{"id":208,"date":"2018-10-30T11:15:52","date_gmt":"2018-10-30T11:15:52","guid":{"rendered":"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=208"},"modified":"2018-10-30T11:27:31","modified_gmt":"2018-10-30T11:27:31","slug":"sampling-and-sampling-distributions-sampling-distribution-of-x-bar","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/chapter\/sampling-and-sampling-distributions-sampling-distribution-of-x-bar\/","title":{"rendered":"Sampling and Sampling Distributions: Sampling Distribution of X bar"},"content":{"raw":" \r\n
\r\n\r\n \r\n\r\n1.\u00a0\u00a0\u00a0\u00a0\u00a0 Learning Outcome\r\n\r\n \r\n\r\n2.\u00a0\u00a0\u00a0\u00a0\u00a0 Introduction\r\n\r\n \r\n\r\n3.\u00a0\u00a0\u00a0\u00a0\u00a0 Central limit theorem\r\n\r\n \r\n\r\n4.\u00a0\u00a0\u00a0\u00a0\u00a0 Sampling distribution of a sampling mean\r\n\r\n \r\n\r\n5.\u00a0\u00a0\u00a0\u00a0\u00a0 Characteristics of the sampling distribution of mean\r\n\r\n \r\n\r\n6.\u00a0\u00a0\u00a0\u00a0\u00a0 Sampling distribution of proportion\r\n\r\n \r\n\r\n7.\u00a0\u00a0\u00a0\u00a0\u00a0 Mean & standard deviation of distribution of proportion\r\n\r\n \r\n\r\nLearning outcomes:<\/strong>\r\n\r\n \r\n\r\nAfter completing this module the students will be able to:<\/span>\r\n\r\n \r\n\r\n1.\u00a0\u00a0\u00a0\u00a0\u00a0 Understand sampling distribution of X bar<\/span>\r\n\r\n2.\u00a0\u00a0\u00a0\u00a0\u00a0 Understand implication of central limit theorem<\/span>\r\n\r\n3.\u00a0\u00a0\u00a0\u00a0\u00a0 Learn sampling from normal population<\/span>\r\n\r\n4.\u00a0\u00a0\u00a0\u00a0\u00a0 Understand sampling distribution of proportion<\/span>\r\n\r\n<\/div>\r\n
\r\n\r\n \r\n\r\n1.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>INTRODUCTION<\/strong>\r\n\r\n \r\n

The sampling distribution of the sample mean, denoted by \u0305 , is a concept that is required to understand right from the sample collection to analysis till the meaningful interpretation drawn from the sample. It relates to introductory statistical inference, which includes normal distribution, confidence intervals and hypothesis testing. Proper analysis and interpretation of a sample statistic requires knowledge of its distribution. If repeated random samples are chosen from the same population, the values of the sample mean, denoted \u00a0\u00a0\u0305will vary from sample to sample. The resulting sampling distribution \u0305 is the distribution of these sample mean \u00a0\u00a0\u0305values, for a large number of samples.<\/p>\r\n\"\"\r\n

Process of Inferential Statistics<\/strong><\/p>\r\n\r\n<\/div>\r\n

\u00a0<\/strong><\/p>\r\n\r\n

\r\n\r\n \r\n\r\n2.\u00a0 Central limit theorem<\/strong>\r\n\r\n \r\n

Suppose we take numerous simple random samples of a given size n from a normal distribution with population mean \u03bc and standard deviation . Then we compute the mean for each of those samples. Some of these sample means will be less than \u03bc and some will be greater than it, thus giving us the sampling distribution. If we plot the sample means using a histogram, we will see that they are normally distributed, where the mean and standard deviation of the sampling distribution X\u0305 are approximately equal to the mean and standard deviation of the population.<\/p>\r\n \r\n\r\nIf the original population has a normal distribution with mean \u03bc and standard deviation , then plotting the sample means for the simple random samples (SRS), each containing n observations, will produce a sampling distribution that also follows a normal distribution. If the original population does not have a normal distribution, but each SRS has a large n (where most texts suggest n > 30), then plotting the sample means will produce a sampling distribution that has an approximate normal distribution. This result is called the Central Limit Theorem.\r\n\r\n \r\n

