{"id":221,"date":"2018-10-30T11:28:19","date_gmt":"2018-10-30T11:28:19","guid":{"rendered":"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=221"},"modified":"2018-10-30T11:42:32","modified_gmt":"2018-10-30T11:42:32","slug":"sampling-and-sampling-distributions-determining-sample-size","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/chapter\/sampling-and-sampling-distributions-determining-sample-size\/","title":{"rendered":"Sampling and Sampling Distributions: Determining sample size"},"content":{"raw":"
\r\n\r\n \r\n\r\n1.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Learning Outcome\r\n\r\n \r\n\r\n2.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Introduction\r\n\r\n \r\n\r\n3.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Data required for computing sample size\r\n\r\n \r\n\r\n4.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Determining sample size for estimating population mean\r\n\r\n \r\n\r\n5.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 When to use population or sample mean\r\n\r\n \r\n\r\n6.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Methods to estimate the\u00a0 population standard deviation\r\n\r\n \r\n\r\n7.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Sample size for estimating population proportion\r\n\r\n \r\n\r\n8.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Determining the sample size with fpc\r\n\r\n \r\n\r\n9.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Summary\r\n\r\n \r\n\r\n \r\n\r\n1.\u00a0\u00a0\u00a0\u00a0 <\/strong>Learning outcomes:<\/strong>\r\n\r\n \r\n\r\n\u00a7\u00a0\u00a0\u00a0\u00a0\u00a0 To realize the importance of estimating optimal sample size.\r\n\r\n \r\n\r\n\u00a7\u00a0\u00a0\u00a0\u00a0\u00a0 Relevance of variables influencing sample size\r\n\r\n \r\n\r\n\u00a7\u00a0\u00a0\u00a0\u00a0\u00a0 To make a preliminary estimate of the appropriate sample size\r\n\r\n \r\n\r\n \r\n\r\n2.\u00a0\u00a0\u00a0\u00a0 <\/strong>Introduction<\/strong>\r\n\r\n \r\n

The most important aspect of any research study is the magnitude of the sample size. If the sample size is very large, it could waste the resources of the researcher and the organization and if the sample size is small, it could not be able to correctly represent the population under study. Therefore it is very important to rightly estimate the sample size for the research under study.<\/p>\r\n \r\n\r\n \r\n\r\n3. Data required for computation of sample size<\/strong>\r\n\r\n \r\n

a)\u00a0\u00a0<\/strong>Magnitude of error (Desired precision (<\/strong>\u00b1<\/span> e) <\/strong>It gives the estimate of how precise we want to be in our measurement. Here the researcher must workout the largest acceptable\u00a0<\/span>difference between the sample mean and the population mean. It is specified by acceptable degree of sampling error. It is termed as standard error (i.e. the standard deviation of the sample means). Therefore the larger the acceptable degree of sampling error, the smaller the sample size must be. For example if we want to estimate the average weight of 60 students in a class within +-4kg. The precision level is +-4kg.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

b)\u00a0\u00a0\u00a0 <\/strong>Value associated with desire confidence level (Z) <\/strong>It states the degree of confidence interval taken in the population mean. It is a percentage or decimal value that tells how confident a researcher can be for his research. It includes the long run percentage of confidence intervals that will include the true population mean. It implies that the greater the desired confidence, the larger the sample size. For example if we want to be 95% confident that the estimate weight of the students in the class should be within +-4kgs, then the desired confidence level is 95%.<\/p>\r\n \r\n\r\nCalculating a Confidence Interval\r\n\r\n \r\n\r\nApproximate location (value) of the population mean\r\n\r\n\u00b5= X\u0305\u00b1 a small sampling error\r\n\r\nEstimation of the sampling error\r\n\r\nSmall sampling error = Z. \u03c3x\u0305\r\n\r\nX\u0305= sample mean\r\n\r\n\u00b5\u00a0\u00a0\u00a0\u00a0\u00a0 = X\u0305 \u00b1 Z. \u03c3x\u0305\r\n\r\n \r\n\r\nc)\u00a0\u00a0\u00a0 Variance (Estimator of the standard deviation of the population, \u03c3) <\/strong>It investigates that\r\n\r\n \r\n

how heterogeneous is the population. This variability is measured by estimating the population standard deviation. A heterogeneous population will have more variance will require a large sample while a homogeneous population having less variance will require a small sample. Therefore a smaller dispersion in the population calls for a smaller sample size while a larger dispersion in the population calls for a larger sample size.<\/p>\r\n\r\n<\/div>\r\n

