Question

A manufacturer of interocular lenses is qualifying a new grinding machine and will qualify the machine if there is evidence that the percentage of polished lenses that contain surface defects does not exceed 2%. A random sample of 250 lenses contains six defective lenses.

Formulate and test an appropriate set of hypotheses to determine if the machine can be qualified. Use α = 0.05.
What is the probability of making a type I error?
What would be the result of making a type II error?

Answer 1

In this scenario, we formulate and test hypotheses to determine if a new grinding machine can be qualified based on the percentage of polished lenses with surface defects.

In this scenario, we are trying to determine if the new grinding machine can be qualified by evaluating the percentage of polished lenses that contain surface defects.

The manufacturer sets a criterion that the percentage should not exceed 2%. We can set up the hypotheses as follows:

Null hypothesis (H0): The percentage of lenses with surface defects is equal to or less than 2%.

Alternative hypothesis (H1): The percentage of lenses with surface defects is greater than 2%.

To test these hypotheses, we can use a hypothesis test for a proportion.

By calculating the test statistic and obtaining the p-value, we can make a decision about whether to reject or fail to reject the null hypothesis.