The central limit theorem states that if a large enough sample is taken (typically n<\/em> > 30); then the sampling distribution of \u0305 is approximately a normal distribution with a mean of \u03bc and a standard deviation of \u03c3\u221an. Therefore \u00b5 \u0305 = \u03bc and \u0305 = \u221a<\/p>\r\n \r\n\r\n \r\n\r\n3.\u00a0 Sampling Distribution of a Sample Mean<\/strong>\r\n\r\n \r\n\r\nThe sample mean \u00a0\u00a0\u0305is a statistic whose value is the average of sample data drawn from a population.\r\n\r\n \r\n

For random samples of size n taken from a given population, the random variable \u0305 is the collection of these sample means, the \u0305\u00a0 \u2019s. Like any random variable, \u0305 has a probability\u00a0\u00a0\u00a0 distribution\u00a0\u00a0\u00a0 associated\u00a0\u00a0\u00a0 with\u00a0\u00a0\u00a0 it;\u00a0\u00a0\u00a0 i.e.,\u00a0\u00a0\u00a0 shape,\u00a0\u00a0\u00a0 mean,\u00a0\u00a0\u00a0 standard\u00a0\u00a0\u00a0 deviation The probability distribution created by plotting sample means, the\u00a0 \u00a0\u0305\u2019s, is the sampling distribution of the mean \u0305.<\/p>\r\n \r\n\r\nThe sampling distribution of \u0305depends on the:\r\n\r\n \r\n\r\ni.\u00a0 distribution of the original population (e.g., normal, skewed, uniform, symmetric)<\/span>\r\n\r\nii. sample size n<\/span>\r\n\r\niii.\u00a0 method of sample selection<\/span>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n3.1 Characteristics of the Sampling Distribution of a mean<\/strong>\r\n\r\n \r\n

When sampling from a normal population whose mean is \u03bc and standard deviation is \u03c3 is taken, than all possible samples of size n are selected from a normal population, then the sampling distribution of the mean has the following three characteristics:<\/p>\r\n \r\n\r\n1.\u00a0\u00a0 The sampling distribution of the mean is a normal distribution, regardless of sample size,n.\r\n\r\n2.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 The mean of the sampling distribution of the mean,\u00a0\u00a0 \u00a0\u00a0\u0305is equal to the mean of the\r\n\r\npopulation, \u03bc:\u00a0\u00a0\u00a0\u00a0 \u00a0 \u0305= \u03bc.\r\n\r\n3. The standard error of the sampling distribution of the mean \u00a0\u00a0\u0305is equal to the standard deviation of the population, \u03c3, divided by the square root of the sample size, n:\r\n\r\n\u0305= \u221a\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

Since in practice we usually do not know \u03bc or \u03c3 we estimate these by \u00a0\u00a0\u0305 and s\u221an respectively. In this case s<\/em> is the estimate of \u03c3 and is the standard deviation of the sample. The expression s\u221an is known as the standard error of the mean, labelled SE (<\/strong>x\u00af).\u00a0<\/strong><\/p>\r\n \r\n

Consider a population of 5 working people who are all neighbours in gurgoan. They are asked to list the no. of kilometres they used to travel daily for work.<\/span><\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

\r\n\r\n \r\n\r\nNow we can have a sample of two person (n=2) and find the mean \u00a0\u00a0\u0305to estimate \u00b5\r\n\r\n \r\n\r\nLike Rakesh = 50km and Aman = 80km, then Rakesh and Suresh and likewise.\r\n\r\nList all possible samples of two people and calculate the mean, \u00a0\u00a0\u0305for each sample.\r\n\r\n \r\n\r\n\"\"<\/div>\r\n
<\/div>\r\n
The data set of all the sample means in column 3 is called a <\/span>sampling distribution of the<\/strong> means<\/strong>.<\/span><\/div>\r\n
<\/div>\r\n
If the population being sampled is a normal distribution, then the sampling distribution of the mean is a normal distribution regardless of the sample size, n.<\/span><\/div>\r\n
\r\n\r\n \r\n\r\n4. Formulas<\/strong>\r\n\r\n \r\n\r\nThe sample standard deviation formula is:\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n

where,<\/p>\r\n \r\n\r\ns = sample standard deviation\r\n\r\n= sum of...\r\n\r\n= sample mean\r\n\r\nn = number of scores in sample.\r\n\r\n \r\n\r\nThe population standard deviation formula is:\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\nwhere,\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n\r\n\r\n\r\n
Confidence\u00a0coefficient<\/td>\r\n50%<\/td>\r\n68.27%<\/td>\r\n90%<\/td>\r\n95%<\/td>\r\n95.45%<\/td>\r\n99%<\/td>\r\n99.73%<\/td>\r\n<\/tr>\r\n
Z<\/td>\r\n0.6745<\/td>\r\n1.00<\/td>\r\n1.645<\/td>\r\n1.96<\/td>\r\n2.00<\/td>\r\n2.58<\/td>\r\n3.00<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\n5.\u00a0 Sample Proportion<\/strong>\r\n\r\n \r\n