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

Sample size compared to margin of error<\/strong><\/p>\r\n \r\n\r\n4.\u00a0 Determining sample size for estimating population mean<\/strong>\r\n\r\n \r\n\r\nWhen sampling distribution of sample mean \u00a0\u00a0\u0305is normal, the standard normal variable z is given by\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\nwhere z = standard normal variate x\u0305= sample mean\r\n\r\n\u00b5\u00a0\u00a0\u00a0\u00a0 = population mean\r\n\r\n \r\n\r\n\u03c3\u00a0\u00a0\u00a0 = population standard deviation n= sample size\r\n\r\n\u0305\u2212\u00b5<\/em>\r\n\r\n\u221a\r\n\r\n<\/div>\r\n
\r\n\r\n\"\"\r\n\r\nConfidence interval for \u00b5 (<\/strong>\u00a0 \u00a0 \u00a0\u00a0)<\/strong>\r\n\r\n \r\n\r\nn\u00a0 Assumptions\r\n\r\n \r\n\r\nn\u00a0\u00a0 Population standard deviation is known\r\n\r\n \r\n\r\nn\u00a0\u00a0 Population is normally distributed\r\n\r\n \r\n\r\nn\u00a0\u00a0 If population is not normal, use large sample\r\n\r\n \r\n\r\nConfidence Interval Estimate\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\n<\/div>\r\nThe difference between x\u0305 and \u00b5 is called the sampling error or margin of error, e.<\/span>\r\n
\r\n\r\nor\u00a0\u00a0\u00a0\u00a0 e=<\/em>\u00a0 \u00a0 \u00a0\u221a\r\n\r\n \r\n

Thus the acceptable margin of error (maximum tolerance difference between unknown population mean \u00b5 and the sample estimate at a particular level of confidence) at the chosen level 1-\u03b1 can be written as<\/p>\r\nwhere\r\n\r\n \r\n\r\nn= sample size\r\n\r\n \r\n\r\nz = standardized value indicating the level of confidence E = accepted magnitude of sampling error (i.e. precision) \u03c3 = estimator of the population standard deviation\r\n\r\n \r\n\r\nor n=2.2\/2\r\n\r\nwhere\r\n\r\ns= sample standard deviation\r\n\r\n \r\n

If population standard deviation \u03c3 is unknown, then sample standard deviation s can be used to find sample size n.<\/p>\r\n \r\n\r\n \r\n

While the first two estimates; the desired precision level and the desired confidence level are at the discretion of the researcher, the standard deviation of the population is bit tough to estimate.<\/p>\r\n \r\n

In case if the true population dispersion is unknown, it could be find, the standard deviation of the sample is used as a proxy figure. This figure could be worked out by any one of the following methods:<\/p>\r\n\r\n<\/div>\r\n \r\n\r\nAny previous research on this topic.<\/span>\r\n\r\nA pilot test or pre test of the data among a sample drawn from the population.<\/span>\r\n\r\nA rule of thumb (one-sixth of the range based on six standard deviations within 99.73 percent confidence.<\/span>\r\n

\r\n\r\n \r\n\r\n5.\u00a0 When to use the sample or population standard deviation<\/strong>\r\n\r\n \r\n

We are normally interested in knowing the population standard deviation because our population contains all the values we are interested in. Therefore, you would normally calculate the population standard deviation if: (1) you have the entire population or (2) you have a sample of a larger population, but you are only interested in this sample and do not wish to generalize our findings to the population. However, in statistics, we are usually presented with a sample from which we wish to estimate (generalize to) a population, and the standard deviation is no exception to this.<\/p>\r\n \r\n

Therefore, if all you have is a sample, but you wish to make a statement about the population standard deviation from which the sample is drawn, you need to use the sample standard deviation. The standard deviation is used in conjunction with the mean to summarise continuous data, not categorical data. In addition, the standard deviation, like the mean, is normally only appropriate when the continuous data is not significantly skewed or has outliers.<\/p>\r\n \r\n

For example a teacher sets an exam for their pupils. The teacher wants to summarize the results the pupils attained as a mean and standard deviation. Which standard deviation should be used? The answer is population standard deviation. The teacher is only interested in this class of pupils' scores and nobody else. On the other end a researcher has interviewed females aged 18 to 30 years old for their view on live in relationships. Which standard deviation would most likely be used? The probable answer is sample standard deviation. Although not explicitly stated, a researcher investigating live in relationship issues will not simply be concerned with just the participants of their study; they will want to show how their sample results can be generalised to the whole population (in this case, females aged 18 to 30 years old).<\/p>\r\n \r\n

