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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.

320 Views

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.

Let $top enclose straight x$ be the mean of the observation x₁, x₂,........ x_n. Then their variances is given by
$space space space space space space straight sigma subscript 1 superscript 2 space equals space 1 over straight n space sum from straight i equals 1 to straight n of space left parenthesis straight x minus top enclose straight x right parenthesis squared$
If a is added to each observation, then the mean of new observations is given by $top enclose straight X space equals space top enclose straight x plus straight a comma$ and therefore the new variance is given by
 $straight sigma subscript 2 superscript 2 space equals space 1 over straight n sum from straight i equals 1 to straight n of left parenthesis straight x subscript straight i plus straight a minus top enclose straight X right parenthesis squared space equals space 1 over straight n sum from straight i equals 1 to straight n of left parenthesis straight x subscript straight i plus straight a minus top enclose straight x minus straight a right parenthesis squared$ [∴ $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ ]
 $equals space 1 over straight n sum from straight i equals 1 to straight n of left parenthesis straight x subscript straight i minus top enclose straight x right parenthesis squared equals straight sigma subscript 1 superscript 2$

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.

183 Views

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.

120 Views