In Exercises 7–10, describe type I and type II errors for a hypothesis test of the claim.


An energy bar maker claims that the mean number of grams of carbohydrates in one bar is less than 25.

Verified step by step guidance

1. Step 1: Understand the hypothesis test setup. The null hypothesis (H₀) represents the claim being tested, which is typically the opposite of the energy bar maker's claim. Here, H₀: μ ≥ 25, where μ is the mean number of grams of carbohydrates in one bar. The alternative hypothesis (H₁) represents the energy bar maker's claim, H₁: μ < 25.
2. Step 2: Define a Type I error. A Type I error occurs when the null hypothesis (H₀) is rejected even though it is true. In this context, a Type I error would mean concluding that the mean number of grams of carbohydrates in one bar is less than 25 (accepting H₁), when in reality, the mean is 25 or greater.
3. Step 3: Define a Type II error. A Type II error occurs when the null hypothesis (H₀) is not rejected even though it is false. In this context, a Type II error would mean failing to conclude that the mean number of grams of carbohydrates in one bar is less than 25 (not accepting H₁), when in reality, the mean is indeed less than 25.
4. Step 4: Relate the errors to the context of the problem. A Type I error might lead to the energy bar maker falsely advertising that their bars have fewer carbohydrates than they actually do. A Type II error might prevent the energy bar maker from promoting their product as having fewer carbohydrates, even though it does.
5. Step 5: Highlight the importance of balancing these errors. In hypothesis testing, the significance level (α) is chosen to control the probability of a Type I error, while the power of the test is related to the probability of avoiding a Type II error. The choice of α and sample size can help balance these errors based on the context and consequences.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (H1), which represents the claim being tested. In this case, the null hypothesis would state that the mean number of grams of carbohydrates is 25 or more, while the alternative hypothesis would claim it is less than 25.

Type I Error

A Type I error occurs when the null hypothesis is incorrectly rejected when it is actually true. In the context of the energy bar maker's claim, this would mean concluding that the mean number of grams of carbohydrates is less than 25 when, in fact, it is 25 or more. The probability of making a Type I error is denoted by alpha (α), which is typically set at a significance level, such as 0.05.

Type II Error

A Type II error happens when the null hypothesis is not rejected when it is false. In this scenario, it would mean failing to recognize that the mean number of grams of carbohydrates is actually less than 25 when the data suggests it is. The probability of making a Type II error is denoted by beta (β), and it reflects the test's ability to detect an effect when there is one.