One of the questions on a national consensus survey asks for respondents' age. Which standard deviation would be used to describe the variation in all ages received from the consensus? The probable answer is population standard deviation. A national consensus is used to find out information about the nation's citizens. By definition, it includes the whole population. Therefore, a population standard deviation would be used.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n6.\u00a0 Methods to estimate the population standard deviation<\/strong>\r\n\r\n \r\n

The standard deviation is a measure of the spread of scores within a set of data. Usually, we are interested in the standard deviation of a population. However, as we are often presented with data from a sample only, we can estimate the population standard deviation from a sample standard deviation. These two standard deviations - sample and population standard deviations - are calculated differently. In statistics, we are usually presented with having to calculate sample standard deviations. But we can also estimate the population standard deviation with the given methods:<\/p>\r\n \r\n

1.\u00a0 Use information from a previous study. If the researchers already conduct such type of study, they can<\/p>\r\n

make use of that information for the current study as well.<\/p>\r\n \r\n\r\n \r\n

2.\u00a0 Use secondary data. Researchers can take help of vast amount of data available with the company, on<\/p>\r\n

their website or their local library. It can also be well estimated with the help of different sources like<\/p>\r\n

industry averages, competitor\u2019s data; government documents some previous surveys and many others. This<\/p>\r\n

existing pool of information may help them to work out the estimate of population standard deviation.<\/p>\r\n \r\n\r\n \r\n\r\n3.\u00a0 Conduct a small study of the population. The researcher may conduct a study of relatively small numbers\r\n\r\nof target population members to better understand the group\u2019s degrees of dispersion from the average for\r\n\r\nthe variable under study.\r\n\r\n \r\n\r\n \r\n\r\n4. Talk to informed people. At last, the researcher could also use the judgment of the experienced managers\r\n\r\nwho are quite knowledgeable about the variable under study.\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

For example a hospital wants an estimate of the mean time that a doctor spends with each patient in the OPD. How large a sample should be taken if the desired margin of error is 2 minutes at a 95 percent level of confidence, assuming population standard deviation of 8 minutes?<\/p>\r\n \r\n\r\nGiven e = 2 minutes\r\n\r\nZ\u00a0\u00a0\u00a0\u00a0 \u221d\/2= 1.96 at 95 percent confidence level\r\n\r\n \r\n\r\n\u03b1 = 8\r\n\r\n2.\u00a0 2\r\n\r\nn=\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2\r\n\r\n= (1.96)2.(8)2\r\n\r\n(2)2\r\n\r\n\u2245\u00a0\u00a0 62\r\n\r\n \r\n\r\n \r\n\r\n7.\u00a0 Sample size for estimating population proportion<\/strong>\r\n\r\n \r\n

The previous discussion was focused on determining sample size for estimating mean values. But at times it is the proportion of population with a particular attribute which is more significant than the mean value. For example one might be interested in knowing the proportion of households who dine outside on weekends rather than enjoying at their home.<\/p>\r\n \r\n\r\n7.1 Calculation of sample size of proportion :<\/strong>\r\n\r\n \r\n\r\nWe already know that the margin of error is 1.96 times the standard error and that the\r\n\r\nstandard error is \u221a\u00a0 ^(1\u2212\u00a0 )^\r\n\r\n \r\n\r\nIn general the formula is<\/span>\r\n\r\ne = z \u221a\u00a0 ^(1\u2212\u00a0 )^<\/span>\r\n\r\nOr e= Z \u221d\/2\u221a( ^ \/\u00a0 )<\/span>\r\n\r\nwhere q=1-p^<\/span>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n\u2022\u00a0\u00a0 e is the desired margin of error, the difference between sample proportion, \u00a0\u0305and population proportion, p.\r\n\r\n \r\n\r\n\u2022\u00a0\u00a0\u00a0 z is the z-score, e.g. 1.645 for a 90% confidence interval, 1.96 for a 95% confidence interval, 2.58 for a 99% confidence interval\r\n\r\n \r\n\r\n\u2022\u00a0 p\u02c6 is our prior judgment of the correct value of p.\r\n\r\n \r\n\r\n\u2022\u00a0 n is the sample size (to be found)\r\n\r\n \r\n\r\n \r\n

