/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q 9E Suppose that each of two statist... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Q 9E (page 479)

Suppose that each of two statisticians, A and B, independently takes a random sample of 20 observations from the normal distribution with unknown mean 渭 and known variance 4. Suppose also that statistician A finds the sample variance in his random sample to be 3.8, and statistician B finds the sample variance in her random sample to be 9.4. For which random sample is the sample mean likely to be closer to the unknown value of 渭?

Short Answer

Expert verified

Both are equally close to the unknown value of 渭.

Step by step solution

01

Given information

Two statisticians, A and B, independently takes a random sample of 20 observations from the normal distribution with unknown mean 渭 and known variance 4 and statistician A finds the sample variance in his random sample to be 3.8, and statistician B finds the sample variance in her random sample to be 9.4. Need to check which random sample is the sample mean likely to be closer to the unknown value of 渭.

02

Checking of which random sample is the sample mean likely to be closer to the unknown value of μ.

Theorem: Independence of Sample mean and sample variance:

Let \({X_1},...,{X_n}\) be a random sample from the normal distribution with mean 渭 and variance \({\sigma ^2}\). Then the sample mean \({\bar X_n} = \frac{1}{n}\sum\limits_{i = 1}^n {{X_i}} \) and sample variance \({\sigma ^2} = \frac{1}{n}\sum\limits_{i = 1}^n {{{\left( {{X_i} - {{\bar X}_n}} \right)}^2}} \) are independent random variables.\(\bar X\) has the normal distribution with mean \(\mu \) and sample variance \(\frac{{{\sigma ^2}}}{n}\)

The theorem indicates the sample mean and sample variance are independent here .So, irrespective of the value of the sample variance the result is the same.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{, }}{\bf{. }}{\bf{. }}{\bf{. , }}{{\bf{X}}_{\bf{n}}}\) form a random sample from the normal distribution with unknown mean \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\), and also that the joint prior distribution of \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\) is the normal-gamma distribution satisfying the following conditions: \({\bf{E}}\left( {\bf{\tau }} \right){\bf{ = 1,Var}}\left( {\bf{\tau }} \right){\bf{ = }}\frac{{\bf{1}}}{{\bf{3}}}\,\,\,{\bf{and}}\,\,{\bf{Pr}}\left( {{\bf{\mu > 3}}} \right){\bf{ = 0}}{\bf{.5}}\,\,{\bf{and}}\,\,{\bf{Pr}}\left( {{\bf{\mu > 0}}{\bf{.12}}} \right){\bf{ = 0}}{\bf{.9}}\,\)

Determine the prior hyper parameters \({{\bf{\mu }}_{\bf{0}}}{\bf{,}}{{\bf{\lambda }}_{\bf{0}}}{\bf{,}}{{\bf{\alpha }}_{\bf{0}}}{\bf{,}}{{\bf{\beta }}_{\bf{0}}}\)

Suppose that\({X_1}...{X_n}\)form a random sample from the normal distribution with mean 0 and unknown standard deviation\(\sigma > 0\). Find the lower bound specified by the information inequality for the variance of any unbiased estimator of\(\log \sigma \).

Show that two random variables \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\)cannot have the joint normal-gamma distribution such that

\({\bf{E}}\left( {\bf{\mu }} \right){\bf{ = 0}}\,\,{\bf{,Var}}\left( {\bf{\mu }} \right){\bf{ = 1,E}}\left( {\bf{\tau }} \right){\bf{ = }}\frac{{\bf{1}}}{{\bf{2}}}\,\,{\bf{and}}\,\,{\bf{Var}}\left( {\bf{\tau }} \right){\bf{ = }}\frac{{\bf{1}}}{{\bf{4}}}\)

Suppose that a random variableXhas the normal distributionwith mean渭and precision\(\tau \). Show that the random variable\({\bf{Y = aX + b}}\;\left( {{\bf{a}} \ne {\bf{0}}} \right)\)has the normal distribution with mean补渭+band precision\(\frac{\tau }{{{{\bf{a}}^{\bf{2}}}}}\).

Sketch the p.d.f. of the\({{\bf{\chi }}^{\bf{2}}}\)distribution withmdegrees of freedom for each of the following values ofm. Locate the mean, the median, and the mode on each sketch. (a)m=1;(b)m=2; (c)m=3; (d)m=4.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.