An LED lamp manufacturer guarantees that the mean life of a certain type of LED lamp is at least 25,000 hours. A random sample of 49 LED lamps has a mean life of 24,800 hours. Assume the population is normally distributed and the population standard deviation is 500 hours. At α = 0.05, do you have enough evidence to reject the manufacturer’s claim?

(a) identify the claim and state H₀ and H_α.
(b) find the standardized test statistic z.
(c) find the corresponding P-value.
(d) decide whether to reject or fail to reject the null hypothesis.
(e) interpret the decision in the context of the original claim.