For example a professor in department of management studies is trying to determine the proportion of students in the department who support late marriages after 30 years of age. He asks, \u201cHow large a sample size do I need?\u201d To answer a question like this we need to ask the researcher certain questions, like 1. How accurately do you need the answer? 2. What level of confidence do you intend to use? 3. What is your current estimate of the proportion of students in the department who support late marriage (approx.)?<\/p>\r\n \r\n\r\nPossible answers might be:\r\n\r\n \r\n\r\n1. \u201cWe need a margin of error less than 2.5%\u201d. Typical surveys have margins of error ranging from less than 1% to something of the order of 4% \u2014 we can choose any margin of error we like but need to specify it.\r\n\r\n \r\n\r\n2.\u00a0\u00a0 95% confidence intervals are typical but not in any way mandatory \u2014 we could do 90%, 99% or something else entirely. For this example, we assume 95%.\r\n\r\n \r\n\r\n3.\u00a0\u00a0\u00a0 May be guided by past surveys or general knowledge of public opinion. Let\u2019s suppose answer is 30%. So in this case we set e equal to 0.025, z = 1.96 and \u02c6p = 0.3, and equation becomes 0.025 = 1.96\u221a0.3\u00a0\u00a0 0.7\u2044\u00a00.3\u00a0 0.7 = (0.025) 2 = .0001617\u00a0<\/span>1.96<\/span>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\nTherefore\r\n\r\nn= .00016170.3X0.7 =1291\r\n\r\n \r\n\r\nTherefore a sample size of 1300 students is required.\r\n\r\n \r\n\r\nWe could clearly try varying any of the elements of this. For example, may be the researcher would be satisfied with a 90% confidence interval, for which z = 1.645. In this case equation becomes\r\n\r\n0.3X0.7\r\n\r\n0.025=1.645\u221a\u00a0 n\r\n\r\n \r\n

for which we can quickly find n = 909. If we are willing to accept a lower confidence level, we can get away with a smaller sample size.<\/p>\r\n \r\n\r\nA different type of variation is \u201cWhat if we have no initial estimate of \u02c6p?\u201d In this case, the\r\n

convention is to assume \u02c6p= 0.5 .The reason is that the standard error formula is\u00a0 \u221ap^(1\u2212p)^ ,n is largest when \u02c6p = 0.5, so this is a conservative assumption that allows for \u02c6p being unknown a priori. If we repeat the calculation with \u02c6p = 0.5 (but returning to z = 1.96), we find<\/p>\r\n0.025 = 1.96\u221a0.5X0n.5 which results in n = 1537.\r\n\r\n \r\n\r\n \r\n

Question\u00a0 :\u00a0 A survey\u00a0 estimated that 20% of\u00a0 all\u00a0 \u00a0Indian\u00a0 aged 16 to 20 are quite concerned for their health. A similar survey is planned for US. They want a 95% confidence\u00a0 interval to have a margin of error of 0.04.<\/p>\r\n \r\n\r\n(a) Find the necessary sample size if they expect to find results similar to those in India\r\n\r\n \r\n\r\n(b)Suppose instead they used the\u00a0 conservative formula based on \u02c6p = 0.5. What is now the required sample size?\r\n\r\n<\/div>\r\n

\r\n\r\nSolution:\r\n\r\n \r\n\r\n(a) The general formula is\r\n\r\nE= \u00a0\u00a0\u221a\u00a0 ^(1\u2212\u00a0 )^\r\n\r\n \r\n\r\nwhich translates to\r\n\r\nn\u00a0\u00a0\u00a0\u00a0 =\u00a0\u00a0 ^(1\u2212\u00a0 ^)\u00a0 2\u00a0\u00a0 2\r\n\r\nn= 0.2X0.8X1.96X1.96 = 384.2\r\n\r\n0.04X0.04\r\n\r\n \r\n\r\nb) With e = 0.04, p^ = 0.5, z= 1.96 we get<\/span>\r\n\r\nn= 0.5 0.5 1.96 1.96 = 600.25\u00a0<\/span>\r\n\r\n0.04 0.04<\/span>\r\n

Thesamplesizeis384 for(a) and 600 for (b), showing the advantage in using the estimated \u02c6p (0.2) so long as we feel\u00a0confident that this is roughly the right guess.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n \r\n\r\n8.\u00a0 Determining the Sample Size with fpc<\/strong>\r\n\r\n \r\n\r\nFinite population correction factor (fpc) is used to determine sample size when sampling without replacement. The use of such factor reduces the standard error by a value equal to\r\n\r\n\u221a(\u00a0\u00a0 \u2212\u00a0\u00a0 )\/\u00a0\u00a0 \u2212 1.\r\n\r\n \r\n\r\nFor example, in estimating the mean, the sampling error is given by\r\n\r\ne =\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u221a\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u221a(\u00a0\u00a0 \u2212\u00a0\u00a0 )\/\u00a0\u00a0 \u2212 1\r\n\r\n \r\n\r\n \r\n\r\n9.\u00a0 Summary<\/strong>\r\n\r\n \r\n

