The quality control manager at a light bulb factory needs to estimate the mean life of a large shipment of light bulbs. The manufacturer's specifications are that the standard deviations is 100 hours. A random sample of 64 light bulbs indicated a sample mean life of 350 hours.

With you limited data, the best estimate is 350 hours.

If you want a probability statement,

P = mean ± Z (SEm)

SEm = SD/√n

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability
(P) and its Z score.

To estimate the mean life of the entire shipment of light bulbs, the quality control manager can use the sample mean as an estimate of the population mean. This estimate is considered a point estimate.

In this case, the sample mean life of the 64 light bulbs is 350 hours. However, to determine the precision of this estimate, we can calculate the margin of error, which helps determine the range within which the true population mean is likely to fall.

To calculate the margin of error, we need to use the formula:

Margin of Error = Critical Value * Standard Error

In this case, the critical value is based on the desired confidence level. Let's assume a 95% confidence level, which is a commonly used confidence level in statistics. To find the critical value, we can use a standard normal distribution table, or use a calculator or software.

The standard error represents the standard deviation of the sample mean and can be calculated using the formula:

Standard Error = Standard Deviation / sqrt(sample size)

In this case, the standard deviation is 100 hours, and the sample size is 64.

Let's calculate the margin of error:

Standard Error = 100 / sqrt(64) = 100 / 8 = 12.5

To find the critical value for a 95% confidence level, we consult a standard normal distribution table or use a calculator, which gives us a value of approximately 1.96.

Margin of Error = 1.96 * 12.5 = 24.5

So, the margin of error is 24.5 hours. This means that we can be 95% confident that the true population mean of the light bulb life falls within a range of 350 ± 24.5 hours, or between 325.5 and 374.5 hours.

Therefore, the quality control manager can estimate that the mean life of the entire shipment of light bulbs is likely to be within this range based on the information from the sample.