Chapter Nine Probability the Normal Curve and Sampling

The Nature and Logic of Probability l Probability – is used to help determine the likelihood that a given outcome from out experiment was probable or not. Anytime we use a sample and not a population, we have to use statistics and probability to draw conclusions. – Subjective Probability – is used in estimations of probability, not using numbers. – Odds – when you hear about the odds of something taking place using numbers; that is a type of probability. – Percentages – are a common way to use numbers to denote probability. – Proportions – also use numbers to indicate probability. Smith/Davis (c) 2005 Prentice Hall

The Nature and Logic of Probability l l The reason probability is so important for researchers is that inferential statistics tests allow us to determine the probability that our results came about by chance. The basic premise behind inferential statistics is that we assume there is no difference between the groups in our experiment (a. k. a. , the null hypothesis). Smith/Davis (c) 2005 Prentice Hall

The Nature and Logic of Probability l With two groups the null hypothesis looks like this: M 1 = M 2 – l In other words, the mean of group 1 is equal to the mean of group 2. The second hypothesis we test in inferential statistics is the alternative (or experimental) hypothesis. Written as: M 1 M 2 – The alternative hypothesis is typically the one the experimenter hopes to support in the experiment. To avoid bias, we start off assuming the null hypothesis and let the data and statistical test demonstrate otherwise. Smith/Davis (c) 2005 Prentice Hall

The Nature and Logic of Probability l l l If the differences between the groups is small or zero, our inferential statistic would also be small. Inferential statistics with small values occur frequently by chance. If it occurs by chance, then we say that the difference between our groups is not significant and conclude that our IV did not affect the DV, and thus accept the null hypothesis. If the difference between the groups is large, our inferential statistic would be large. Inferential statistics with large values occur rarely by chance, and thus, we would say that the difference between our group sis significant and conclude that some factor other than chance (our IV) is at work (affecting the DV). Smith/Davis (c) 2005 Prentice Hall

The Nature and Logic of Probability l What is considered chance in psychology? – – – Typically psychologists say that any event that occurs by chance 5 times or fewer in 100 occasions is a rare event. Thus, the common phrase mentioned in publications is “. 05 level of significance. ” Meaning, that a result is considered significant if it would occur 5 or fewer times by chance in 100 replications of the experiment when the null hypothesis is actually true. Smith/Davis (c) 2005 Prentice Hall

A Conceptual Statistical Note l l Although with our hypotheses we discuss comparisons of sample means, researchers and statisticians want to compare the two population means represented by the two groups so they can draw conclusions about the effects of the IV in the populations of interest. Thus, the conceptual null hypothesis is μ 1 = μ 2, and the conceptual alternative hypothesis is μ 1 μ 2. Because we rarely can test populations, we resort to sampling from those populations and end up comparing sample means. Although we use sample means in our formulas, the statistical null and alternative hypotheses use Smith/Davis (c) 2005 Prentice Hall population means.

Probability and the Normal Curve l l l The shape and spread of the normal distribution never changes, so the probability associated with a z score of a certain size will never change. Thus, to use the normal distribution to find probabilities, you can use the z score formula: This concept is true for statistical tests in general. We use probabilities from statistical tests because they are objective, which allows us to avoid making subjective guesses about the outcomes of our research. Smith/Davis (c) 2005 Prentice Hall

Probability and Decisions l The process of using probability to make decisions concerning experimental findings is fairly straightforward: – – We use a statistical test to find the probability of a given event occurring by chance. We then compare that probability to the critical probability for making our decisions – the. 05 level of significance. If the probability of the statistic occurring by chance is above. 05, then we decide that chance is still a possible explanation for the finding, and thus are uncertain about the outcome and conclude that is possible that the finding is due to chance (i. e. , accept the null hypothesis). If the probability of our statistic test is less than. 05, we believe that chance is not a likely explanation for the finding. Thus, the null hypothesis is rejected and the alternative hypothesis is accepted. Smith/Davis (c) 2005 Prentice Hall

Comparing a Sample to a Population: The One-Sample t Test l l The One-Sample t test – in some instances we would like to compare a sample mean to a mean of a population. To do such a problem, rather than using the normal distribution as we did for the z scores, we need to use a new statistical distribution: the t distribution. The t distribution is similar to the normal shape, but the t distribution is used for smaller samples than we usually would have for the normal distribution. The t distribution is flatter in the middle and higher on the tails compared to the normal distribution. Smith/Davis (c) 2005 Prentice Hall

Comparing a Sample to a Population: The One-Sample t Test l l Being higher on the tails is a concern because the critical region for a statistical test (the. 05 level) is located in the tails of the curves. Using a normal distribution for small samples can give us misleading probabilities and, therefore, result in misleading conclusions. Smith/Davis (c) 2005 Prentice Hall