One of the most common requests that researchers get from investigating agencies are sample size calculations or sample size justifications for the proposed study. The sample size is the number of experimental units or samples included in a study. Determining the sample size to answer the research question is one of the important requirements in designing a study. In order to calculate the sample size, it is required to have some idea of the results expected in a\u00a0study. In general, the greater the variability in the outcome variable, the larger the sample size required\u00a0<\/span>to assess whether an observed effect is a true effect.<\/p>\r\n\r\n<\/div>\r\n

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

    \r\n \t
  1. Tulsian P.C. and Pandey V. (2002). Quantitative Techniques, Theory & Problems (1st<\/em> edition)<\/em>. New Delhi: Pearson India.<\/li>\r\n \t
  2. Black. K (2013) Business Statistics For Contemporary Decision Making (8th Edition) New Delhi: Wiley<\/li>\r\n \t
  3. 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
  4. https:\/\/www.unc.edu\/~rls\/s151-2010\/class23.pdf<\/li>\r\n<\/ol>\r\n
    \r\n\r\n \r\n\r\n<\/div>","rendered":"
    \n

     <\/p>\n

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

     <\/p>\n

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

     <\/p>\n

    3.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Data required for computing sample size<\/p>\n

     <\/p>\n

    4.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Determining sample size for estimating population mean<\/p>\n

     <\/p>\n

    5.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 When to use population or sample mean<\/p>\n

     <\/p>\n

    6.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Methods to estimate the\u00a0 population standard deviation<\/p>\n

     <\/p>\n

    7.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Sample size for estimating population proportion<\/p>\n

     <\/p>\n

    8.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Determining the sample size with fpc<\/p>\n

     <\/p>\n

    9.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Summary<\/p>\n

     <\/p>\n

     <\/p>\n

    1.\u00a0\u00a0\u00a0\u00a0 <\/strong>Learning outcomes:<\/strong><\/p>\n

     <\/p>\n

    \u00a7\u00a0\u00a0\u00a0\u00a0\u00a0 To realize the importance of estimating optimal sample size.<\/p>\n

     <\/p>\n

    \u00a7\u00a0\u00a0\u00a0\u00a0\u00a0 Relevance of variables influencing sample size<\/p>\n

     <\/p>\n

    \u00a7\u00a0\u00a0\u00a0\u00a0\u00a0 To make a preliminary estimate of the appropriate sample size<\/p>\n

     <\/p>\n

     <\/p>\n

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

     <\/p>\n

    The most important aspect of any research study is the magnitude of the sample size. If the sample size is very large, it could waste the resources of the researcher and the organization and if the sample size is small, it could not be able to correctly represent the population under study. Therefore it is very important to rightly estimate the sample size for the research under study.<\/p>\n

     <\/p>\n

     <\/p>\n

    3. Data required for computation of sample size<\/strong><\/p>\n

     <\/p>\n

    a)\u00a0\u00a0<\/strong>Magnitude of error (Desired precision (<\/strong>\u00b1<\/span> e) <\/strong>It gives the estimate of how precise we want to be in our measurement. Here the researcher must workout the largest acceptable\u00a0<\/span>difference between the sample mean and the population mean. It is specified by acceptable degree of sampling error. It is termed as standard error (i.e. the standard deviation of the sample means). Therefore the larger the acceptable degree of sampling error, the smaller the sample size must be. For example if we want to estimate the average weight of 60 students in a class within +-4kg. The precision level is +-4kg.<\/span><\/p>\n<\/div>\n

    \n

     <\/p>\n

    b)\u00a0\u00a0\u00a0 <\/strong>Value associated with desire confidence level (Z) <\/strong>It states the degree of confidence interval taken in the population mean. It is a percentage or decimal value that tells how confident a researcher can be for his research. It includes the long run percentage of confidence intervals that will include the true population mean. It implies that the greater the desired confidence, the larger the sample size. For example if we want to be 95% confident that the estimate weight of the students in the class should be within +-4kgs, then the desired confidence level is 95%.<\/p>\n

     <\/p>\n

    Calculating a Confidence Interval<\/p>\n

     <\/p>\n

    Approximate location (value) of the population mean<\/p>\n

    \u00b5= X\u0305\u00b1 a small sampling error<\/p>\n

    Estimation of the sampling error<\/p>\n

    Small sampling error = Z. \u03c3x\u0305<\/p>\n

    X\u0305= sample mean<\/p>\n

    \u00b5\u00a0\u00a0\u00a0\u00a0\u00a0 = X\u0305 \u00b1 Z. \u03c3x\u0305<\/p>\n

     <\/p>\n

    c)\u00a0\u00a0\u00a0 Variance (Estimator of the standard deviation of the population, \u03c3) <\/strong>It investigates that<\/p>\n

