The lifetimes of a certain brand of light bulbs are known to be normally distributed with a mean of 1600 hours and a standard deviation of 400 hours. A random sample of 64 of these light bulbs is taken. The probability is 0.10 that the sample mean lifetime differs from the population mean lifetime by at least how many hours?
To find the difference between the sample mean and the population mean that corresponds to a probability of 0.10, we need to find the z-score that corresponds to this probability.
First, we calculate the standard error of the mean:
Standard Error = σ / sqrt(n)
Standard Error = 400 / sqrt(64)
Standard Error = 50
Now, we need to find the z-score that corresponds to a probability of 0.90 (1-0.10) on the standard normal distribution. We can use a z-table or a calculator to find that the z-score is approximately 1.28.
Finally, we convert the z-score back to the difference in hours:
Difference in hours = z-score * Standard Error
Difference in hours = 1.28 * 50
Difference in hours = 64
Therefore, the sample mean lifetime differs from the population mean lifetime by at least 64 hours with a probability of 0.10.