Type I & Type II Errors: Videos & Practice Problems

When conducting a hypothesis test, we calculate a probability from sample data to decide whether to reject or fail to reject the null hypothesis. Typically, this decision aligns with reality, but occasionally, errors occur where our conclusion does not match the true state of nature. These errors are classified as Type I and Type II errors, which are crucial concepts in statistical hypothesis testing.

Consider a scenario where a treatment is claimed to lower a patient's blood pressure to 120. The null hypothesis (\(H_0\)) states that the mean blood pressure \(\mu = 120\), implying the treatment works as advertised. The alternative hypothesis (\(H_a\)) posits that \(\mu > 120\), meaning the treatment does not work effectively.

In hypothesis testing, two common outcomes occur without error: rejecting a false null hypothesis (correctly concluding the treatment does not work) and failing to reject a true null hypothesis (correctly concluding the treatment works). However, errors arise in two other cases. A Type I error happens when we reject a true null hypothesis, mistakenly concluding the treatment does not work when it actually does. Conversely, a Type II error occurs when we fail to reject a false null hypothesis, incorrectly concluding the treatment works when it does not.

To remember these errors, the mnemonic “rat fluff” is helpful: Reject True corresponds to Type I error, and Fail to reject False corresponds to Type II error.

The probability of committing a Type I error is denoted by the significance level \(\alpha\), which is the threshold for rejecting the null hypothesis based on the p-value. The p-value represents the probability of observing the sample data assuming the null hypothesis is true. Thus, \(\alpha\) is the maximum risk of rejecting a true null hypothesis that we are willing to accept. Reducing \(\alpha\) decreases the chance of a Type I error.

The probability of a Type II error is denoted by \(\beta\), which is not simply \$1 - \alpha\( but a separate value dependent on factors like sample size and effect size. Minimizing \)\beta\( involves increasing \)\alpha\(, which reduces the chance of failing to reject a false null hypothesis.

This inverse relationship means that lowering the risk of one type of error increases the risk of the other. Therefore, deciding which error to minimize depends on the context and consequences of each error. In the blood pressure treatment example, a Type II error (concluding the treatment works when it does not) is more serious and potentially unethical, so increasing \)\alpha\( to reduce \)\beta$ might be preferred.

Understanding and balancing Type I and Type II errors is essential for making informed decisions in hypothesis testing, ensuring that conclusions drawn from data are as accurate and ethical as possible.

When testing a claim about a population proportion, such as a smartphone company asserting that 90% of customers are satisfied with their newest model, hypothesis testing provides a structured approach to evaluate this assertion. The null hypothesis (\(H_0\)) represents the company's claim, stating that the true population proportion \(p\) equals 0.9. The alternative hypothesis (\(H_a\)) reflects the opposing view—in this case, that the true satisfaction rate is lower, so \(p < 0.9\).

To assess this, a random sample of 100 customers is surveyed, with 84% reporting satisfaction. This sample proportion, denoted as \(\hat{p} = 0.84\), is compared against the claimed proportion \(p = 0.9\). The significance level, or probability of a Type I error (\(\alpha\)), is set at 0.1, meaning there is a 10% risk of incorrectly rejecting the null hypothesis when it is actually true.

The test statistic used is the z-score, which measures how many standard deviations the sample proportion is from the hypothesized population proportion. The formula for the z-score in a one-proportion z-test is:

Here, \(n\) is the sample size. Substituting the values:

A z-score of -2 indicates the sample proportion is two standard deviations below the claimed population proportion. To determine the statistical significance, the p-value is calculated as the probability of observing a z-score less than -2 under the null hypothesis. This left-tail probability is approximately 0.0228.

Since the p-value (≈ 0.0228) is less than the significance level \(\alpha = 0.1\), there is sufficient evidence to reject the null hypothesis. This means the data supports the consumer advocacy group's belief that the true satisfaction rate is less than 90%. In practical terms, the survey results suggest the company's claim may be overstated, and the actual customer satisfaction is likely lower.

Understanding this process highlights the importance of hypothesis testing in making data-driven decisions, especially when evaluating claims about population parameters. The use of the z-test for proportions, the interpretation of p-values, and the role of significance levels are fundamental concepts in inferential statistics that enable confident conclusions about real-world phenomena.