     <\/p>\n

    how heterogeneous is the population. This variability is measured by estimating the population standard deviation. A heterogeneous population will have more variance will require a large sample while a homogeneous population having less variance will require a small sample. Therefore a smaller dispersion in the population calls for a smaller sample size while a larger dispersion in the population calls for a larger sample size.<\/p>\n<\/div>\n

    \n

    \"\"<\/p>\n

    Sample size compared to margin of error<\/strong><\/p>\n

     <\/p>\n

    4.\u00a0 Determining sample size for estimating population mean<\/strong><\/p>\n

     <\/p>\n

    When sampling distribution of sample mean \u00a0\u00a0\u0305is normal, the standard normal variable z is given by<\/p>\n<\/div>\n

    \n

     <\/p>\n

    where z = standard normal variate x\u0305= sample mean<\/p>\n

    \u00b5\u00a0\u00a0\u00a0\u00a0 = population mean<\/p>\n

     <\/p>\n

    \u03c3\u00a0\u00a0\u00a0 = population standard deviation n= sample size<\/p>\n

    \u0305\u2212\u00b5<\/em><\/p>\n

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

    \n

    \"\"<\/p>\n

    Confidence interval for \u00b5 (<\/strong>\u00a0 \u00a0 \u00a0\u00a0)<\/strong><\/p>\n

     <\/p>\n

    n\u00a0 Assumptions<\/p>\n

     <\/p>\n

    n\u00a0\u00a0 Population standard deviation is known<\/p>\n

     <\/p>\n

    n\u00a0\u00a0 Population is normally distributed<\/p>\n

     <\/p>\n

    n\u00a0\u00a0 If population is not normal, use large sample<\/p>\n

     <\/p>\n

    Confidence Interval Estimate<\/p>\n

     <\/p>\n

    \"\"<\/p>\n

     <\/p>\n<\/div>\n

    The difference between x\u0305 and \u00b5 is called the sampling error or margin of error, e.<\/span><\/p>\n

    \n

    or\u00a0\u00a0\u00a0\u00a0 e=<\/em>\u00a0 \u00a0 \u00a0\u221a<\/p>\n

     <\/p>\n

    Thus the acceptable margin of error (maximum tolerance difference between unknown population mean \u00b5 and the sample estimate at a particular level of confidence) at the chosen level 1-\u03b1 can be written as<\/p>\n

    where<\/p>\n

     <\/p>\n

    n= sample size<\/p>\n

     <\/p>\n

    z = standardized value indicating the level of confidence E = accepted magnitude of sampling error (i.e. precision) \u03c3 = estimator of the population standard deviation<\/p>\n

     <\/p>\n

    or n=2.2\/2<\/p>\n

    where<\/p>\n

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

     <\/p>\n

    If population standard deviation \u03c3 is unknown, then sample standard deviation s can be used to find sample size n.<\/p>\n

     <\/p>\n

     <\/p>\n

    While the first two estimates; the desired precision level and the desired confidence level are at the discretion of the researcher, the standard deviation of the population is bit tough to estimate.<\/p>\n

     <\/p>\n

    In case if the true population dispersion is unknown, it could be find, the standard deviation of the sample is used as a proxy figure. This figure could be worked out by any one of the following methods:<\/p>\n<\/div>\n

     <\/p>\n

    Any previous research on this topic.<\/span><\/p>\n

    A pilot test or pre test of the data among a sample drawn from the population.<\/span><\/p>\n

    A rule of thumb (one-sixth of the range based on six standard deviations within 99.73 percent confidence.<\/span><\/p>\n

    \n

     <\/p>\n

    5.\u00a0 When to use the sample or population standard deviation<\/strong><\/p>\n

     <\/p>\n

    We are normally interested in knowing the population standard deviation because our population contains all the values we are interested in. Therefore, you would normally calculate the population standard deviation if: (1) you have the entire population or (2) you have a sample of a larger population, but you are only interested in this sample and do not wish to generalize our findings to the population. However, in statistics, we are usually presented with a sample from which we wish to estimate (generalize to) a population, and the standard deviation is no exception to this.<\/p>\n

     <\/p>\n

    Therefore, if all you have is a sample, but you wish to make a statement about the population standard deviation from which the sample is drawn, you need to use the sample standard deviation. The standard deviation is used in conjunction with the mean to summarise continuous data, not categorical data. In addition, the standard deviation, like the mean, is normally only appropriate when the continuous data is not significantly skewed or has outliers.<\/p>\n

