Hypothesis Testing

Full Answer Section

Increasing the sample size directly increases the power of a statistical test. With a larger sample, we obtain a more precise estimate of the population parameters, and the standard error of the statistic decreases. This makes it easier to detect a true effect if one exists.

Consider a study comparing the growth rates of two groups of children. If the actual growth rates are slightly different in the community (the null hypothesis of no difference is false), a small sample might not provide enough statistical evidence to detect this difference due to random variability. However, as the sample size increases, the random variability averages out, and the true difference becomes more apparent, leading to a higher probability of correctly rejecting the false null hypothesis (i.e., reduced probability of a Type II error).

Scholarly Support:

According to Gravetter and Wallnau (2017) in Essentials of Statistics for the Behavioral Sciences (9th ed., p. 257):

"The power of the statistical test is the probability that the test will correctly reject a false null hypothesis. That is, power is the probability that the test will identify a treatment effect if one actually exists... One way to increase the power of a test is to increase the sample size (n). A larger sample provides a more accurate representation of the population, and therefore, it is more likely to detect a real treatment effect."

Type I Error and Sample Size:

A Type I error occurs when we reject the null hypothesis even though it is actually true in the population (a false positive). The probability of making a Type I error is denoted by α (alpha), which is the significance level set by the researcher before conducting the study (commonly α = 0.05).

The significance level (α) is the pre-determined threshold for considering a result statistically significant. It represents the probability of rejecting the null hypothesis if the null hypothesis is true. This probability is set by the researcher and is independent of the sample size.

Increasing the sample size does not change the researcher's pre-set tolerance for making a Type I error. If the null hypothesis is true, a significance level of 0.05 means there is a 5% chance of observing data that leads to rejecting the null hypothesis, regardless of whether the sample size is large or small (assuming all other assumptions of the statistical test are met). While a very large sample size might detect even trivial deviations from the null hypothesis as statistically significant, this does not increase the probability of making a Type I error if the null hypothesis is exactly true. The significance level (α) remains the defined probability of a false positive.

Scholarly Support:

As stated by Field (2018) in Discovering Statistics Using IBM SPSS Statistics (5th ed., p. 88):

"The probability of a Type I error is determined by the α level (the significance level) that you choose. Traditionally, this is .05, meaning that there is a 5% chance that we will obtain a significant result when the null hypothesis is actually true. This probability is set by the experimenter and is not affected by the size of the sample."

In summary, increasing the sample size provides more statistical power to detect a true effect (reducing Type II error), but the risk of incorrectly rejecting a true null hypothesis (Type I error) is controlled by the pre-determined significance level (α) and is not altered by the sample size itself.

Response to Classmate's Post (Example):

Hi [Classmate's Name],

Your explanation of power and Type II error is clear. I agree that a larger sample provides a more accurate reflection of the population, making it easier to detect a real effect.

However, I wanted to add a point regarding the potential for confusion with very large samples and Type I errors. While the probability of a Type I error (α) remains fixed, a very large sample size can lead to statistically significant results even for very small and practically unimportant effects. This doesn't increase the likelihood of a false positive if the null hypothesis is precisely true, but it can lead to rejecting a null hypothesis when the observed difference is trivial in the real world. This highlights the importance of considering effect size and practical significance alongside statistical significance, especially with large samples.

Sample Answer

An increase in sample size significantly reduces the probability of a Type II error (failing to reject a false null hypothesis), but it does not impact the probability of a Type I error (rejecting a true null hypothesis), for the following reasons:

Type II Error and Sample Size:

A Type II error occurs when there is a real effect or difference in the population, but the study fails to detect it, leading to the acceptance of a false null hypothesis. The probability of a Type II error is denoted by β (beta), and the power of a statistical test is defined as 1 - β, which is the probability of correctly rejecting a false null hypothesis.