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 x, the mean coating thickness for the sample. (a) Describe the sampling distribution of x for a random sample of size 16. The distribution of x is with mean mm and standard deviation 0.0125 mm. (b) When no unusual circumstances are present, we expect x to be within 2? x of 3 mm, the desired value. An x value farther from 3 mm than 2? x is interpreted as an indication of a problem that needs attention. Calculate 3 ± 2? x. 3 − 2? x = mm 3 + 2? x = mm (c) Referring to part (b), what is the probability that a sample mean will be outside 3 ± 2? x just by chance (that is, when there are no unusual circumstances)? (Round your answer to four decimal places.)

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

the mean coating thickness for the sample.

(a)

Describe the sampling distribution of

for a random sample of size 16.

The distribution of x is with mean mm and standard deviation 0.0125 mm.

(b)

When no unusual circumstances are present, we expect

to be within

2? _x

of 3 mm, the desired value. An

value farther from 3 mm than

2? _x

is interpreted as an indication of a problem that needs attention. Calculate

3 ± 2? _x.

3 − 2? _x

= mm

3 + 2? _x

= mm

(c)

Referring to part (b), what is the probability that a sample mean will be outside

3 ± 2? _x

just by chance (that is, when there are no unusual circumstances)? (Round your answer to four decimal places.)

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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