A manufacturer produces high-precision pressure sensors. Historical quality control data indicates that 1 in 80 sensors produced has a calibration error.
In a random sample of 800 sensors, determine:
(i) the mean number of sensors with a calibration error,
(ii) the standard deviation of the number of sensors with a calibration error, giving your answer to 3 decimal places.
A quality analyst believes that, due to new maintenance protocols, the actual proportion of sensors with calibration errors is lower than the historical data suggests.
A random sample of 1200 sensors is taken and 8 are found to have a calibration error.
A test of the analyst's claim is to be carried out at the 5% level of significance.
(i) State the hypotheses for this test.
(ii) Using a suitable approximation, carry out the test.
It is further claimed that 15% of sensors with calibration errors also possess a specific micro-fracture defect.
To test this claim, a random sample of n n\,n sensors with calibration errors is inspected. The random variable Y Y\,Y represents the number of sensors in the sample that possess the micro-fracture defect.
A two-tailed test, at the 5% level of significance, is carried out to determine if the proportion differs from 15%.
The critical region for the test is Y=0Y = 0Y=0 or Y≥wY \ge wY≥w.
Find the smallest possible value of n n\,n and the corresponding value of www.