Identifying Type I and Type II Errors In Exercises 31–36, describe type I and type II errors for a hypothesis test of the indicated claim.


Chess A local chess club claims that the length of time to play a game has a standard deviation of more than 12 minutes.

Verified step by step guidance

1. Understand the null and alternative hypotheses: The null hypothesis (H₀) is that the standard deviation of the length of time to play a game is less than or equal to 12 minutes (σ ≤ 12). The alternative hypothesis (H₁) is that the standard deviation is greater than 12 minutes (σ > 12).
2. Define a Type I error: A Type I error occurs when the null hypothesis is rejected even though it is true. In this context, it means concluding that the standard deviation of the length of time to play a game is greater than 12 minutes when, in reality, it is 12 minutes or less.
3. Define a Type II error: A Type II error occurs when the null hypothesis is not rejected even though it is false. In this context, it means failing to conclude that the standard deviation of the length of time to play a game is greater than 12 minutes when, in reality, it is greater than 12 minutes.
4. Relate the errors to the context: A Type I error might lead the chess club to believe that games are more variable in duration than they actually are, potentially influencing scheduling or tournament planning. A Type II error might lead the chess club to overlook the variability in game durations, which could also affect planning.
5. Summarize the importance of balancing 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 (1 - β) is related to the probability of avoiding a Type II error. The chess club should consider the consequences of both errors when interpreting the results of the test.

Key Concepts

Type I Error

A Type I error occurs when a null hypothesis is incorrectly rejected when it is actually true. In the context of the chess club's claim, this would mean concluding that the standard deviation of game time is greater than 12 minutes when, in fact, it is not. This type of error is often denoted by the symbol alpha (α) and represents a false positive in hypothesis testing.

Type II Error

A Type II error happens when a null hypothesis is not rejected when it is false. For the chess club's claim, this would mean failing to conclude that the standard deviation of game time is greater than 12 minutes when it actually is. This error is represented by the symbol beta (β) and indicates a false negative in hypothesis testing.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to determine whether to reject H0. In this case, the null hypothesis would state that the standard deviation of game time is 12 minutes or less, while the alternative would assert that it is greater.