ED 602

Statistical Research for Behavioral Sciences

Brian G. Smith, Ph.D.
Lesson - 11

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Homework - Lesson 11

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Assessment - Lesson 11

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Assignments and Information
 
Reading: Chapter 9
  Definition Page: Contains definitions arranged alphabetically.

Notes
Statistical Hypothesis

  • A statement about a population parameter
  • May or may not be true
  • Established to set up a testable condition
  • Two hypotheses are formed
    • Null hypothesis the hypothesis that states there is no difference
    • Alternative hypothesis is the hypothesis that must be true if the null hypothesis turns out to be false

Z tests

  • A test of significance
  • Uses z scores for hypothesis testing
  • Based on many samples
  • You need to know the standard error
  • Uses the formula

 

Significance levels

  • It is possible to have a measurable difference in means, and still keep the null hypothesis
  • There is a critical value, or level, you need to reach to reject the null hypothesis
  • Most common significance level is an a of 0.05 or less
  • 2-tailed tests have rejection regions at both ends of the distribution
  • 1-tailed tests can be used if you are using a hypothesis that predicts a directional relationship among the means. In other words there is a rejection region at only one end of the distribution.

One sample t test

  • A test of significance
  • Used when a population s is not known
  • A test for significant difference between sample mean and hypothesized population mean
  • Uses degrees of freedom rather than sample size
  • Measures the relative size of the difference between means
  • Uses degrees of freedom to find significance levels
  • Uses the formula
when


Significance levels in a one-sample t test

  • Both two-tailed and one-tailed tests tables are on page 491
  • Find your degrees of freedom
  • Choose an a level of 0.05 or lower for most research
  • Table lists only positive values, but t is significant if it is plus or minus that number

Assumptions of the one sample t test

  • Subjects selected through random sampling
  • The population has a normal distribution of scores
  • Scores measured on an interval or ratio scale

Types of errors

  • Type I
    • A false positive, finding significant difference that isn’t there
    • Happens if you are overly aggressive in your research
    • a is the probability of making this type of error
  • Type II
    • A false negative, not finding the significant difference that is there
    • Happens if you are overly conservative in your research
    • b is the probability of making this type of error
    • In statistics, it is better to have this type of error rather than Type I

Confidence intervals

It is possible to find confidence intervals using the t statistic rather than z scores using the formula

X – (t.05)(sx-bar) to X + (t.05)(sX-bar)

Now we can use this new information to look at our BASC data. The complaining parent in our study has given us the statistical hypothesis that there is a significant difference between the scores of Hispanic youth on the BASC versus the scores of non-Hispanic youth on the BASC. Our team is hoping that the null hypothesis is true. There is no significant difference between the scores of the Hispanic and non-Hispanic groups.

We can start with a z test using the BASC data from the Hispanic group (mean 51.4, standard deviation 11.12) and national norms (mean = 50.00, standard deviation 7.50) to find out if the BASC is biased.

To reject the null hypothesis, and call the BASC assessment biased, we would need a z of at least 1.96 to reach an alpha level of 0.05 or less. 0.93 is less than 1.96, so we would keep the null hypothesis. Based on this sample, there was not a significant difference between the BASC scores of the normative sample and the BASC scores of our sample of Hispanic youth.

If we are really nervous about a lawsuit, we can double-check our results by trying a t test.
Remember that


and


Now we look at the table of critical values (page 491) for t when we have 25 degrees of freedom (df = N-1 = 25-1=24). We would need a t of at least 2.064 to reject the null hypothesis and call the BASC measure biased.

Because we kept the null hypothesis, we have to be aware that there is a small possibility that we have made a Type II error.

SPSS Tips:

 

 

Vocabulary

Parametric test – A statistical test involving hypotheses that state a relationship about a population parameter.

Nonparametric test – A statistical test involving hypotheses that do not state a relationship about a population parameter.

Statistical hypothesis – A statement about a population parameter (for a parametric test).

Null hypothesis – a statement of a condition that a scientist tentatively holds to be true about a population; it is the hypothesis that is tested by a statistical test. For our BASC study we hope to show that there is no difference between the scores of students who are Caucasian, and scores of students who are Hispanic.

Alternative hypothesis – A statement of what must be true if the null hypothesis is false. The alternative hypothesis is that there is a significant difference between the scores of students who are Caucasian and the scores of students who are Hispanic, and the test is biased.

Test statistic – A number calculated from the scores of the sample that allows testing a statistical null hypothesis. z and t are examples of test statistics.

Significance level – A probability value that provides the criterion for rejecting a null hypothesis in a statistical test.

Alpha (α) – The value of the significance level stated as a probability. In research we often want an alpha of 0.05 or less.

Rejection region – Values on the sampling distribution of the test statistic that have a probability equal to or less than a if the null hypothesis is true. If the test statistic falls in to the rejection region, the null hypothesis is rejected.

Two-tailed test – A statistical test using rejection regions in both tails of the sampling distribution of the test statistic.

One-tailed test – a statistical test using a rejection region in only one tail of the sampling distribution of the test statistic. Also called a directional test.

Critical value – The specific numerical values that define the boundaries of the rejection region.

Statistically significant difference – The observed value of the test statistic falls into the rejection region and the null hypothesis is rejected.

Nonsignificant difference – The observed value of the test statistic does not fall into a rejection region and the null hypothesis is not rejected.

One-sample t test – A t test used to test the difference between a sample mean and a hypothesized population mean for statistical significance when S is estimated by σ.

Degrees of freedom – the number of scores free to vary when calculating a statistic.

Type I error – The error in statistical decision making that occurs if the null hypothesis is rejected when actually true of the population.

Type II error – The error in statistical decision making that occurs if the null hypothesis is not rejected when it is false and the alternative hypothesis is true.

Beta (β) – The probability of a Type II error.

Power – The probability of rejecting the null hypothesis when the null hypothesis is false and the alternative hypothesis is true. The power of a statistical test is given by 1-β.