The probability of a type II error for a hypothesis test

In summary, a college student takes a random sample of 15 rest stops and records the data in hours to estimate how long she drove between stops. Given that the true mean is 2.5 hours, she wants to know the probability of making a type II error and the power of the test. The calculated P(type II error) is 0.221 and the power of the test is 0.779, after rejecting the null hypothesis when t < -3.463 or when t > 0.7991 with a significance level of 0.05.
  • #1
Mr Davis 97
1,462
44

Homework Statement


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1. After a long road trip during summer break, a college student wants to estimate how long she drove between rest stops. She takes a random sample of 15 stops and records this data in hours: 1.5, 2.6, 1.75, 3.5, 2.25, 3.75, 1.8, 2.9, 3.2, 3.15, 1.5, 2, 3, 2.75, and 2.2. She thinks that the real mean is not 2.25 hours. Given that the true mean is 2.5 hours, what is the probability that she will make a type II error? What is the power of the test?

Homework Equations



The equation for a t statistic

The Attempt at a Solution



With my calculator, I found that we reject the null hypothesis when t < -3.463 or when t > .7991. I found P(type II error) = .221, and the power of the test to be .779. Is this correct?
 
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  • #2
Why are your critical values not equally spaced either side of 0? What significance level were you using? What work did you do to get the power?
 

1. What is a type II error in a hypothesis test?

A type II error, also known as a false negative, occurs when the null hypothesis is actually false, but the test fails to reject it. In other words, the test incorrectly concludes that there is no significant difference or relationship between the variables being tested.

2. How is the probability of a type II error calculated?

The probability of a type II error is calculated as the area under the alternative hypothesis curve that falls within the critical region of the null hypothesis. This can be determined using statistical software or by looking up the corresponding value on a standard normal distribution table.

3. What factors affect the probability of a type II error?

The probability of a type II error can be affected by several factors, including the sample size, the significance level (alpha), the effect size, and the variability of the data. As the sample size and the effect size increase, the probability of a type II error decreases. On the other hand, a higher significance level and greater variability in the data can increase the probability of a type II error.

4. How does the probability of a type II error relate to the power of a hypothesis test?

The power of a hypothesis test is equal to 1 minus the probability of a type II error. In other words, the power of a test is the probability of correctly rejecting the null hypothesis when it is false. Therefore, as the probability of a type II error decreases, the power of the test increases.

5. Can the probability of a type II error be reduced?

Yes, the probability of a type II error can be reduced by increasing the sample size, selecting a higher significance level, and choosing a more appropriate test for the research question. It is important to find a balance between the probability of a type II error and the probability of a type I error (rejecting a true null hypothesis), as both errors can have significant consequences in the interpretation of research results.

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