A sample proportion is where a random sample of objects n<\/em> is taken from a population P; if x objects have a certain characteristic then the sample proportion \u201cp\u201d is: p = x\/n. For example: 100 people are asked if they are non-vegetarian. If 40 people respond \u201cyes\u201d then the sample proportion p = 40\/100.<\/p>\r\n \r\n\r\n5.1 Sampling Distribution of a Proportion<\/strong>\r\n\r\n \r\n

The sampling distribution of a proportion is when you repeat your survey for all possible samples of the population. For example: instead of polling 100 people once to ask if they are non-vegetarian, you\u2019ll poll them multiple times to get a better estimate of your statistic.<\/p>\r\n \r\n\r\n5.2 Mean of Sampling Distribution of the Proportion<\/strong>\r\n\r\n \r\n

The mean of sampling distribution of the proportion, P, is a special case of the sampling distribution of the mean. The mean of the sampling distribution of the proportion is related to the binomial distribution.<\/p>\r\n \r\n\r\n5.3 Standard Deviation of Sampling Distribution of the Proportion<\/strong>\r\n\r\n \r\n

If a random sample of n observations is taken from a binomial population with parameter p, the sampling distribution (i.e. all possible samples taken from the population) will have a standard deviation of:<\/p>\r\n \r\n\r\nStandard deviation of binomial distribution = \u03c3p = \u221a [pq\/n] where q=1-p.\r\n\r\n<\/div>\r\n

    \r\n \t
  1. Summary<\/strong><\/li>\r\n<\/ol>\r\n \r\n\r\nTo summarize:\r\n\r\n \r\n\r\n1.) The sampling distribution is a theoretical distribution of a sample statistic.\r\n\r\n \r\n\r\n2.) There is a different sampling distribution for each sample statistic.\r\n\r\n \r\n\r\n3.) Each sampling distribution is characterized by parameters, two of which known \u03bc<\/em> and \u03c3<\/em>.\r\n\r\n \r\n\r\nThe latter is called the standard error.\r\n\r\n \r\n\r\n4.) The sampling distribution of the mean is a special case of the sampling distribution.\r\n\r\n \r\n

    5.) The Central Limit Theorem relates the parameters of the sampling distribution of the mean to the population model and is very important in statistical thinking.<\/p>\r\n \r\n

    Learn More:<\/strong><\/p>\r\n \r\n

      \r\n \t
    1. Black. K (2013) Business Statistics For Contemporary Decision Making (8th Edition) New Delhi: Wiley<\/li>\r\n \t
    2. Cooper D.R., Schindler P. S. and Sharma J.K. (2012). Business Research Methods (11<\/em>th<\/em> Edition) <\/em>New Delhi: Mc Graw Hill Education<\/li>\r\n \t
    3. Vohra N.D. (2009). Quantitative Techniques in Management (4<\/em>th<\/em> Edition)<\/em> New Delhi: Mc Graw Hill Publication.<\/li>\r\n \t
    4. Tulsian P.C. and Pandey V. (2002). Quantitative Techniques, Theory & Problems (1st<\/em> edition)<\/em>. New Delhi: Pearson India.<\/li>\r\n \t
    5. http:\/\/www.statisticssolutions.com\/standard-error\/<\/a><\/li>\r\n \t
    6. http:\/\/www.statisticshowto.com\/probability-and-statistics\/sampling-in-statistics\/<\/a><\/li>\r\n \t
    7. Bunnies, Dragons and the \u2018Normal\u2019 World: https:\/\/www.youtube.com\/watch?v=jvoxEYmQHNM<\/a><\/li>\r\n \t
    8. http:\/\/www.statisticshowto.com\/sampling-distribution\/<\/li>\r\n<\/ol>","rendered":"

