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Statistics

Long Answer Type

491. The diameters of circles (in mm) drawn in a design are given below:

Calculate the standard deviation and mean diameter of the circles.

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492. The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation 8 was incorrect. Calculate the correct mean and S.D. in each of the following cases: (i) If the wrong item is omitted (ii) If it is replaced by 12.

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Short Answer Type

493.

If the observations x₁, x₂,........ x_n are changed to x₁ + a, x₂ + a, x₃ + a, x_n+ a, where a is a negative or positive number, show that the variance remains unchanged.

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Long Answer Type

494. The mean of 5 observations is 4.4 and their variance is 8.24. If three of the observations are 1, 2 and 6. Find the other two observations.

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495.
Given that $top enclose straight x$ is the mean and $straight sigma squared$ is the variance of n observations $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ Prove that the mean and variance of observations $ax subscript 1 comma space ax subscript 2 comma space ax subscript 3 comma space....... comma space ax subscript straight n$ are $straight a space top enclose straight x space and space straight a squared space straight sigma squared$ respectively, $left parenthesis straight a not equal to 0 right parenthesis.$

$top enclose straight x space equals space fraction numerator begin display style sum from blank to blank of end style straight x over denominator straight n end fraction space equals space fraction numerator straight x subscript 1 plus straight x subscript 2 plus straight x subscript 3 plus....... plus straight x subscript straight n over denominator straight n end fraction$ ...(i)

Also, $fraction numerator begin display style sum from blank to blank of end style straight x squared over denominator straight n end fraction equals fraction numerator straight x subscript 1 superscript 2 plus straight x subscript 2 superscript 2 plus straight x subscript 3 superscript 2 plus......... plus straight x subscript straight n superscript 2 over denominator straight n end fraction$ ...(ii)

For the new data $straight y equals ax subscript 1 comma space ax subscript 2 comma space ax subscript 3 comma space....... comma space ax subscript straight n$

$rightwards double arrow$ $sum from blank to blank of straight y space equals space ax subscript 1 plus ax subscript 2 plus ax subscript 3 plus......... plus ax subscript straight n$

∴ Mean, $space space space top enclose straight y space equals space fraction numerator ax subscript 1 plus ax subscript 2 plus ax subscript 3 plus...... plus ax subscript straight n over denominator straight n end fraction space equals space fraction numerator straight a left parenthesis straight x subscript 1 plus straight x subscript 2 plus straight x subscript 3 plus... plus straight x subscript straight n right parenthesis over denominator straight n end fraction equals straight a top enclose straight x$
Hence, the new mean, $top enclose straight y space equals space straight a top enclose straight x$ [By using (i)]

Variance of first data, $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
....(iii)
Variance of the new data, $space space space space space space space space space space straight sigma subscript 1 superscript 2 space equals space fraction numerator straight n begin display style sum from blank to blank of end style straight y squared minus left parenthesis begin display style sum from blank to blank of end style straight y right parenthesis squared over denominator straight n squared end fraction$
$<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
$space space space equals space straight a squared open square brackets fraction numerator straight n left parenthesis straight x subscript 1 superscript 2 plus straight x subscript 2 superscript 2 plus straight x subscript 3 superscript 2 plus.... plus straight x subscript straight n superscript 2 right parenthesis space minus space left parenthesis straight x subscript 1 plus straight x subscript 2 plus straight x subscript 3 plus........ plus straight x subscript straight n right parenthesis squared over denominator straight n squared end fraction close square brackets space equals space straight a squared straight sigma squared$
[By using (iii)]
Hence, the variance $straight sigma subscript 1 superscript 2 space equals space straight a squared straight sigma squared$

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496. The mean and standard deviation of a group of 100 observations was found to be 20 and 3 respectively. Later on it was found that three observations were incorrect, which were recorded as 21,21 and 18. Find the standard deviation, if the incorrect observations were omitted.

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