PSY292 - Chapter 9

Degrees of Freedom

Describe the number of scores in a sample that are independent and free to vary. Because the sample mean places a restriction on the value of one score in the sample, there are n - 1 degrees of freedom for a sample with n score. The greater the value of df for a sample, the better the sample variance, s^2, represents the population variance, σ^2, and the better the t statistic approximates the z-score. How well a t distribution approximates a normal distributor is determined by degrees of freedom. In general, the greater the sample size (n) is, the larger the degrees of freedom (n - 1) are, and the better the t distribution approximates the normal distribution.

Problem with using z-scores....

The shortcoming of using a z-score for hypothesis testing is that the z-score formula requires more information than is usually available. Specifically, a z-score requires that we know the value of the population standard deviation (or variance), which is needed to compute the standard error. In most situations, however, the standard deviation for the population is not known.

Difference between the t formula and the z formula...

The z-score uses the actual population variance, σ^2 (or the standard deviation) The t formula uses the corresponding sample variance (or standard deviation) when the population value is not known.

The t statistic is used instead of a z-score for hypothesis testing when the population standard deviation is........

UNKNOWN Z-scores are used when the population standard deviation (sigma) is known. The t statistic uses the sample variance or standard deviation in place of the unknown population values.

If other factors are held constant, explain how each of the following influences the value of the independent-measures t statistic and the likelihood of rejecting the null hypothesis: a. Increasing the number of scores in each sample. b. Increasing the variance for each sample.

A) The size of the two samples influences the magnitude of the estimated standard error in the denominator of the t-statistic. As sample size increases, the value of t also increases (moves farther away from zero), and the likelihood of rejecting H0 also increases. B) The variability of the score influences the estimated standard error in the denominator of the t-statistic. As the variability of a scores increases the value of t decreases (becomes closer to zero) and the likelihood of rejecting the H0 decreases.

T/F: Sample size affects Cohen's d.

FALSE

Why t distributions tend to be flatter and more spread out than the normal distribution. Pt.2

For both formulas, z and t, the top of the formula, M - μ, can take on different values because the sample mean (M) varies from one sample to another. For z-scores, however, the bottom of the formula does not vary, provided that all of the samples are the same size and are selected from the same population. Specifically, all of the z-scores have the same standard error in the denominator, M 2/n, because the population variance and the sample size are the same for every sample. For t statistics the bottom of the formula varies from one sample to another

How does sample size influence the outcome of a hypothesis test and measures of effect size?

Increasing sample size increases the likelihood of rejecting the null hypothesis but has little or no effect on measures of effect size.

How does the standard deviation influence the outcome of a hypothesis test and measures of effect size?

Increasing the sample variance reduces the likelihood of rejecting the null hypothesis and reduces measures of effect size.

T/F: An increase in variance leads to a decrease in effect size.

TRUE

T/F: As df increases, the shape of the t distribution approaches a normal distribution

TRUE

Describe the homogeneity of variance assumption and explain why it is important for the independent measures t test.

The homogeneity of variance assumption specifies that the variances are equal for the two populations from which the samples are obtained. If this assumption is violated, the t-statistic can cause misleading conclusions for a hypothesis test.

Explain why t distributions tend to be flatter and more spread out than the normal distribution.

The sample variance (s^2) in the t formula changes from one sample to another and contributes to the variability of the t statistic. A z-score uses the population variance which is constant from one sample to another. Normal distributions are based on large sample sizes while t-distributions are based on smaller sample sizes.