# What factors influence statistical power?

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espond to the following in a minimum of 175 words:

Read the following scenario and explain what power issues may arise. What factors influence statistical power?

A researcher is exploring differences between men and women on ‘number of different recreational drugs used.’ The researcher collects data on a sample of 50 men and 50 women between the ages of 18-25. Each participant is asked ‘how many different recreational drugs have you tried in your life?’ The IV is gender (male/female) and the DV is ‘number of reported drugs.’

Part2-PLEASE SEE ATTACHMENT

PART3-PLEASE SEE ATTACHMENT…THIS IS A GROUP ASSIGNMENT I ONLY HAVE TO COMPLETE A PART OF THE TABLE. I WILL POST MY PART ON TUESDAY

REFERENCE

CHAPTER 13

LEARNING OBJECTIVES

Explain how researchers use inferential statistics to evaluate sample data.

Distinguish between the null hypothesis and the research hypothesis.

Discuss probability in statistical inference, including the meaning of statistical significance.

Describe the t test and explain the difference between one-tailed and two-tailed tests.

Describe the F test, including systematic variance and error variance.

Describe what a confidence interval tells you about your data.

Distinguish between Type I and Type II errors.

Discuss the factors that influence the probability of a Type II error.

Discuss the reasons a researcher may obtain nonsignificant results.

Define power of a statistical test.

Describe the criteria for selecting an appropriate statistical test.

Page 267IN THE PREVIOUS CHAPTER, WE EXAMINED WAYS OF DESCRIBING THE RESULTS OF A STUDY USING DESCRIPTIVE STATISTICS AND A VARIETY OF GRAPHING TECHNIQUES. In addition to descriptive statistics, researchers use inferential statistics to draw more general conclusions about their data. In short, inferential statistics allow researchers to (a) assess just how confident they are that their results reflect what is true in the larger population and (b) assess the likelihood that their findings would still occur if their study was repeated over and over. In this chapter, we examine methods for doing so.

SAMPLES AND POPULATIONS

Inferential statistics are necessary because the results of a given study are based only on data obtained from a single sample of research participants. Researchers rarely, if ever, study entire populations; their findings are based on sample data. In addition to describing the sample data, we want to make statements about populations. Would the results hold up if the experiment were conducted repeatedly, each time with a new sample?

In the hypothetical experiment described in Chapter 12 (see Table 12.1), mean aggression scores were obtained in model and no-model conditions. These means are different: Children who observe an aggressive model subsequently behave more aggressively than children who do not see the model. Inferential statistics are used to determine whether the results match what would happen if we were to conduct the experiment again and again with multiple samples. In essence, we are asking whether we can infer that the difference in the sample means shown in Table 12.1 reflects a true difference in the population means.

Recall our discussion of this issue in Chapter 7 on the topic of survey data. A sample of people in your state might tell you that 57% prefer the Democratic candidate for an office and that 43% favor the Republican candidate. The report then says that these results are accurate to within 3 percentage points, with a 95% confidence level. This means that the researchers are very (95%) confident that, if they were able to study the entire population rather than a sample, the actual percentage who preferred the Democratic candidate would be between 60% and 54% and the percentage preferring the Republican would be between 46% and 40%. In this case, the researcher could predict with a great deal of certainty that the Democratic candidate will win because there is no overlap in the projected population values. Note, however, that even when we are very (in this case, 95%) sure, we still have a 5% chance of being wrong.

Inferential statistics allow us to arrive at such conclusions on the basis of sample data. In our study with the model and no-model conditions, are we confident that the means are sufficiently different to infer that the difference would be obtained in an entire population?

Page 268

INFERENTIAL STATISTICS

Much of the previous discussion of experimental design centered on the importance of ensuring that the groups are equivalent in every way except the independent variable manipulation. Equivalence of groups is achieved by experimentally controlling all other variables or by randomization. The assumption is that if the groups are equivalent, any differences in the dependent variable must be due to the effect of the independent variable.

This assumption is usually valid. However, it is also true that the difference between any two groups will almost never be zero. In other words, there will be some difference in the sample means, even when all of the principles of experimental design are rigorously followed. This happens because we are dealing with samples, rather than populations. Random or chance error will be responsible for some difference in the means, even if the independent variable had no effect on the dependent variable.

Therefore, the difference in the sample means does show any true difference in the population means (i.e., the effect of the independent variable) plus any random error. Inferential statistics allow researchers to make inferences about the true difference in the population on the basis of the sample data. Specifically, inferential statistics give the probability that the difference between means reflects random error rather than a real difference.

