Question

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?

Answer 1

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

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,

`\mu_(\bar x)=\bar x`

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

`\sigma_(\bar x)=(\sigma)/(√(n))`

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,

`\mu\pm 2\cdot (s)/(√(n))`

Compute the upper bound as follows:

`\text{Upper Bound}=\mu\pm 2\cdot (s)/(√(n))`

`=2+(2*(0.002)/(√(100)))\\\\=2+0.0004\\\\=2.004`

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