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Chapter 7: Q2E (page 284)

Each of the following is a confidence interval for \({\rm{\mu = }}\) true average (i.e., population mean) resonance frequency (Hz) for all tennis rackets of a certain type: \({\rm{(114}}{\rm{.4,115}}{\rm{.6)(114}}{\rm{.1,115}}{\rm{.9)}}\) a. What is the value of the sample mean resonance frequency? b. Both intervals were calculated from the same sample data. The confidence level for one of these intervals is \({\rm{90\% }}\) and for the other is \({\rm{99\% }}\). Which of the intervals has the \({\rm{90\% }}\) confidence level, and why?

Short Answer

Expert verified

(a) The value of sample mean resonance is \({\rm{115}}\).

(b) Interval \({\rm{(114}}{\rm{.4,115}}{\rm{.6)}}\) as narrower.

Step by step solution

01

Define frequency

The number of times a repeated event occurs per unit of time is known as frequency. It's also referred to as temporal frequency to distinguish it from spatial frequency, and ordinary frequency to distinguish it from angular frequency.

02

Explanation


(a) When a normal population is given,

\({\rm{100(1 - \alpha )\% confidence interval}}\)

the mean is calculated using,

\(\left( {{\rm{\bar x - }}{{\rm{z}}_{{\rm{\alpha /2}}}}{\rm{ \times }}\frac{{\rm{\sigma }}}{{\sqrt {\rm{n}} }}{\rm{,\bar x + }}{{\rm{z}}_{{\rm{\alpha /2}}}}{\rm{ \times }}\frac{{\rm{\sigma }}}{{\sqrt {\rm{n}} }}} \right)\)

when it is known what the value\({{\rm{\sigma }}^{\rm{2}}}\)is.

So, because sample mean is the middle of the provided interval,

\(\begin{array}{c}\left( {{\rm{\bar x + }}{{\rm{z}}_{{\rm{\alpha /2}}}}{\rm{ \times }}\frac{{\rm{\sigma }}}{{\sqrt {\rm{n}} }}{\rm{ + \bar x - }}{{\rm{z}}_{{\rm{\alpha /2}}}}{\rm{ \times }}\frac{{\rm{\sigma }}}{{\sqrt {\rm{n}} }}} \right){\rm{/2 = }}\frac{{{\rm{2\bar x}}}}{{\rm{2}}}\\{\rm{ = \bar x}}{\rm{.}}\end{array}\)

As a result, the sample mean,

\(\begin{array}{c}{\rm{\bar x = }}\frac{{{\rm{114}}{\rm{.4 + 115}}{\rm{.6}}}}{{\rm{2}}}\\{\rm{ = 115}}\\{\rm{ = }}\frac{{{\rm{114}}{\rm{.1 + 115}}{\rm{.9}}}}{{\rm{2}}}\end{array}\)

Therefore, the value is \({\rm{115}}\).

03

Explanation

(b) The \({\rm{90}}\) percent confidence level refers to the narrower confidence interval \({\rm{(114}}{\rm{.4,115}}{\rm{.6)}}\), because the broader the interval, the greater the probability (the higher confidence level).

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Most popular questions from this chapter

A journal article reports that a sample of size 5 was used as a basis for calculating a 95% CI for the true average natural frequency (Hz) of delaminated beams of a certain type. The resulting interval was (229.764, 233.504). You decide that a confidence level of 99% is more appropriate than the 95% level used. What are the limits of the 99% interval? (Hint: Use the center of the interval and its width to determine \(\overline x \) and s.)

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The following observations are lifetimes (days) subsequent to diagnosis for individuals suffering from blood cancer (鈥淎 Goodness of Fit Approach to the Class of Life Distributions with Unknown Age,鈥� Quality and Reliability Engr. Intl., \({\rm{2012: 761--766):}}\)

\(\begin{array}{*{20}{l}}{{\rm{115 181 255 418 441 461 516 739 743 789 807}}}\\{{\rm{865 924 983 1025 1062 1063 1165 1191 1222 1222 1251}}}\\{{\rm{1277 1290 1357 1369 1408 1455 1478 1519 1578 1578 1599}}}\\{{\rm{1603 1605 1696 1735 1799 1815 1852 1899 1925 1965}}}\end{array}\)

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