A can of soda is labeled as containing 16 fluid ounces. The quality control manager wants to verify that the filling machine is neither over-filling nor under-filling the cans. Complete parts (a) through (d) below. (a) Determine the null and alternative hypotheses that would be used to determine if the filling machine is calibrated correctly. $H_{0}: \mu=16$ $H_{1}: \mu \neq 16$ (Type integers or decimals. Do not round.) (b) The quality control manager obtains a sample of 85 cans and measures the contents. The sample evidence leads the manager to reject the null hypothesis. Write a conclusion for this hypothesis test. There is sufficient evidence to conclude that the machine is out of calibration (c) Suppose, in fact, the machine is not out of calibration. Has a Type I or Type If error been made? A has been made since the sample evidence led the quality-control manager to the null hypothesis, when the is true.
Solution
Step 1 ：(a) Determine the null and alternative hypotheses that would be used to determine if the filling machine is calibrated correctly. The null hypothesis \(H_{0}: \mu=16\) states that the machine is calibrated correctly and fills the cans with 16 fluid ounces. The alternative hypothesis \(H_{1}: \mu \neq 16\) states that the machine is not calibrated correctly and does not fill the cans with 16 fluid ounces.
Step 2 ：(b) The quality control manager obtains a sample of 85 cans and measures the contents. The sample evidence leads the manager to reject the null hypothesis. This means there is sufficient evidence to conclude that the machine is out of calibration.
Step 3 ：(c) Suppose, in fact, the machine is not out of calibration. This means a Type I error has been made since the sample evidence led the quality-control manager to reject the null hypothesis, when the null hypothesis is true. A Type I error is made when the null hypothesis is true but is rejected. In this case, the machine is not out of calibration (which means the null hypothesis is true), but the sample evidence led the quality-control manager to reject the null hypothesis. Therefore, a Type I error has been made. Final Answer: \(\boxed{\text{Type I}}\)
