Suppose a hardware manufacturer is checking its nails to make sure they are of the right length. A quality control investigator collects a sample of 100 nails and measures their lengths, finding that their mean is 2.000cm with a sample standard deviation of 0.002cm. Suppose the investigator knows that nearly all of the nail population produced will be within 2 standard deviations. What will be the most likely upper bound on the length of a randomly chosen nail from all nails manufactured by the company?

The upper bound on the length of a randomly chosen nail from all nails manufactured by the company is 2.004 cm.

Step-by-step explanation:

According to the Central Limit Theorem if we have an unknown population with mean μ and standard deviation σ and appropriately huge random samples (n > 30) are selected from the population with replacement, then the distribution of the sample mean will be approximately normally distributed.  

Then, the mean of the sample means is given by,

And the standard deviation of the sample means (also known as the standard error) is given by,

In this case the sample of nails selected is quite large, i.e. n = 100 > 30.

So, the sampling distribution of sample mean length of nails will be approximately normal.

Then according to the Empirical rule, 95% of the normal distribution is contained in the range,

Compute the upper bound as follows:

Thus, the upper bound on the length of a randomly chosen nail from all nails manufactured by the company is 2.004 cm.

step-by-step explanation:

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