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for the following situation, find the mean and standard deviation of th…

Question

for the following situation, find the mean and standard deviation of the population. list all samples (with replacement) of the given size from that population and find the mean of each. find the mean and standard deviation of the sampling distribution and compare them with the mean and standard deviation of the population. the word counts of five essays are 508, 639, 552, 612, and 575. use a sample size of 2. identify all samples of size 2 with the correct accompanying means below. a. 575, 508, $\bar{x}=541.5$ b. 612, 612, $\bar{x}=612$ c. 612, 575, $\bar{x}=593.5$ d. 612, 552, $\bar{x}=582$ e. 552, 639, $\bar{x}=595.5$ f. 508, 639, $\bar{x}=573.5$ g. 508, 552, $\bar{x}=530$ h. 639, 508, $\bar{x}=573.5$ i. 552, 508, $\bar{x}=530$ j. 639, 575, $\bar{x}=607$ k. 575, 552, $\bar{x}=563.5$ l. 575, 640, $\bar{x}=607.5$ m. 575, 575, $\bar{x}=575$ n. 508, 612, $\bar{x}=560$ o. 639, 552, $\bar{x}=595.5$ p. 552, 575, $\bar{x}=563.5$ q. 575, 612, $\bar{x}=593.5$ r. 508, 575, $\bar{x}=541.5$ s. 612, 639, $\bar{x}=625.5$ t. 639, 612, $\bar{x}=625.5$ u. 639, 639, $\bar{x}=639$ v. 612, 639, $\bar{x}=559.5$ w. 552, 552, $\bar{x}=552$ x. 612, 508, $\bar{x}=560$ y. 552, 612, $\bar{x}=582$ z. 575, 639, $\bar{x}=607$ . 508, 508, $\bar{x}=508$

Explanation:

Step1: Calculate sample mean formula

The formula for the mean of a sample of size $n = 2$ (two - value sample) is $\bar{x}=\frac{x_1 + x_2}{2}$, where $x_1$ and $x_2$ are the values in the sample.

Step2: Check each option

  • Option A: $\bar{x}=\frac{575 + 508}{2}=\frac{1083}{2}=541.5$
  • Option B: $\bar{x}=\frac{612+612}{2}=612$
  • Option C: $\bar{x}=\frac{612 + 575}{2}=\frac{1187}{2}=593.5$
  • Option D: $\bar{x}=\frac{612+552}{2}=\frac{1164}{2}=582$
  • Option E: $\bar{x}=\frac{552 + 639}{2}=\frac{1191}{2}=595.5$
  • Option F: $\bar{x}=\frac{508+639}{2}=\frac{1147}{2}=573.5$
  • Option G: $\bar{x}=\frac{508 + 552}{2}=\frac{1060}{2}=530$
  • Option H: $\bar{x}=\frac{639+508}{2}=\frac{1147}{2}=573.5$
  • Option I: $\bar{x}=\frac{552 + 508}{2}=\frac{1060}{2}=530$
  • Option J: $\bar{x}=\frac{639+575}{2}=\frac{1214}{2}=607$
  • Option K: $\bar{x}=\frac{575+552}{2}=\frac{1127}{2}=563.5$
  • Option L: There is no 640 in the population, so this option is incorrect.
  • Option M: $\bar{x}=\frac{575+575}{2}=575$
  • Option N: $\bar{x}=\frac{508+612}{2}=\frac{1120}{2}=560$
  • Option O: $\bar{x}=\frac{639+552}{2}=\frac{1191}{2}=595.5$
  • Option P: $\bar{x}=\frac{552+575}{2}=\frac{1127}{2}=563.5$
  • Option Q: $\bar{x}=\frac{575+612}{2}=\frac{1187}{2}=593.5$
  • Option R: $\bar{x}=\frac{508+575}{2}=\frac{1083}{2}=541.5$
  • Option S: $\bar{x}=\frac{612+639}{2}=\frac{1251}{2}=625.5$
  • Option T: $\bar{x}=\frac{639+612}{2}=\frac{1251}{2}=625.5$
  • Option U: $\bar{x}=\frac{639+639}{2}=639$
  • Option V: There is a calculation error, $\bar{x}=\frac{612 + 639}{2}=625.5

eq559.5$

  • Option W: $\bar{x}=\frac{552+552}{2}=552$
  • Option X: $\bar{x}=\frac{612+508}{2}=\frac{1120}{2}=560$
  • Option Y: $\bar{x}=\frac{552+612}{2}=\frac{1164}{2}=582$
  • Option Z: $\bar{x}=\frac{575+639}{2}=\frac{1214}{2}=607$
  • Option [. $\bar{x}=\frac{508+508}{2}=508$

The correct samples and their means are: A. 575, 508, $\bar{x}=541.5$; B. 612, 612, $\bar{x}=612$; C. 612, 575, $\bar{x}=593.5$; D. 612, 552, $\bar{x}=582$; E. 552, 639, $\bar{x}=595.5$; F. 508, 639, $\bar{x}=573.5$; G. 508, 552, $\bar{x}=530$; H. 639, 508, $\bar{x}=573.5$; I. 552, 508, $\bar{x}=530$; J. 639, 575, $\bar{x}=607$; K. 575, 552, $\bar{x}=563.5$; M. 575, 575, $\bar{x}=575$; N. 508, 612, $\bar{x}=560$; O. 639, 552, $\bar{x}=595.5$; P. 552, 575, $\bar{x}=563.5$; Q. 575, 612, $\bar{x}=593.5$; R. 508, 575, $\bar{x}=541.5$; S. 612, 639, $\bar{x}=625.5$; T. 639, 612, $\bar{x}=625.5$; U. 639, 639, $\bar{x}=639$; W. 552, 552, $\bar{x}=552$; X. 612, 508, $\bar{x}=560$; Y. 552, 612, $\bar{x}=582$; Z. 575, 639, $\bar{x}=607$; [. 508, 508, $\bar{x}=508$

Answer:

A. 575, 508, $\bar{x}=541.5$; B. 612, 612, $\bar{x}=612$; C. 612, 575, $\bar{x}=593.5$; D. 612, 552, $\bar{x}=582$; E. 552, 639, $\bar{x}=595.5$; F. 508, 639, $\bar{x}=573.5$; G. 508, 552, $\bar{x}=530$; H. 639, 508, $\bar{x}=573.5$; I. 552, 508, $\bar{x}=530$; J. 639, 575, $\bar{x}=607$; K. 575, 552, $\bar{x}=563.5$; M. 575, 575, $\bar{x}=575$; N. 508, 612, $\bar{x}=560$; O. 639, 552, $\bar{x}=595.5$; P. 552, 575, $\bar{x}=563.5$; Q. 575, 612, $\bar{x}=593.5$; R. 508, 575, $\bar{x}=541.5$; S. 612, 639, $\bar{x}=625.5$; T. 639, 612, $\bar{x}=625.5$; U. 639, 639, $\bar{x}=639$; W. 552, 552, $\bar{x}=552$; X. 612, 508, $\bar{x}=560$; Y. 552, 612, $\bar{x}=582$; Z. 575, 639, $\bar{x}=607$; [. 508, 508, $\bar{x}=508$