The lifespans of high-performance LED modules tested under extreme thermal stress are recorded. A random sample of 15 modules yields a mean lifespan of 1250 hours1250\text{ hours}1250 hours. The lifespans are known to follow a normal distribution with a population standard deviation of 45 hours45\text{ hours}45 hours.
Estimate the bounds within which 90%90\%90% of the lifespans of individual modules are expected to fall.
Construct a 95%95\%95% confidence interval for the population mean lifespan μ\muμ of these modules.
An engineer asserts that the true mean lifespan of these modules under these conditions is 1280 hours1280\text{ hours}1280 hours.
Evaluate the engineer's assertion based on your result in part (b). Provide a mathematical justification.
