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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.

We have , n = 20, $space top enclose straight x space equals space 10 comma space space space straight sigma space equals space 2 comma$

Now, $space space space space top enclose straight x space equals space fraction numerator begin display style sum from blank to blank of straight x end style over denominator straight n end fraction equals fraction numerator begin display style sum from blank to blank of straight x end style over denominator 20 end fraction equals 10$ $rightwards double arrow$ $sum from blank to blank of straight x space equals space 20 space cross times space 10 space equals space 200$
So, wrong $left parenthesis straight sigma squared right parenthesis space equals space fraction numerator straight n begin display style sum from blank to blank of end style straight x squared minus left parenthesis begin display style sum from blank to blank of end style straight x right parenthesis squared over denominator straight n squared end fraction$

$rightwards double arrow$ $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

$rightwards double arrow$ $stack space sum with blank below and blank on top straight x squared space equals space fraction numerator 1600 plus 40000 over denominator 20 end fraction equals 41600 over 20 equals 2080$

(a) Let $stack space space sum with blank below and blank on top straight X$ be the correct sum of observations
So, $sum from blank to blank of straight X space equals space 200 minus 8 space equals space 192$
Now, $n subscript italic 1$ (number of observations) = 19
So, correct mean $top enclose straight X space equals space fraction numerator begin display style sum from blank to blank of end style straight X over denominator n subscript italic 1 end fraction equals 192 over 19 equals 10.10$
Let $sum from blank to blank of straight X squared$ be the correct sum of squares of the observations = $sum from blank to blank of straight x squared minus 64 equals 2080 minus 64 equals 2016$
So, Correct $space space space straight sigma space equals space square root of fraction numerator straight n subscript 1 begin display style sum from blank to blank of end style straight X squared minus left parenthesis begin display style sum from blank to blank of end style straight X right parenthesis squared over denominator straight n subscript 1 squared end fraction end root space equals space square root of fraction numerator 19 cross times 2016 minus left parenthesis 192 right parenthesis squared over denominator 361 end fraction end root equals square root of 1440 over 361 end root equals 1.997$

(b) Let $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ be correct sum of observation if entry 8 is replaced by 12,
then $sum from blank to blank of straight y space equals space sum from blank to blank of straight x minus 8 plus 12 equals 200 plus 4 equals 204$
and $sum from blank to blank of straight y squared$ is correct sum of the squares of the observations,
then $sum from blank to blank of straight y squared space equals space sum from blank to blank of straight x squared plus 12 squared minus 8 squared equals 2080 plus 144 minus 64 equals 2160$

Now, correct mean, $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

Correct, $straight sigma squared space equals space fraction numerator straight n begin display style sum from blank to blank of end style straight y squared minus left parenthesis begin display style sum from blank to blank of end style straight y right parenthesis squared over denominator straight n squared end fraction equals fraction numerator 20 cross times 2160 minus left parenthesis 204 right parenthesis squared over denominator left parenthesis 20 right parenthesis squared end fraction equals 1584 over 400$

So, correct $straight sigma space equals space square root of 1584 over 400 end root space equals space fraction numerator square root of 1584 over denominator 20 end fraction equals 1.99.$

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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.

137 Views

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, $left parenthesis straight a not equal to 0 right parenthesis.$

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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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