     <\/p>\n

    For example a teacher sets an exam for their pupils. The teacher wants to summarize the results the pupils attained as a mean and standard deviation. Which standard deviation should be used? The answer is population standard deviation. The teacher is only interested in this class of pupils’ scores and nobody else. On the other end a researcher has interviewed females aged 18 to 30 years old for their view on live in relationships. Which standard deviation would most likely be used? The probable answer is sample standard deviation. Although not explicitly stated, a researcher investigating live in relationship issues will not simply be concerned with just the participants of their study; they will want to show how their sample results can be generalised to the whole population (in this case, females aged 18 to 30 years old).<\/p>\n

     <\/p>\n

    One of the questions on a national consensus survey asks for respondents’ age. Which standard deviation would be used to describe the variation in all ages received from the consensus? The probable answer is population standard deviation. A national consensus is used to find out information about the nation’s citizens. By definition, it includes the whole population. Therefore, a population standard deviation would be used.<\/span><\/p>\n<\/div>\n

    \n

     <\/p>\n

    6.\u00a0 Methods to estimate the population standard deviation<\/strong><\/p>\n

     <\/p>\n

    The standard deviation is a measure of the spread of scores within a set of data. Usually, we are interested in the standard deviation of a population. However, as we are often presented with data from a sample only, we can estimate the population standard deviation from a sample standard deviation. These two standard deviations – sample and population standard deviations – are calculated differently. In statistics, we are usually presented with having to calculate sample standard deviations. But we can also estimate the population standard deviation with the given methods:<\/p>\n

     <\/p>\n

    1.\u00a0 Use information from a previous study. If the researchers already conduct such type of study, they can<\/p>\n

    make use of that information for the current study as well.<\/p>\n

     <\/p>\n

     <\/p>\n

    2.\u00a0 Use secondary data. Researchers can take help of vast amount of data available with the company, on<\/p>\n

    their website or their local library. It can also be well estimated with the help of different sources like<\/p>\n

    industry averages, competitor\u2019s data; government documents some previous surveys and many others. This<\/p>\n

    existing pool of information may help them to work out the estimate of population standard deviation.<\/p>\n

     <\/p>\n

     <\/p>\n

    3.\u00a0 Conduct a small study of the population. The researcher may conduct a study of relatively small numbers<\/p>\n

    of target population members to better understand the group\u2019s degrees of dispersion from the average for<\/p>\n

    the variable under study.<\/p>\n

     <\/p>\n

     <\/p>\n

    4. Talk to informed people. At last, the researcher could also use the judgment of the experienced managers<\/p>\n

    who are quite knowledgeable about the variable under study.<\/p>\n<\/div>\n

    \n

     <\/p>\n

    For example a hospital wants an estimate of the mean time that a doctor spends with each patient in the OPD. How large a sample should be taken if the desired margin of error is 2 minutes at a 95 percent level of confidence, assuming population standard deviation of 8 minutes?<\/p>\n

     <\/p>\n

    Given e = 2 minutes<\/p>\n

    Z\u00a0\u00a0\u00a0\u00a0 \u221d\/2= 1.96 at 95 percent confidence level<\/p>\n

     <\/p>\n

    \u03b1 = 8<\/p>\n

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

    n=\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2<\/p>\n

    = (1.96)2.(8)2<\/p>\n

    (2)2<\/p>\n

    \u2245\u00a0\u00a0 62<\/p>\n

     <\/p>\n

     <\/p>\n

    7.\u00a0 Sample size for estimating population proportion<\/strong><\/p>\n

     <\/p>\n

    The previous discussion was focused on determining sample size for estimating mean values. But at times it is the proportion of population with a particular attribute which is more significant than the mean value. For example one might be interested in knowing the proportion of households who dine outside on weekends rather than enjoying at their home.<\/p>\n

     <\/p>\n

    7.1 Calculation of sample size of proportion :<\/strong><\/p>\n

     <\/p>\n

    We already know that the margin of error is 1.96 times the standard error and that the<\/p>\n

    standard error is \u221a\u00a0 ^(1\u2212\u00a0 )^<\/p>\n

     <\/p>\n

    In general the formula is<\/span><\/p>\n

    e = z \u221a\u00a0 ^(1\u2212\u00a0 )^<\/span><\/p>\n

    Or e= Z \u221d\/2\u221a( ^ \/\u00a0 )<\/span><\/p>\n

    where q=1-p^<\/span><\/p>\n<\/div>\n

    \n

     <\/p>\n

    \u2022\u00a0\u00a0 e is the desired margin of error, the difference between sample proportion, \u00a0\u0305and population proportion, p.<\/p>\n

