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# A machine produces parts with lengths that are normally distributed with displaystylesigma={0.66}. A sample of 20 parts has a mean length of 75.98. a) Give a point estimate for displaystylemu. (Give your answer correct to two decimal places.) b) Find the 95% confidence maximum error of estimate for displaystylemu. (Give your answer correct to two decimal places.) c) Find the 95% confidence interval for displaystylemu. (Give your answer correct to two decimal places.) Lower limit displaystyle=? Upper limit displaystyle=? # A machine produces parts with lengths that are normally distributed with displaystylesigma={0.66}. A sample of 20 parts has a mean length of 75.98. a) Give a point estimate for displaystylemu. (Give your answer correct to two decimal places.) b) Find the 95% confidence maximum error of estimate for displaystylemu. (Give your answer correct to two decimal places.) c) Find the 95% confidence interval for displaystylemu. (Give your answer correct to two decimal places.) Lower limit displaystyle=? Upper limit displaystyle=?

Question
Confidence intervals asked 2020-12-12
A machine produces parts with lengths that are normally distributed with $$\displaystyle\sigma={0.66}$$. A sample of 20 parts has a mean length of 75.98.
a)
Give a point estimate for $$\displaystyle\mu$$. (Give your answer correct to two decimal places.)
b)
Find the $$95\%$$ confidence maximum error of estimate for $$\displaystyle\mu$$. (Give your answer correct to two decimal places.)
c)
Find the $$95\%$$ confidence interval for $$\displaystyle\mu$$. (Give your answer correct to two decimal places.)
Lower limit $$\displaystyle=?$$
Upper limit $$\displaystyle=?$$

## Answers (1) 2020-12-13
Step 1
Given:
A machine produces parts with lengths that are normally distributed with $$\displaystyle\sigma={0.66}$$. A sample of 20 parts has a mean length of 75.98.
To Determine:
a) A point estimate for $$\displaystyle\mu.$$
b) The $$95\%$$ confidence maximum error of estimate for $$\displaystyle\mu.$$
c) The $$95\%$$ confidence interval for $$\displaystyle\mu$$
Step 2
Solution:
A machine produces parts with lengths that are normally distributed with the given values,
$$\displaystyle\sigma={0.66},{n}={20},\overline{{x}}={75.98}$$
a) A point estimate for $$\displaystyle\mu$$ is
$$\displaystyle\overline{{\mu}}=\overline{{x}}={75.98}$$
b) For $$95\%$$ confidence intervals,
Z - critical value is 1.96.
So, The maximum confidence error of estimate for $$\displaystyle\mu$$ is,
$$\displaystyle{M}.{E}.=\frac{{{Z}\times\sigma}}{\sqrt{{{n}}}}$$
$$\displaystyle{M}.{E}.=\frac{{{1.96}\times{0.66}}}{\sqrt{{{20}}}}$$
$$\displaystyle{M}.{E}.=\frac{{{1.96}\times{0.66}}}{{4.4721}}$$
$$\displaystyle{M}.{E}.=\frac{1.2936}{{4.4721}}={0.289260}$$
c) The $$95\%$$ confidence interval for $$\displaystyle\mu$$,
Lower Limit $$\displaystyle=\overline{{{x}}}-{M}.{E}.$$
L.L.,
$$\displaystyle={75.98}-{0.289260}$$
$$\displaystyle={75.69074}$$
Upper Limit $$\displaystyle=\overline{{{x}}}+{M}.{E}.$$
U.L.,
$$\displaystyle={75.98}+{0.289260}$$
$$\displaystyle={76.26926}$$

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