truth, but the truth cannot be certain even if the P value is less than the α. The conclusion drawn from data analysis may not be the truth, and this is the error previously mentioned. Type I and Type II errors occur once we set a critical value; they show a pattern of trade-off between α and β, the probabilities of error. As can be seen in Fig. 1, the power means the probability of rejecting the null hypothesis when the alternative hypothesis is true; the power increases as the sample size increases.

The same rule applies to the normality test. The conditions required to conduct a t-test include the measured values in ratio scale or interval scale, simple random extraction, homogeneity of variance, appropriate sample size, and normal distribution of data. The normality assumption means that the collected data follows a normal distribution, which is essential for parametric assumption. Most statistical programs basically support the normality test, but the results only include P values and not the power of the normality test. Is it possible to conclude that the data follows a normal distribution if the P value is greater than or equal to α in the normality test? This article starts from this question and discusses the relationships between the sample size, normality, and power.