A factory makes propeller drive shafts for ships. A quality assurance engineer at the factory needs to estimate the true mean length of the shafts. She randomly selects four drive shafts made at the factory, measures their lengths, and finds their sample mean to be 1000 mm. The lengths are known to have a normal distribution with a standard deviation is 2 mm.
Calculate a 95% confidence interval for the true mean length of the shafts.

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joaobezerra joaobezerra · Answer 1 · 2020-03-17T15:01:20+01:00

Answer:

Step-by-step explanation:

We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:

[tex]\alpha = \frac{1-0.95}{2} = 0.025[/tex]

Now, we have to find z in the Ztable as such z has a pvalue of [tex]1-\alpha[/tex].

So it is z with a pvalue of [tex]1-0.025 = 0.975[/tex], so [tex]z = 1.96[/tex]

Now, find M as such

[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]

In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.

[tex]M = 1.96*\frac{2}{\sqrt{4}}[/tex]

The lower end of the interval is the sample mean subtracted by M. So it is 1000 - 1.96 = 998.04 mm

The upper end of the interval is the sample mean added to M. So it is 1000 + 1.96 = 1001.96 mm