Question

The thickness (in millimeters) of the coating applied to hard drives is one characteristic that determines the usefulness of the product. When no unusual circumstances are present, the thickness (x) has a normal distribution with a mean of 3 mm and a standard deviation of 0.05 mm. Suppose that the process will be monitored by selecting a random sample of 16 drives from each shift's production and determining $\bar{x}$, the mean coating thickness for the sample. (a) Describe the sampling distribution of $\bar{x}$ for a random sample of size 16. The distribution of $\bar{x}$ is normal with mean 3 mm and standard deviation 0.0125 mm. (b) When no unusual circumstances are present, we expect $\bar{x}$ to be within $2\sigma_{\bar{x}}$ of 3 mm, the desired value. An $\bar{x}$ value farther from 3 mm than $2\sigma_{\bar{x}}$ is interpreted as an indication of a problem that needs attention. Calculate $3 \pm 2\sigma_{\bar{x}}$. $3 - 2\sigma_{\bar{x}} = 2.9$ mm $3 + 2\sigma_{\bar{x}} = 3.025$ mm (c) Referring to part (b), what is the probability that a sample mean will be outside $3 \pm 2\sigma_{\bar{x}}$ just by chance (that is, when there are no unusual circumstances)? (Round your answer to four decimal places.) You may need to use the appropriate table in the appendix to answer this question.

          The thickness (in millimeters) of the coating applied to hard drives is one characteristic that determines the usefulness of the product. When no unusual circumstances are present, the thickness (x) has a normal distribution with a mean of 3 mm and a standard deviation of 0.05 mm. Suppose that the process will be monitored by selecting a random sample of 16 drives from each shift's production and determining $\bar{x}$, the mean coating thickness for the sample.
(a) Describe the sampling distribution of $\bar{x}$ for a random sample of size 16.
The distribution of $\bar{x}$ is normal with mean 3 mm and standard deviation 0.0125 mm.
(b) When no unusual circumstances are present, we expect $\bar{x}$ to be within $2\sigma_{\bar{x}}$ of 3 mm, the desired value. An $\bar{x}$ value farther from 3 mm than $2\sigma_{\bar{x}}$ is interpreted as an indication of a problem that needs attention. Calculate $3 \pm 2\sigma_{\bar{x}}$.
$3 - 2\sigma_{\bar{x}} = 2.9$ mm
$3 + 2\sigma_{\bar{x}} = 3.025$ mm
(c) Referring to part (b), what is the probability that a sample mean will be outside $3 \pm 2\sigma_{\bar{x}}$ just by chance (that is, when there are no unusual circumstances)? (Round your answer to four decimal places.)
You may need to use the appropriate table in the appendix to answer this question.

The thickness (in millimeters) of the coating applied to hard drives is one characteristic that determines the usefulness of the product. When no unusual circumstances are present, the thickness (x) has a normal distribution with a mean of 3 mm and a standard deviation of 0.05 mm. Suppose that the process will be monitored by selecting a random sample of 16 drives from each shift's production and determining x̅, the mean coating thickness for the sample.
(a) Describe the sampling distribution of x̅ for a random sample of size 16.
The distribution of x̅ is normal with mean 3 mm and standard deviation 0.0125 mm.
(b) When no unusual circumstances are present, we expect x̅ to be within 2σx̅ of 3 mm, the desired value. An x̅ value farther from 3 mm than 2σx̅ is interpreted as an indication of a problem that needs attention. Calculate 3 ± 2σx̅.
3 - 2σx̅ = 2.9 mm
3 + 2σx̅ = 3.025 mm
(c) Referring to part (b), what is the probability that a sample mean will be outside 3 ± 2σx̅ just by chance (that is, when there are no unusual circumstances)? (Round your answer to four decimal places.)
You may need to use the appropriate table in the appendix to answer this question.

Added by Nicholas H.

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