BHS 204-01 Methods in Behavioral Sciences I April 23, 2003 Chapter 5 (Ray) Experimental Decision-Making

Inferential Statistics We use information about groups (samples) to make inferences about populations. Population – all possible cases. Sample – a subset of cases drawn as randomly as possible from the larger population. Probability tells us whether the sample is representative of the larger population. If the experiment were repeated would we get the same result?

Examples With Coins Observed coin tosses are a sample of all possible coin tosses. Probability rules: P(A & B) = P(A) x P(B) P(A or B) = P(A) + P(B) Figure it out from the rules – or figure it out by listing all possible events and seeing how many produce the expected outcome.

Probabilities can be Observed Probabilities depend on the frequencies of events possible within a population: Probability of selecting a male subject from a group is % of males within that group. Probability of observing H on a coin toss is the frequency of H among the coins sides (H & T). Probabilities can be combined: Probability of drawing someone male or under 20 is 20/40 + 15/40 – (overlap) .375 = .825

Normal Distribution The normal distribution consists of the frequencies likely to be observed with repeated sampling from a population. It is useful because we can use it to know what percentage of cases are likely to fall within different portions of the curve. Central Limit Theorem – most of variability will fall within 2 standard deviations of the mean.

Figure 5.3. (p. 116) Probability distribution for combinations resulting from tossing 10 coins.

Figure 5.5. (p. 117) Normal distribution showing standard deviations.

Standard Error of the Mean With a normal distribution, the means of most sampled groups are likely to fall within a small range of the observed sample mean. Standard error of the mean – the standard deviation of all means possible to be observed with repeated sampling. Divide standard deviation by square root of the number of scores (cases). Standard error defines a confidence interval.

Figure 5.7. (p. 119) Hypothetical distribution of means derived from giving a standardized test to either a large or a small number of students a large number of times.

Hypothesis Testing We wish to know where the observed mean of a sample falls within the distribution of means from all possible samples. With two scores, we want to know whether the difference between them is due to sampling variation or the manipulation. T-test – a widely used statistic for testing differences between means of two groups.

Population vs Sample Statistics Population statistics: Mean m Variance s2 Standard deviation s Sample statistics: Mean M or X Variance S2 Standard deviation S or SD

Adjusting for the Population When inferential statistics are used, an adjustment is made to allow the sample mean to more closely approximate the population mean. Population variance is calculated by dividing by N-1, not N. Other statistics show this same adjustment – t-test uses N-1 not N in denominator.

Degrees of Freedom Degrees of freedom (df) – how many numbers can vary and still produce the observed result. Population statistics include the degrees of freedom. Calculated differently depending upon the experimental design – based on the number of groups. T-Test df = (Ngroup1 -1) + (Ngroup2 -1)

Reporting T-Test Results Include a sentence that gives the direction of the result, the means, and the t-test results: Example: The experimental group showed significantly greater weight gain (M = 55) compared to the control group (M = 21), t(12) = 3.97, p=.0019, two-tailed. Give the exact probability of the t value. Underline all statistics.

When to Use a T-Test When two independent groups are compared. When sample sizes are small (N< 30). When the actual population distribution is unknown (not known to be normal). When the variances within the two groups are unequal. When sample sizes are unequal.

Using Error Bars in Graphs Error bars show the standard error of the mean for the observed results. To visually assess statistical significance, see whether: The mean (center point of error bar) for one group falls outside the error bars for the other group. Also compare how large the error bars are for the two groups.

Figure 5.8. (p. 124) Graphic illustration of cereal experiment.