PSY 3213a.0W59
Research Methods (Class E-mail)
Dr. Jay Brophy

Introduction:

    Previously, we have discussed descriptive statistics where you simply describe a sample with statistics such as the mean, mode and variance. Later, we addressed describing the relationship between two variables using statistics such as correlation and regression.    Now, we are starting to discuss inferential statistics which are used when you want to apply (infer) the results from a sample to a population. This is also called hypothesis testing since you are using a sample to test a hypothesis in hopes of inferring the conclusions about the hypothesis to the population.

    Warning: From here on out, if we talk about a standard deviation, it is calculated using n - 1 (not n) in the denominator.

    There are always two statistical hypotheses for an experiment. Of these two, one will be supported at the end of the experiment. The other will not be supported. The two hypotheses must cover all possible outcomes of the experiment. Put another way, the hypotheses need to be mutually exclusive and exhaustive.

   The first hypothesis is called the alternative (or experimental) hypothesis. This represents what the experimenter is hoping to find or prove. It is symbolized as "H_A" or "H₁". When creating hypotheses you should always write the experimental hypothesis first.

   The second hypothesis is called the null hypothesis and represents all other possibilities than what the experimenter is hoping to find. It is important to make sure that this hypothesis encompasses all possible outcomes other than the alternative hypothesis. This hypothesis must be written after the alternative hypothesis. The null hypothesis is symbolized as "H₀".

  An example of a set of hypotheses for an experiment could be:

              H₁: Prozac affects mood.
              H₀: Prozac does not affect mood.

   Unidirectional versus non-directional hypotheses:    A directional hypothesis is one that specifies the direction of the relationship. So, if the hypothesis says "the drug will increase mood", it is a unidirectional hypothesis. This is also known as a 1-tailed test since you are interested either in an increase or a decrease in the dependent variable (DV), but not both.

  Alternatively, if a hypothesis does not specify a direction it is a non-directional hypothesis. So, if the hypothesis is "the drug affects mood", this would be a non-directional hypothesis. This is also known as a two tailed test since you are interested in whether there is both an increase or decrease due to the DV.

   When trying to determine a hypothesis in a published study, you look for the words relating the IV and DV. Words like "increase, decrease, raise, lower, augment, diminish"  represent directional hypotheses. Words such as "affect, alter, change, influence" represent non directional hypotheses.

  Hypothesis Matching:

    When writing hypotheses, you must make sure that the two hypotheses cover all possibilities. This is easier with non-directional hypothesis since it is normally accomplished through the addition of the word "not" to the null hypothesis.

    For a non-directional experimental hypothesis might be:

          H₁: Prozac affects mood.

    The corresponding non-directional null hypothesis:

           H₀: Prozac does not affect mood.

          ( Note: all possible outcomes to the experiment are represented by one of the two hypotheses).

    Writing hypotheses for unidirectional hypotheses can be more complicated since the null hypothesis must appropriately match the experimental hypothesis. Always refer to the alternative hypothesis before writing the null hypothesis.

    For a unidirectional (positive) experimental hypothesis might be:

           H₁: Prozac increases mood.

     The corresponding null hypothesis would be:

           H₀: Prozac does not increase mood.

            or, equivalently,

          H₀: Prozac decreases mood or does not affect mood.

   Note that a null hypothesis that said:

          H₀: Prozac decreases mood

would not be correct since the possible outcome of Prozac not affecting mood is not accounted for in either the experimental or null hypothesis.

   For a unidirectional (negative) experimental hypothesis, an example might be:

          H₁: Prozac decreases mood.

    The corresponding null hypothesis would be:

           H₀: Prozac does not decrease mood.

    Once you have conducted an experiment, you have the scores on the DV for the sample. In general, you want to compare the mean of the (treated) sample that received the IV to the mean of the (untreated) population that did not receive the IV. If the treated sample mean is significantly different than the untreated population mean, then you know that the IV had an effect on the DV.

   To determine whether there is a difference between the population and the sample, you cannot directly compare the sample mean to the population mean, however, due to sampling error.

   Sampling error occurs since your sample does not perfectly represent the population. For example, if you select 30 people for your sample from a population of 100,000 people to study whether a drug increases intelligence, the sample could just by random chance have the 30 people with the lowest intelligence or the highest intelligence.

   If your sample has, by chance, the 30 lowest intelligence people from the population, and you compare the mean intelligence of the sample to the mean intelligence of the population, it will always almost look like the drug did not work since the sample mean will be lower than the population mean even if the drug was effective. Alternatively, if your sample has the 30 most intelligent people from the population it will almost always look like the drug worked since the sample mean will be above the population mean.

  The problem with sampling error is that you can never know how much error is created by who was selected for the sample. Just remember, due to sampling error, it is not possible to directly compare the sample mean to the population mean.  Instead, you must compare the sample mean to the sampling distribution.

    A sampling distribution is created by taking a sample of a given size from the population and calculating the mean for that sample. Then you take another sample of  the same size and calculate the mean for this sample. Then, you continue taking samples and calculating the means until you have created every possible sample of this size and determined their means.

   This sampling distribution tells how frequently means of each level occur for samples of a given size. In essence, the sampling distribution shows how often you expect a mean to occur for a sample of a given size due to sampling error (who was in the sample).

   Conveniently, the mean of the sampling distribution is the same as the mean of the population. Also, the mean of the sampling distribution (know as the standard error) is the standard deviation of the population divided by the square root of the sample size  (n). The standard error is symbolized as sigma sub xbar. So,

           sigma sub xbar = sigma / square root n

   Once you have the sampling distribution, you can compare the sample (treated) mean to the sampling distribution. If the treatment was effective, you would expect the sample mean to occur very infrequently in the sampling distribution.

   Now for the good news, you do not have to calculate the sampling distribution really. It is just conceptual and the formulas we will use for inferential statistics already take this into account.

   There are two statistics that we will cover in the next set of lecture notes. These, and all statistics for the rest of the semester, are used in specific circumstances to determine whether the sample mean is far enough away from the mean of the sampling distribution (which is the same as the population mean).    The statistical formulas are not particularly complex. What is more important is to understand when each of the statistics is used. The first two statistics that we will discuss next week are as follows:

   The z test is used when you have a single sample in the experiment and the population mean and the population standard deviation are known.

   The one sample t test is used when you have a single sample in the experiment and the population mean (but not the population standard deviation) is known.
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This Page Maintained by: Dr. Jay Brophy
This page was last revised on March 18, 2007.