       <\/p>\n

      \n

       <\/p>\n

      1.\u00a0\u00a0\u00a0\u00a0\u00a0 Learning Outcome<\/p>\n

       <\/p>\n

      2.\u00a0\u00a0\u00a0\u00a0\u00a0 Introduction<\/p>\n

       <\/p>\n

      3.\u00a0\u00a0\u00a0\u00a0\u00a0 Central limit theorem<\/p>\n

       <\/p>\n

      4.\u00a0\u00a0\u00a0\u00a0\u00a0 Sampling distribution of a sampling mean<\/p>\n

       <\/p>\n

      5.\u00a0\u00a0\u00a0\u00a0\u00a0 Characteristics of the sampling distribution of mean<\/p>\n

       <\/p>\n

      6.\u00a0\u00a0\u00a0\u00a0\u00a0 Sampling distribution of proportion<\/p>\n

       <\/p>\n

      7.\u00a0\u00a0\u00a0\u00a0\u00a0 Mean & standard deviation of distribution of proportion<\/p>\n

       <\/p>\n

      Learning outcomes:<\/strong><\/p>\n

       <\/p>\n

      After completing this module the students will be able to:<\/span><\/p>\n

       <\/p>\n

      1.\u00a0\u00a0\u00a0\u00a0\u00a0 Understand sampling distribution of X bar<\/span><\/p>\n

      2.\u00a0\u00a0\u00a0\u00a0\u00a0 Understand implication of central limit theorem<\/span><\/p>\n

      3.\u00a0\u00a0\u00a0\u00a0\u00a0 Learn sampling from normal population<\/span><\/p>\n

      4.\u00a0\u00a0\u00a0\u00a0\u00a0 Understand sampling distribution of proportion<\/span><\/p>\n<\/div>\n

      \n

       <\/p>\n

      1.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>INTRODUCTION<\/strong><\/p>\n

       <\/p>\n

      The sampling distribution of the sample mean, denoted by \u0305 , is a concept that is required to understand right from the sample collection to analysis till the meaningful interpretation drawn from the sample. It relates to introductory statistical inference, which includes normal distribution, confidence intervals and hypothesis testing. Proper analysis and interpretation of a sample statistic requires knowledge of its distribution. If repeated random samples are chosen from the same population, the values of the sample mean, denoted \u00a0\u00a0\u0305will vary from sample to sample. The resulting sampling distribution \u0305 is the distribution of these sample mean \u00a0\u00a0\u0305values, for a large number of samples.<\/p>\n

      \"\"<\/p>\n

      Process of Inferential Statistics<\/strong><\/p>\n<\/div>\n

      \u00a0<\/strong><\/p>\n

      \n

       <\/p>\n

      2.\u00a0 Central limit theorem<\/strong><\/p>\n

       <\/p>\n

      Suppose we take numerous simple random samples of a given size n from a normal distribution with population mean \u03bc and standard deviation . Then we compute the mean for each of those samples. Some of these sample means will be less than \u03bc and some will be greater than it, thus giving us the sampling distribution. If we plot the sample means using a histogram, we will see that they are normally distributed, where the mean and standard deviation of the sampling distribution X\u0305 are approximately equal to the mean and standard deviation of the population.<\/p>\n

       <\/p>\n

      If the original population has a normal distribution with mean \u03bc and standard deviation , then plotting the sample means for the simple random samples (SRS), each containing n observations, will produce a sampling distribution that also follows a normal distribution. If the original population does not have a normal distribution, but each SRS has a large n (where most texts suggest n > 30), then plotting the sample means will produce a sampling distribution that has an approximate normal distribution. This result is called the Central Limit Theorem.<\/p>\n

       <\/p>\n

      The central limit theorem states that if a large enough sample is taken (typically n<\/em> > 30); then the sampling distribution of \u0305 is approximately a normal distribution with a mean of \u03bc and a standard deviation of \u03c3\u221an. Therefore \u00b5 \u0305 = \u03bc and \u0305 = \u221a<\/p>\n

       <\/p>\n

       <\/p>\n

      3.\u00a0 Sampling Distribution of a Sample Mean<\/strong><\/p>\n

       <\/p>\n

      The sample mean \u00a0\u00a0\u0305is a statistic whose value is the average of sample data drawn from a population.<\/p>\n