Comparing a Sample to a Population: The One-Sample t Test l l l Another characteristic of the t distribution that is different from the normal distribution is that the t distribution does not retain its shape; it changes shape based on the size of the sample. The t distribution changes shape based on its number of degrees of freedom (i. e. , the ability of a number in a given set to assume any value). This ability is influence by the restrictions imposed on the set of numbers. For every restriction, one number is determine and must assume a fixed or specified value. Smith/Davis (c) 2005 Prentice Hall

Comparing a Sample to a Population: The One-Sample t Test l l When examining a critical value table, you will see that the values in the df column are continuous until 30, at which point they jump by larger and larger increments. Should your value for df not appear in the table, you should bias the test against yourself. – – – That is, never give yourself degrees of freedom that you don’t actually have. This decreases the chance of you making a statistical error. You will also notice in the critical value table, that there are more probabilities listed than just the. 05 level. Having additional levels in the table allows us to get a better idea of the probability of chance of our results. Smith/Davis (c) 2005 Prentice Hall

Comparing a Sample to a Population: The One-Sample t Test l The statistical formula for the one-sample t test is: l Marginal significance is often labeled by the area of probability between. 05 and. 10. The conclusion is that your results were almost significant, but not quite. l – Typically, researchers will discuss results that are marginally significant in their articles. However, they will likely use “hedge words” (e. g. , “these results may indicate, ” rather than “these results indicate”) in their discussions. Smith/Davis (c) 2005 Prentice Hall

One-Tailed and Two-Tailed Tests of Significance l l l You can state your experimental hypotheses in a direction or a nondirectional manner. Non-directional would look like this: M 1 M 2 Directional would look like this: either M 1 < M 2 or M 1 > M 2 A one-tailed t test evaluates the probability of an outcome in only one direction (greater than or less than; a directional hypothesis), whereas the two-tailed t test evaluates the outcome in both possible directions (a non-directional hypothesis). For non-directional hypotheses, the probability of the result occurring by chance alone is split in half and distributed equally in the two tails of the distribution. Although the test is calculated the same, you would consult different columns in the t table. Because the probability is not split for the one-tailed test, the critical value is lower, and thus it is easier to find a significant result. The main reason researchers don’t use the one-tailed test is because they don’t exactly know how an experiment will turn out, and thus are cautious in their predictions. Smith/Davis (c) 2005 Prentice Hall

When Statistics Go Astray: Type I and Type II Errors l When we conduct research and use probability in determining significance of our tests, there is always the possibility that your experiment represents one of those 5 times in 100 when the results did occur by chance. – – Type I Error – occurs when the null hypothesis is true and you make an error in accepting the experimental hypothesis. The experimenter directly controls the probability of making a Type I error by setting the significance level (e. g. , switching from. 05 to. 01) Type II error – occurs when we reject a true experimental hypothesis. This type of errors is not under the control of the researcher. We can cut down on Type II errors by implementing techniques that will cause our groups to differ as much as possible (e. g. , using a strong IV and larger groups of participants are two techniques that can help avoid Type II errors). Smith/Davis (c) 2005 Prentice Hall

When Statistics Go Astray: Type I and Type II Errors Smith/Davis (c) 2005 Prentice

Sampling Considerations and Basic Research Strategies l Sampling – – When we select a group to represent the population we can that group a sample. Techniques you can use to obtain a sample include: l l l Random sampling - ensures that every member of the population has an equal chance of being selected for inclusion in the sample. Thus giving us a representative sample of the population. Random sampling without replacement – occurs when a participant is not eligible to be chosen again once you’ve selected him/her. This is the technique psychologists prefer. Random sampling with replacement – occurs when you return a participant to the population and have him/her eligible for selection again. Smith/Davis (c) 2005 Prentice Hall

Sampling Considerations and Basic Research Strategies l Sampling – – To increase representativeness of our sample, we can increase a larger sample size. Generally, the larger the sample, the more representative it will be of the population. Stratified random sampling is another technique we can use to increase representativeness – it involves dividing the population into subpopulations or strata and then drawing a random sample from one or more of these strata. Smith/Davis (c) 2005 Prentice Hall

Basic Research Strategies l l l Single-strata approach – seeks to acquire data from a single, specified segment of the population. Cross-sectional research – involves the comparison of two or more groups of participants during the same time, rather limited, time span. Longitudinal research – involves obtaining a random sample from the population of interest; then this sample (or cohort) would be contacted periodically over an extended period of time to determine if any changes had occurred during the time of interest. Smith/Davis (c) 2005 Prentice Hall