a manufacturing plant uses 3000 light bulbs whose lifetimes are independently normally distributed with mean 500 hrs, and standard deviation 50 hrs. to minimize the number of bulbs that burn out during production hours, all bulbs are replaced after given period of operation. how often should the bulbs be replaced if you want no more than 1% to burn out between replacements?
Should I just look up 1% under the z-distribution and find corresponding z-score to it, and with that find the X-value in hours?
Yes.
Yes, you are on the right track! To determine how often the bulbs should be replaced in order to have no more than 1% burn out between replacements, you can use the z-score to find the corresponding lifetime value.
First, let's find the z-score corresponding to the 1% burnout rate. The burnout rate of 1% corresponds to the area under the normal distribution curve to the left of the z-score. You can use a z-table or a statistical software to find the z-score.
Using a standard normal distribution table or calculator, you will find that the z-score corresponding to the 1% burnout rate is approximately -2.33. This means that 1% of the bulbs will burn out before approximately -2.33 standard deviations from the mean.
Next, you need to find the corresponding X-value in hours. Since the mean lifetime of the bulbs is 500 hours and the standard deviation is 50 hours, you can use the formula:
X = mean + (z-score * standard deviation)
X = 500 + (-2.33 * 50) = 376.5
Therefore, if you want no more than 1% of the bulbs to burn out between replacements, you should replace the bulbs after approximately 376.5 hours of operation.