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Statistics for the Behavioral Sciences, 4/e

Michael Thorne, Mississippi State University -- Mississippi State
Martin Giesen, Mississippi State University -- Mississippi State

Confidence Intervals and Hypothesis Testing

Chapter Overview

The sampling distribution of means is derived by taking successive, same-sized random samples from some population, computing a mean for a measurable characteristic of each sample, and plotting the means on a frequency polygon. The first property of the sampling distribution is that its mean is the mean of the population or µ. The second property, which is a simplified version of the central limit theorem, is that the larger the size of each sample, the more nearly the sampling distribution will approximate the normal curve. The third property is that the larger the sample size, the smaller the standard deviation of the sampling distribution. The standard deviation is called the standard error of the mean.

When population parameters (µ and σ) are known, z scores are appropriate. However, we often do not know them, so they must be estimated from sample values. The sample mean (

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A confidence interval is a range of values within which the population mean almost certainly lies. The confidence intervals usually computed are the 95% and the 99% confidence intervals. Equations for the confidence intervals are derived in the chapter from the formula introduced in Chapter 6 to convert z scores to raw scores.

t scores are estimated z scores. They are used in place of z scores when population parameters are estimated from the sample. t scores correspond to the t distribution, and the t scores used in the confidence interval equations are determined from Table B (see Appendix 2), which contains values of t cutting off deviant portions of the distribution. In order to use the table of critical t scores, we need to know the df. For confidence intervals and the one-sample t test, df = N - 1. A confidence interval is an interval estimate of the population mean.

Another important use of the distribution of t is to test hypotheses about the value of µ. The seven-step procedure introduced for testing the null hypothesis is as follows:

State the null hypothesis in symbols (H₀: µ = µ₀) and in words in the context of the problem.
State the alternative hypothesis in symbols (e.g., H₁: µ ≠ µ₀) and in words.
Choose an a level, the level at which you will reject or fail to reject the null hypothesis. Set a = .05, if there are no specific instructions in the problem.
State the rejection rule. For example, the rule for a particular problem may be as follows: If |t_comp| is ≥ t_crit, then reject the null hypothesis. This is the rejection rule for a nondirectional hypothesis.
Compute the test statistic. In this chapter, the equation for the test statistic is
Make a decision by applying the rejection rule.
Write a conclusion statement in the context of the problem.

For a directional test, H₀ is rejected if t_comp is of the same sign but is more extreme than t_crit. A Type I, or a, error (a "false claim") is defined as rejecting the null hypothesis when it is really true. The probability of an a error is equal to a, the level at which we are trying to reject the null hypothesis. Failing to reject H₀ when it is false is called a Type II, or β error (a "failure of detection"). Lowering the value of a increases the probability of a β error. The power of a statistical test is the probability that the test will detect a false null hypothesis. Factors affecting the power of a test are

The size of a. The smaller the a level, the less powerful the test will be.
The sample size. The larger the sample size, the greater the power of the test will be.
The distance between the hypothesized mean and the true mean. The greater the distance, the greater the power of the test will be.

In analyzing the results from large numbers of research studies, meta-analysis uses quantitative procedures to integrate the findings. Meta-analysis uses the effect size - the size of the difference between the null hypothesis and the alternative hypothesis in standardized units - results of studies rather than simply reporting whether or not the results were statistically significant.

Some researchers argue that hypothesis testing should be abandoned, claiming that the procedures are misleading and we should instead report confidence intervals and effect sizes. People opposed to hypothesis testing hold that many research studies don't have enough power to find what they are looking for, with a corresponding increase in Type II errors. This point is valuable if it forces experimenters to be more attentive to having sufficient power to detect an effect if it is present.

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