The production manager of a manufacturer wants to evaluate a modified ball bearing production process. When the process is operating properly, the process produces ball bearings whose weights have a mean of 5 ounces and a standard deviation of 0.1 ounce. A new raw-material supplier was used for a recent production run, and the manager wants to know if that change has caused any problem. If there was a problem, the mean weight of a ball bearing would be different. The manager will test the following hypotheses and From a random sample of 100 ball bearings, the sample mean was 5.03. The significance level is specified as 5%. What is the probability of type I errors in this hypothesis testing? Question 9 options: a) 5.03% b) 1.00% c) 1.96% d) 0.03% e) 5.00%
The correct answer and explanation is:
The correct answer is: e) 5.00%
Explanation:
In hypothesis testing, the probability of a Type I error is the probability of rejecting the null hypothesis when it is actually true. This probability is also known as the significance level, denoted by α (alpha).
Here is the context:
- The null hypothesis H0H_0 states that the mean weight of ball bearings is 5 ounces.
- The alternative hypothesis HaH_a states that the mean weight is different from 5 ounces (two-tailed test).
- The significance level α is given as 5%, or 0.05.
- The sample size is 100, and the sample mean observed is 5.03 ounces.
The significance level α of 5% means that there is a 5% chance that the test will wrongly reject the null hypothesis, even if the true mean is indeed 5 ounces. This error is a Type I error.
- The Type I error probability is chosen by the researcher before conducting the test.
- It does not depend on the sample data but is fixed by the significance level.
- In this problem, the significance level is explicitly stated as 5%, so the Type I error probability is 5%.
The other answer choices are percentages that might confuse with p-values or test statistics, but the question specifically asks for the probability of Type I error, which directly corresponds to the significance level.
Summary:
- Type I error probability = significance level α
- α = 0.05 or 5%
- Therefore, the probability of Type I error in this hypothesis test is 5.00%.
This means if the process is operating correctly, there is a 5% chance that the manager would incorrectly conclude the new supplier has caused a change in the mean weight.