Statistics Ch. 9 Clarifying the concepts

Explain what each part of the following statistic means, as it would be reported in APA

(1) t is the symbol for the test statistic. (2) The degrees of freedom is listed inside the parentheses. (3) The value of the test statistic, typically to two decimal places, is written on the right side of the equal sign. (4) The p value, 0.032, is that associated with the test statistic. It is less than the cutoff p level of 0.05; thus, the null hypothesis was rejected.

What information does a dot plot provide

A dot plot depicts the distribution of the sample data and thus provides information similar to that provided by a histogram. The dot plot, however, contains a separate dot for each observation.

Why do the t distributions merge with the z distributions as sample size increases

As the sample size increases, we can feel more confident in the estimate of the variability in the population. Remember, this estimate of variability (s) is calculated with N - 1 in the denominator in order to inflate the estimate somewhat. As the sample increases from 10 to 100, for example, and then up to 1000, subtracting 1 from N has less of an impact on the overall calculation. As this happens, the t distributions approach the z distribution, where we in fact knew the population standard deviation and did not need to estimate it.

How is the calculation of standard error different for a t test than for a z test?

For both tests, standard error is calculated as the standard deviation divided by the square root of N. For the z test, the population standard deviation is calculated with N in the denominator. For the t test, the standard deviation for the population is estimated by dividing the sum of squared deviations by N -1.

What does free to vary mean

Free to vary refers to the number of scores that can take on different values if a given parameter is known.

How is the critical t value affected by sample size and degrees of freedom

The critical t value gets smaller, or closer to the mean, as sample size and degrees of freedom increase.

Explain why the standard error for the distribution of sample means is smaller than the standard deviation of sample scores

The distribution of sample means is always smaller than the distribution of sample scores because the effect of an extreme score is reduced when a mean is calculated based on many scores. Because of this, the standard error of the means is smaller than the standard deviation of the scores.

Why do we modify the formula for calculating standard deviation when usinf t test (and divide by N-1)

To conduct a t test, we use the sample data to estimate variability—the standard deviation—of the entire population. The estimated standard deviation from the sample is not likely to be exactly equal to the population standard deviation; because the sample is smaller and likely less variable, the estimate of the standard deviation is likely to be somewhat smaller. The mathematical calculation that corrects for the error we accumulate when using t tests is subtraction; that is, we divide by N - 1 rather than N, which makes the estimated standard deviation slightly larger.

When should we use a t distribution

We should use a t distribution when we do not know the population standard deviation and are comparing two groups.

When is it appropriate to use a single samples t test

We use a single-sample t test when we want to compare the mean of a sample to that of a population but we do not know the population standard deviation.

Define the symbols

t stands for the t statistic, M is the sample mean, µM is the mean of the distribution of means, and sM is the standard error as estimated from a sample.