NULL AND RESEARCH HYPOTHESES

Statistical inference begins with a statement of the null hypothesis and a research (or alternative) hypothesis. The null hypothesis is simply that the population means are equal—the observed difference is due to random error. The research hypothesis is that the population means are, in fact, not equal. The null hypothesis states that the independent variable had no effect; the research hypothesis states that the independent variable did have an effect. In the aggression modeling experiment, the null and research hypotheses are:

H0 (null hypothesis): The population mean of the no-model group is equal to the population mean of the model group.

H1 (research hypothesis): The population mean of the no-model group is not equal to the population mean of the model group.

The logic of the null hypothesis is this: If we can determine that the null hypothesis is incorrect, then we accept the research hypothesis as correct. Acceptance of the research hypothesis means that the independent variable had an effect on the dependent variable.

The null hypothesis is used because it is a very precise statement—the population means are exactly equal. This permits us to know precisely the Page 269probability of obtaining our results if the null hypothesis is correct. Such precision is not possible with the research hypothesis, so we infer that the research hypothesis is correct only by rejecting the null hypothesis. We reject the null hypothesis when we find a very low probability that the obtained results could be due to random error. This is what is meant by statistical significance: A significant result is one that has a very low probability of occurring if the population means are equal. More simply, significance indicates that there is a low probability that the difference between the obtained sample means was due to random error. Significance, then, is a matter of probability.

PROBABILITY AND SAMPLING DISTRIBUTIONS

Probability is the likelihood of the occurrence of some event or outcome. We all use probabilities frequently in everyday life. For example, if you say that there is a high probability that you will get an A in this course, you mean that this outcome is likely to occur. Your probability statement is based on specific information, such as your grades on examinations. The weather forecaster says there is a 10% chance of rain today; this means that the likelihood of rain is very low. A gambler gauges the probability that a particular horse will win a race on the basis of the past records of that horse.

Probability in statistical inference is used in much the same way. We want to specify the probability that an event (in this case, a difference between means in the sample) will occur if there is no difference in the population. The question is: What is the probability of obtaining this result if only random error is operating? If this probability is very low, we reject the possibility that only random or chance error is responsible for the obtained difference in means.

Probability: The Case of ESP

The use of probability in statistical inference can be understood intuitively from a simple example. Suppose that a friend claims to have ESP (extrasensory perception) ability. You decide to test your friend with a set of five cards commonly used in ESP research; a different symbol is presented on each card. In the ESP test, you look at each card and think about the symbol, and your friend tells you which symbol you are thinking about. In your actual experiment, you have 10 trials; each of the five cards is presented two times in a random order. Your task is to know whether your friend’s answers reflect random error (guessing) or whether they indicate that something more than random error is occurring. The null hypothesis in your study is that only random error is operating. In this case, the research hypothesis is that the number of correct answers shows more than random or chance guessing. (Note, however, that accepting the research hypothesis could mean that your friend has ESP ability, but it could also mean that the cards were marked, that you had somehow cued your friend when thinking about the symbols, and so on.)

Page 270You can easily determine the number of correct answers to expect if the null hypothesis is correct. Just by guessing, 1 out of 5 answers (20%) should be correct. On 10 trials, 2 correct answers are expected under the null hypothesis. If, in the actual experiment, more (or less) than 2 correct answers are obtained, would you conclude that the obtained data reflect random error or something more than merely random guessing?

Suppose that your friend gets 3 correct. Then you would probably conclude that only guessing is involved, because you would recognize that there is a high probability that there would be 3 correct answers even though only 2 correct are expected under the null hypothesis. You expect that exactly 2 answers in 10 trials would be correct in the long run, if you conducted this experiment with this subject over and over again. However, small deviations away from the expected 2 are highly likely in a sample of 10 trials.

Suppose, though, that your friend gets 7 correct. You might conclude that the results indicate more than random error in this one sample of 10 observations. This conclusion would be based on your intuitive judgment that an outcome of 70% correct when only 20% is expected is very unlikely. At this point, you would decide to reject the null hypothesis and state that the result is significant. A significant result is one that is very unlikely if the null hypothesis is correct.

A key question then becomes: How unlikely does a result have to be before we decide it is significant? A decision rule is determined prior to collecting the data. The probability required for significance is called the alpha level. The most common alpha level probability used is .05. The outcome of the study is considered significant when there is a .05 or less probability of obtaining the results; that is, there are only 5 chances out of 100 that the results were due to random error in one sample from the population. If it is very unlikely that random error is responsible for the obtained results, the null hypothesis is rejected.

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