       <\/p>\n

      For random samples of size n taken from a given population, the random variable \u0305 is the collection of these sample means, the \u0305\u00a0 \u2019s. Like any random variable, \u0305 has a probability\u00a0\u00a0\u00a0 distribution\u00a0\u00a0\u00a0 associated\u00a0\u00a0\u00a0 with\u00a0\u00a0\u00a0 it;\u00a0\u00a0\u00a0 i.e.,\u00a0\u00a0\u00a0 shape,\u00a0\u00a0\u00a0 mean,\u00a0\u00a0\u00a0 standard\u00a0\u00a0\u00a0 deviation The probability distribution created by plotting sample means, the\u00a0 \u00a0\u0305\u2019s, is the sampling distribution of the mean \u0305.<\/p>\n

       <\/p>\n

      The sampling distribution of \u0305depends on the:<\/p>\n

       <\/p>\n

      i.\u00a0 distribution of the original population (e.g., normal, skewed, uniform, symmetric)<\/span><\/p>\n

      ii. sample size n<\/span><\/p>\n

      iii.\u00a0 method of sample selection<\/span><\/p>\n<\/div>\n

      \n

       <\/p>\n

      3.1 Characteristics of the Sampling Distribution of a mean<\/strong><\/p>\n

       <\/p>\n

      When sampling from a normal population whose mean is \u03bc and standard deviation is \u03c3 is taken, than all possible samples of size n are selected from a normal population, then the sampling distribution of the mean has the following three characteristics:<\/p>\n

       <\/p>\n

      1.\u00a0\u00a0 The sampling distribution of the mean is a normal distribution, regardless of sample size,n.<\/p>\n

      2.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 The mean of the sampling distribution of the mean,\u00a0\u00a0 \u00a0\u00a0\u0305is equal to the mean of the<\/p>\n

      population, \u03bc:\u00a0\u00a0\u00a0\u00a0 \u00a0 \u0305= \u03bc.<\/p>\n

      3. The standard error of the sampling distribution of the mean \u00a0\u00a0\u0305is equal to the standard deviation of the population, \u03c3, divided by the square root of the sample size, n:<\/p>\n

      \u0305= \u221a<\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

      Since in practice we usually do not know \u03bc or \u03c3 we estimate these by \u00a0\u00a0\u0305 and s\u221an respectively. In this case s<\/em> is the estimate of \u03c3 and is the standard deviation of the sample. The expression s\u221an is known as the standard error of the mean, labelled SE (<\/strong>x\u00af).\u00a0<\/strong><\/p>\n

       <\/p>\n

      Consider a population of 5 working people who are all neighbours in gurgoan. They are asked to list the no. of kilometres they used to travel daily for work.<\/span><\/p>\n<\/div>\n

      \"\"<\/p>\n

      \n

       <\/p>\n

      Now we can have a sample of two person (n=2) and find the mean \u00a0\u00a0\u0305to estimate \u00b5<\/p>\n

       <\/p>\n

      Like Rakesh = 50km and Aman = 80km, then Rakesh and Suresh and likewise.<\/p>\n

      List all possible samples of two people and calculate the mean, \u00a0\u00a0\u0305for each sample.<\/p>\n

       <\/p>\n

      \"\"<\/div>\n

      <\/div>\n
      The data set of all the sample means in column 3 is called a <\/span>sampling distribution of the<\/strong> means<\/strong>.<\/span><\/div>\n
      <\/div>\n
      If the population being sampled is a normal distribution, then the sampling distribution of the mean is a normal distribution regardless of the sample size, n.<\/span><\/div>\n
      \n

       <\/p>\n

      4. Formulas<\/strong><\/p>\n

       <\/p>\n

      The sample standard deviation formula is:<\/p>\n

      \"\"<\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      where,<\/p>\n

       <\/p>\n

      s = sample standard deviation<\/p>\n

      = sum of…<\/p>\n

      = sample mean<\/p>\n

      n = number of scores in sample.<\/p>\n

       <\/p>\n

      The population standard deviation formula is:<\/p>\n

       <\/p>\n

      \"\"<\/p>\n

       <\/p>\n

      where,\"\"<\/p>\n<\/div>\n

      \n\n\n\n\n
      Confidence\u00a0coefficient<\/td>\n50%<\/td>\n68.27%<\/td>\n90%<\/td>\n95%<\/td>\n95.45%<\/td>\n99%<\/td>\n99.73%<\/td>\n<\/tr>\n
      Z<\/td>\n0.6745<\/td>\n1.00<\/td>\n1.645<\/td>\n1.96<\/td>\n2.00<\/td>\n2.58<\/td>\n3.00<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

