Elementary Statistics: A Step-by-Step Approach with Formula Card 9th Edition

Published by McGraw-Hill Education
ISBN 10: 0078136334
ISBN 13: 978-0-07813-633-7

Chapter 7 - Confidence Intervals and Sample - 7-1 Confidence Intervals for the Mean When σ Is Known - Exercises 7-1 - Page 379: 12

Answer

a. 7.2 b. $6.6\lt\mu\lt7.8$ c. $6.4\lt\mu\lt8.0$ d. the 99% confidence, see explanations below.

Work Step by Step

Given $\bar X=7.2, \sigma=2.1,n=50$, a. The best point estimate of the population mean is the sample mean 7.2 b. At the 95% confidence, the critical z-value is $z_{\alpha/2}=1.96$ and the error can be found as $E=1.96×\frac{2.1}{\sqrt {50}}=0.58$ so the interval of the mean can be found as $\bar X−E\lt\mu\lt\bar X+E$ which gives $6.6\lt\mu\lt7.8$ c. At the 99% confidence, the critical z-value is $z_{\alpha/2}=2.575$ and the error can be found as $E=2.575\times\frac{2.1}{\sqrt {50}}=0.76$ so the interval of the mean can be found as $\bar X−E\lt\mu\lt\bar X+E$ which gives $6.4\lt\mu\lt8.0$ d. The interval for the 99% confidence is larger because a higher confidence level requires a higher critical z-value and a larger error tolerance.
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