Final answer:
The question involves calculating the proportion of flood lamps that will last for certain durations using the normal distribution characteristics of the product, with a mean of 3,750 hours and a standard deviation of 300 hours. Z-scores are computed to determine the percentages.
Step-by-step explanation:
The question relates to the normal distribution of a brand of flood lamps that has a mean lifetime of 3,750 hours and a standard deviation of 300 hours. To answer the various parts regarding the proportion of lamps that will last for different durations, the standard normal distribution (Z-score) will be used. The Z-score is calculated using the formula Z = (X - μ) / σ, where X is the value of interest, μ is the mean, and σ is the standard deviation.
- (a) For lamps lasting at least 4,000 hours, the Z-score would be (4,000 - 3,750) / 300 = 0.83. We can then look up this Z-score in a standard normal table or use a calculator to find the corresponding percentage.
- (b) To find the proportion between 3,600 and 4,000 hours, calculate the Z-scores for both 3,600 ((3,600 - 3,750) / 300 = -0.50) and 4,000, and find the area under the normal curve between these two Z-scores.
- (c) For lamps lasting less than 3,800 hours, calculate the Z-score (-0.50) and lookup or calculate the cumulative probability.
The calculated proportions will provide the required probabilities for the respective timeframes.