       <\/p>\n

      5.\u00a0 Sample Proportion<\/strong><\/p>\n

       <\/p>\n

      A sample proportion is where a random sample of objects n<\/em> is taken from a population P; if x objects have a certain characteristic then the sample proportion \u201cp\u201d is: p = x\/n. For example: 100 people are asked if they are non-vegetarian. If 40 people respond \u201cyes\u201d then the sample proportion p = 40\/100.<\/p>\n

       <\/p>\n

      5.1 Sampling Distribution of a Proportion<\/strong><\/p>\n

       <\/p>\n

      The sampling distribution of a proportion is when you repeat your survey for all possible samples of the population. For example: instead of polling 100 people once to ask if they are non-vegetarian, you\u2019ll poll them multiple times to get a better estimate of your statistic.<\/p>\n

       <\/p>\n

      5.2 Mean of Sampling Distribution of the Proportion<\/strong><\/p>\n

       <\/p>\n

      The mean of sampling distribution of the proportion, P, is a special case of the sampling distribution of the mean. The mean of the sampling distribution of the proportion is related to the binomial distribution.<\/p>\n

       <\/p>\n

      5.3 Standard Deviation of Sampling Distribution of the Proportion<\/strong><\/p>\n

       <\/p>\n

      If a random sample of n observations is taken from a binomial population with parameter p, the sampling distribution (i.e. all possible samples taken from the population) will have a standard deviation of:<\/p>\n

       <\/p>\n

      Standard deviation of binomial distribution = \u03c3p = \u221a [pq\/n] where q=1-p.<\/p>\n<\/div>\n

        \n
      1. Summary<\/strong><\/li>\n<\/ol>\n

         <\/p>\n

        To summarize:<\/p>\n

         <\/p>\n

        1.) The sampling distribution is a theoretical distribution of a sample statistic.<\/p>\n

         <\/p>\n

        2.) There is a different sampling distribution for each sample statistic.<\/p>\n

         <\/p>\n

        3.) Each sampling distribution is characterized by parameters, two of which known \u03bc<\/em> and \u03c3<\/em>.<\/p>\n

         <\/p>\n

        The latter is called the standard error.<\/p>\n

         <\/p>\n

        4.) The sampling distribution of the mean is a special case of the sampling distribution.<\/p>\n

         <\/p>\n

        5.) The Central Limit Theorem relates the parameters of the sampling distribution of the mean to the population model and is very important in statistical thinking.<\/p>\n

         <\/p>\n

        Learn More:<\/strong><\/p>\n

         <\/p>\n

          \n
        1. Black. K (2013) Business Statistics For Contemporary Decision Making (8th Edition) New Delhi: Wiley<\/li>\n
        2. Cooper D.R., Schindler P. S. and Sharma J.K. (2012). Business Research Methods (11<\/em>th<\/em> Edition) <\/em>New Delhi: Mc Graw Hill Education<\/li>\n
        3. Vohra N.D. (2009). Quantitative Techniques in Management (4<\/em>th<\/em> Edition)<\/em> New Delhi: Mc Graw Hill Publication.<\/li>\n
        4. Tulsian P.C. and Pandey V. (2002). Quantitative Techniques, Theory & Problems (1st<\/em> edition)<\/em>. New Delhi: Pearson India.<\/li>\n
        5. http:\/\/www.statisticssolutions.com\/standard-error\/<\/a><\/li>\n
        6. http:\/\/www.statisticshowto.com\/probability-and-statistics\/sampling-in-statistics\/<\/a><\/li>\n
        7. Bunnies, Dragons and the \u2018Normal\u2019 World: https:\/\/www.youtube.com\/watch?v=jvoxEYmQHNM<\/a><\/li>\n
        8. http:\/\/www.statisticshowto.com\/sampling-distribution\/<\/li>\n<\/ol>\n","protected":false},"author":3,"menu_order":18,"template":"","meta":{"_acf_changed":false,"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["dr-uday-khanna"],"pb_section_license":""},"chapter-type":[],"contributor":[63],"license":[],"class_list":["post-208","chapter","type-chapter","status-publish","hentry","contributor-dr-uday-khanna"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/208","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":5,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/208\/revisions"}],"predecessor-version":[{"id":220,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/208\/revisions\/220"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/208\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/media?parent=208"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapter-type?post=208"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/contributor?post=208"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/license?post=208"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}