In this problem, the null hypothesis ($H_0$) is that the standard deviation of the length of time to play a game is less than or equal to 15 minutes ($H_0: \sigma \leq 15$).

Type I Error:

A Type I error occurs when we reject the null hypothesis when it is actually true.

• Type I error statement: A type I error will occur if the actual standard deviation of the length of time to play a game is 15 minutes or less, but you reject the null hypothesis $H_0: \sigma \leq 15$.

Type II Error:

A Type II error occurs when we fail to reject the null hypothesis when it is actually false.

• Type II error statement: A type II error will occur if the actual standard deviation of the length of time to play a game is greater than 15 minutes, but you fail to reject the null hypothesis $H_0: \sigma \leq 15$.

Summary:

• Type I error: The actual standard deviation is 15 minutes or less, but you incorrectly reject $H_0: \sigma \leq 15$.
• Type II error: The actual standard deviation is greater than 15 minutes, but you incorrectly fail to reject $H_0: \sigma \leq 15$.

Would you like more details or further explanation?

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Here are 5 related questions:

1. What is the difference between a type I and type II error in hypothesis testing?
2. How does the choice of significance level (alpha) affect the probability of a type I error?
3. What is the power of a test, and how does it relate to type II errors?
4. Can you explain how to calculate the critical value for this hypothesis test?
5. How do you determine whether to use a one-tailed or two-tailed test?

Tip: Increasing the sample size in hypothesis testing can help reduce the likelihood of a type II error.