10 Statistical Inference and Hypothesis Testing
I Nature and Purpose of Statistical Inference
A Differences between Deductive and Inductive Reasoning
If both propositions are true, then the following deduction must be true:
Deductive reasoning is of special use in science after hypotheses are formed. Using deductive reasoning, an investigator can say, “If the following hypothesis is true, then the following prediction or predictions also should be true.” If a prediction can be tested empirically, the hypothesis may be rejected or not rejected on the basis of the findings. If the data are inconsistent with the predictions from the hypothesis, the hypothesis must be rejected or modified. Even if the data are consistent with the hypothesis, however, they cannot prove that the hypothesis is true, as shown in Chapter 4 (see Fig. 4-2).
II Process of Testing Hypotheses
A False-Positive and False-Negative Errors
Science is based on the following set of principles:
Previous experience serves as the basis for developing hypotheses.
Hypotheses serve as the basis for developing predictions.
Predictions must be subjected to experimental or observational testing.
If the predictions are consistent with the data, they are retained, but if they are inconsistent with the data, they are rejected or modified.
4 Compare p Value Obtained with Alpha
After the p value is obtained, it is compared with the alpha level previously chosen.
B Variation in Individual Observations and in Multiple Samples
1 Standard Deviation and Standard Error
Chapter 9 focused on individual observations and the extent to which they differed from the mean. One assertion was that a normal (gaussian) distribution could be completely described by its mean and standard deviation. Figure 9-6 showed that, for a truly normal distribution, 68% of observations fall within the range described as the mean ± 1 standard deviation, 95.4% fall within the range of the mean ± 2 standard deviations, and 95% fall within the range of the mean ± 1.96 standard deviations. This information is useful in describing individual observations (raw data), but it is not directly useful when comparing means or proportions.
2 Confidence Intervals
The SD shows the variability of individual observations, whereas the SE shows the variability of means. The mean ± 1.96 SD estimates the range in which 95% of individual observations would be expected to fall, whereas the mean ± 1.96 SE estimates the range in which 95% of the means of repeated samples of the same size would be expected to fall. If the value for the mean ± 1.96 SE is known, it can be used to calculate the 95% confidence interval, which is the range of values in which the investigator can be 95% confident that the true mean of the underlying population falls. Other confidence intervals, such as the 99% confidence interval, also can be determined easily. Box 10-2 shows the calculation of the SE and the 95% confidence interval for the systolic blood pressure data in Table 10-1.
