**C8.** Which of the following situations would yield the greatest power (assuming alpha and sample size are held constant)?

Null mean = 500, Alternative mean = 510, Standard Deviation = 40

Null mean = 500, Alternative mean = 540, Standard Deviation = 160

Null mean = 500, Alternative mean = 520, Standard Deviation = 60

Check your answers:

#### Show Hint

We have three sets of means, each with a different standard deviation. Recall that effect size is determined by the difference between the means divided by the standard deviation. Larger effect sizes produce greater power. Try using the formula below for each set of means and standard deviation.

#### Null mean = 500, Alternative mean = 510, Standard Deviation = 40

Sorry, this answer is incorrect! Your answer situation reflects the smallest standard deviation. However, recall that power is also related the magnitude of differences between means.

For this example, we have an effect size of .25 (see calculation below)

#### Null mean = 500, Alternative mean = 540, Standard Deviation = 160

*Sorry, this answer is incorrect!* Your answer reflects the situation in which we have the largest raw difference between the means. However, recall that power is also related the standard deviation.

For this example, we have an effect size of .25 (see calculation below)

#### Null mean = 500, Alternative mean = 520, Standard Deviation = 60

**Correct!**

This situation has the largest effect size, .33. If alpha and sample size are held constant, the effect size is the only other determinant of power, and a larger effect size produces greater power.

**C9. **Consider the shape of the sampling distributions for samples of size n = 4, n = 25, and n = 100. What happens to the sampling distribution of the sample mean when n is increased (assuming nothing else changes)?

Sampling distribution becomes more disperse.

Sampling distribution becomes less disperse.

Sampling distribution remains the same.

Check your answers:

#### Show Hint

Recall the dispersion of a sampling distribution is defined by the standard error of the mean as defined by the formula below:

Given a larger sample size (with a constant standard deviation) what is the effect of sample size on the standard error?

#### Sampling distribution becomes more disperse.

*Sorry, this answer is incorrect!* As the sample size rises the dispersion of the sampling distribution does not increase. Recall this dispersion is defined by the standard error of the mean as shown in the formula below:

#### Sampling distribution becomes less disperse.

**Correct!** As the sample size is increased the dispersion of the sampling distribution of the mean is reduced. As seen in the formula below, a larger sample size leads to a smaller standard error of the mean.

#### Sampling distribution remains the same.

*Sorry, this answer is incorrect!* As the sample size rises the dispersion of a sampling distribution does not stay the same. The dispersion, defined by the standard error of the mean (below), changes as a function of the sample size.

**C10. **So far you have examined the effect of magnitude of difference between the null mean and the alternative mean, standard deviation, sample size, and alpha level on power. Which of the answers below best summarizes the effect of each on power?

More power = large magnitude of difference, larger standard deviation, larger sample, larger alpha.

More power = large magnitude of difference, smaller standard deviation, larger sample, smaller alpha.

More power = large magnitude of difference, smaller standard deviation, larger sample, larger alpha.

More power = smaller magnitude of difference, smaller standard deviation, larger sample, smaller alpha.

Check your answers:

#### More power = large magnitude of difference, larger standard deviation, larger sample, larger alpha.

*Sorry, this answer is incorrect!* Remember that the effect size (the ratio of magnitude of difference to standard deviation) is reduced by larger standard deviations.

#### More power = large magnitude of difference, smaller standard deviation, larger sample, smaller alpha.

*Sorry, this answer is incorrect!* Remember that a smaller alpha indicates a more stringent criteria, reducing power.

#### More power = large magnitude of difference, smaller standard deviation, larger sample, larger alpha.

**Correct!** Power is greater when the effect size, sample size, and alpha are larger, and variability is minimized.

#### More power = smaller magnitude of difference, smaller standard deviation, larger sample, smaller alpha.

*Sorry, this answer is incorrect!* Remember that the effect size (the ratio of magnitude of difference to standard deviation) is reduced by a smaller magnitude of differences between means.

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