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Statistics / Statistics for Engineers and Scientists 4 / Chapter 5 / Problem 13SE

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

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A 99% confidence interval for a population mean based on a

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QUESTION:

A 99% confidence interval for a population mean based on a sample of size 64 is computed to be (16.3, 18.7). How large a sample is needed so that a 99% confidence interval will specify the mean to within \(\pm\)1.0?

Equation transcription:

$\pm{}$

Text transcription:

\pm

Questions & Answers

QUESTION:

A 99% confidence interval for a population mean based on a sample of size 64 is computed to be (16.3, 18.7). How large a sample is needed so that a 99% confidence interval will specify the mean to within \(\pm\)1.0?

Equation transcription:

$\pm{}$

Text transcription:

\pm

ANSWER:

Answer:

Step 1 of 1:

In this question, we are asked to find the number of samples needed so that a $99\%$ confidence interval will specify the mean to within $\pm{}1.0$ .

A $99\%$ confidence interval for a population mean based on a sample of size 64 is computed to be $(16.3,\ 18.7)$ .

Let ${X}_{1}...........{X}_{{n}_{x}}$ be a large sample of size ${n}_{X}$ from a population with mean ${\mu{}}_{X}$ and standard deviation ${\sigma{}}_{X}$

then a level $100(1\ -\ \alpha{})\%$ confidence interval for ${\mu{}}_{X}\$ is

$\overline{X\ }\ \pm{}\ {z}_{\alpha{}/2}\times{}\frac{{\sigma{}}_{X}^{\ }}{\sqrt{n}}$ ……………..(1)

Given level of confidence $=\ 99\%$ , sample size $=\ 64$ , and confidence interval $\ =\ (16.3,\ 18.7)$ .

An upper confidence bound = $18.7,$ and lower confidence bound $=\ 16.3$

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