What is the purpose of using inferential statistics in psychological research

As we have seen throughout this book, many studies in psychology focus on the difference between two means. The most common null hypothesis test for this type of statistical relationship is the t testA family of null hypothesis tests used to compare two means.. In this section, we look at three types of t tests that are used for slightly different research designs: the one-sample t test, the dependent-samples t test, and the independent-samples t test.

One-Sample t Test

The one-sample t testA null hypothesis test used to compare one sample mean with a hypothetical population mean that provides an interesting standard of comparison. is used to compare a sample mean (M) with a hypothetical population mean (μ0) that provides some interesting standard of comparison. The null hypothesis is that the mean for the population (µ) is equal to the hypothetical population mean: μ = μ0. The alternative hypothesis is that the mean for the population is different from the hypothetical population mean: μ ≠ μ0. To decide between these two hypotheses, we need to find the probability of obtaining the sample mean (or one more extreme) if the null hypothesis were true. But finding this p value requires first computing a test statistic called t. (A test statisticIn null hypothesis testing, a statistic such as t or F that is computed only to help find the p value for the sample result. is a statistic that is computed only to help find the p value.) The formula for t is as follows:

Again, M is the sample mean and µ0 is the hypothetical population mean of interest. SD is the sample standard deviation and N is the sample size.

The reason the t statistic (or any test statistic) is useful is that we know how it is distributed when the null hypothesis is true. As shown in , this distribution is unimodal and symmetrical, and it has a mean of 0. Its precise shape depends on a statistical concept called the degrees of freedom, which for a one-sample t test is N − 1. (There are 24 degrees of freedom for the distribution shown in .) The important point is that knowing this distribution makes it possible to find the p value for any t score. Consider, for example, a t score of +1.50 based on a sample of 25. The probability of a t score at least this extreme is given by the proportion of t scores in the distribution that are at least this extreme. For now, let us define extreme as being far from zero in either direction. Thus the p value is the proportion of t scores that are +1.50 or above or that are −1.50 or below—a value that turns out to be .14.

Figure 13.1 Distribution of t Scores (With 24 Degrees of Freedom) When the Null Hypothesis Is True

The red vertical lines represent the two-tailed critical values, and the green vertical lines the one-tailed critical values when α = .05.

Fortunately, we do not have to deal directly with the distribution of t scores. If we were to enter our sample data and hypothetical mean of interest into one of the online statistical tools in or into a program like SPSS (Excel does not have a one-sample t test function), the output would include both the t score and the p value. At this point, the rest of the procedure is simple. If p is less than .05, we reject the null hypothesis and conclude that the population mean differs from the hypothetical mean of interest. If p is greater than .05, we retain the null hypothesis and conclude that there is not enough evidence to say that the population mean differs from the hypothetical mean of interest. (Again, technically, we conclude only that we do not have enough evidence to conclude that it does differ.)

If we were to compute the t score by hand, we could use a table like to make the decision. This table does not provide actual p values. Instead, it provides the critical valuesIn null hypothesis testing, the value or values of a test statistic that correspond to a p value of .05 and therefore serve as a cutoff for deciding to reject the null hypothesis. of t for different degrees of freedom (df) when α is .05. For now, let us focus on the two-tailed critical values in the last column of the table. Each of these values should be interpreted as a pair of values: one positive and one negative. For example, the two-tailed critical values when there are 24 degrees of freedom are +2.064 and −2.064. These are represented by the red vertical lines in . The idea is that any t score below the lower critical value (the left-hand red line in ) is in the lowest 2.5% of the distribution, while any t score above the upper critical value (the right-hand red line) is in the highest 2.5% of the distribution. This means that any t score beyond the critical value in either direction is in the most extreme 5% of t scores when the null hypothesis is true and therefore has a p value less than .05. Thus if the t score we compute is beyond the critical value in either direction, then we reject the null hypothesis. If the t score we compute is between the upper and lower critical values, then we retain the null hypothesis.

Table 13.2 Table of Critical Values of t When α = .05

Critical valuedfOne-tailedTwo-tailed32.3533.18242.1322.77652.0152.57161.9432.44771.8952.36581.8602.30691.8332.262101.8122.228111.7962.201121.7822.179131.7712.160141.7612.145151.7532.131161.7462.120171.7402.110181.7342.101191.7292.093201.7252.086211.7212.080221.7172.074231.7142.069241.7112.064251.7082.060301.6972.042351.6902.030401.6842.021451.6792.014501.6762.009601.6712.000701.6671.994801.6641.990901.6621.9871001.6601.984

Thus far, we have considered what is called a two-tailed testA null hypothesis test (e.g., a t test or test of Pearson’s r) in which the null hypothesis is rejected if the sample result is extreme in either direction. Used when the researcher does not have a strong expectation about the direction of the relationship., where we reject the null hypothesis if the t score for the sample is extreme in either direction. This makes sense when we believe that the sample mean might differ from the hypothetical population mean but we do not have good reason to expect the difference to go in a particular direction. But it is also possible to do a one-tailed testA null hypothesis test (e.g., a t test or test of Pearson’s r) in which the null hypothesis is rejected only if the sample result is extreme in one direction specified before the data are collected. Used when the researcher has a strong expectation about the direction of the relationship., where we reject the null hypothesis only if the t score for the sample is extreme in one direction that we specify before collecting the data. This makes sense when we have good reason to expect the sample mean will differ from the hypothetical population mean in a particular direction.

Here is how it works. Each one-tailed critical value in can again be interpreted as a pair of values: one positive and one negative. A t score below the lower critical value is in the lowest 5% of the distribution, and a t score above the upper critical value is in the highest 5% of the distribution. For 24 degrees of freedom, these values are −1.711 and +1.711. (These are represented by the green vertical lines in .) However, for a one-tailed test, we must decide before collecting data whether we expect the sample mean to be lower than the hypothetical population mean, in which case we would use only the lower critical value, or we expect the sample mean to be greater than the hypothetical population mean, in which case we would use only the upper critical value. Notice that we still reject the null hypothesis when the t score for our sample is in the most extreme 5% of the t scores we would expect if the null hypothesis were true—so α remains at .05. We have simply redefined extreme to refer only to one tail of the distribution. The advantage of the one-tailed test is that critical values are less extreme. If the sample mean differs from the hypothetical population mean in the expected direction, then we have a better chance of rejecting the null hypothesis. The disadvantage is that if the sample mean differs from the hypothetical population mean in the unexpected direction, then there is no chance at all of rejecting the null hypothesis.

Example One-Sample t Test

Imagine that a health psychologist is interested in the accuracy of college students’ estimates of the number of calories in a chocolate chip cookie. He shows the cookie to a sample of 10 students and asks each one to estimate the number of calories in it. Because the actual number of calories in the cookie is 250, this is the hypothetical population mean of interest (µ0). The null hypothesis is that the mean estimate for the population (μ) is 250. Because he has no real sense of whether the students will underestimate or overestimate the number of calories, he decides to do a two-tailed test. Now imagine further that the participants’ actual estimates are as follows:

250, 280, 200, 150, 175, 200, 200, 220, 180, 250.

The mean estimate for the sample (M) is 212.00 calories and the standard deviation (SD) is 39.17. The health psychologist can now compute the t score for his sample:

If he enters the data into one of the online analysis tools or uses SPSS, it would also tell him that the two-tailed p value for this t score (with 10 − 1 = 9 degrees of freedom) is .013. Because this is less than .05, the health psychologist would reject the null hypothesis and conclude that college students tend to underestimate the number of calories in a chocolate chip cookie. If he computes the t score by hand, he could look at and see that the critical value of t for a two-tailed test with 9 degrees of freedom is ±2.262. The fact that his t score was more extreme than this critical value would tell him that his p value is less than .05 and that he should reject the null hypothesis.

Finally, if this researcher had gone into this study with good reason to expect that college students underestimate the number of calories, then he could have done a one-tailed test instead of a two-tailed test. The only thing this would change is the critical value, which would be −1.833. This slightly less extreme value would make it a bit easier to reject the null hypothesis. However, if it turned out that college students overestimate the number of calories—no matter how much they overestimate it—the researcher would not have been able to reject the null hypothesis.

What is the purpose of using inferential statistics in psychological research quizlet?

What is the main purpose of inferential statistics?

What are the inferential statistics used in psychological research?

What are the benefits of inferential statistics in psychology?