     <\/p>\n

    \u2022\u00a0\u00a0\u00a0 z is the z-score, e.g. 1.645 for a 90% confidence interval, 1.96 for a 95% confidence interval, 2.58 for a 99% confidence interval<\/p>\n

     <\/p>\n

    \u2022\u00a0 p\u02c6 is our prior judgment of the correct value of p.<\/p>\n

     <\/p>\n

    \u2022\u00a0 n is the sample size (to be found)<\/p>\n

     <\/p>\n

     <\/p>\n

    For example a professor in department of management studies is trying to determine the proportion of students in the department who support late marriages after 30 years of age. He asks, \u201cHow large a sample size do I need?\u201d To answer a question like this we need to ask the researcher certain questions, like 1. How accurately do you need the answer? 2. What level of confidence do you intend to use? 3. What is your current estimate of the proportion of students in the department who support late marriage (approx.)?<\/p>\n

     <\/p>\n

    Possible answers might be:<\/p>\n

     <\/p>\n

    1. \u201cWe need a margin of error less than 2.5%\u201d. Typical surveys have margins of error ranging from less than 1% to something of the order of 4% \u2014 we can choose any margin of error we like but need to specify it.<\/p>\n

     <\/p>\n

    2.\u00a0\u00a0 95% confidence intervals are typical but not in any way mandatory \u2014 we could do 90%, 99% or something else entirely. For this example, we assume 95%.<\/p>\n

     <\/p>\n

    3.\u00a0\u00a0\u00a0 May be guided by past surveys or general knowledge of public opinion. Let\u2019s suppose answer is 30%. So in this case we set e equal to 0.025, z = 1.96 and \u02c6p = 0.3, and equation becomes 0.025 = 1.96\u221a0.3\u00a0\u00a0 0.7\u2044\u00a00.3\u00a0 0.7 = (0.025) 2 = .0001617\u00a0<\/span>1.96<\/span><\/p>\n<\/div>\n

    \n

     <\/p>\n

    Therefore<\/p>\n

    n= .00016170.3X0.7 =1291<\/p>\n

     <\/p>\n

    Therefore a sample size of 1300 students is required.<\/p>\n

     <\/p>\n

    We could clearly try varying any of the elements of this. For example, may be the researcher would be satisfied with a 90% confidence interval, for which z = 1.645. In this case equation becomes<\/p>\n

    0.3X0.7<\/p>\n

    0.025=1.645\u221a\u00a0 n<\/p>\n

     <\/p>\n

    for which we can quickly find n = 909. If we are willing to accept a lower confidence level, we can get away with a smaller sample size.<\/p>\n

     <\/p>\n

    A different type of variation is \u201cWhat if we have no initial estimate of \u02c6p?\u201d In this case, the<\/p>\n

    convention is to assume \u02c6p= 0.5 .The reason is that the standard error formula is\u00a0 \u221ap^(1\u2212p)^ ,n is largest when \u02c6p = 0.5, so this is a conservative assumption that allows for \u02c6p being unknown a priori. If we repeat the calculation with \u02c6p = 0.5 (but returning to z = 1.96), we find<\/p>\n

    0.025 = 1.96\u221a0.5X0n.5 which results in n = 1537.<\/p>\n

     <\/p>\n

     <\/p>\n

    Question\u00a0 :\u00a0 A survey\u00a0 estimated that 20% of\u00a0 all\u00a0 \u00a0Indian\u00a0 aged 16 to 20 are quite concerned for their health. A similar survey is planned for US. They want a 95% confidence\u00a0 interval to have a margin of error of 0.04.<\/p>\n

     <\/p>\n

    (a) Find the necessary sample size if they expect to find results similar to those in India<\/p>\n

     <\/p>\n

    (b)Suppose instead they used the\u00a0 conservative formula based on \u02c6p = 0.5. What is now the required sample size?<\/p>\n<\/div>\n

    \n

    Solution:<\/p>\n

     <\/p>\n

    (a) The general formula is<\/p>\n

    E= \u00a0\u00a0\u221a\u00a0 ^(1\u2212\u00a0 )^<\/p>\n

     <\/p>\n

    which translates to<\/p>\n

    n\u00a0\u00a0\u00a0\u00a0 =\u00a0\u00a0 ^(1\u2212\u00a0 ^)\u00a0 2\u00a0\u00a0 2<\/p>\n

    n= 0.2X0.8X1.96X1.96 = 384.2<\/p>\n

    0.04X0.04<\/p>\n

     <\/p>\n

    b) With e = 0.04, p^ = 0.5, z= 1.96 we get<\/span><\/p>\n

    n= 0.5 0.5 1.96 1.96 = 600.25\u00a0<\/span><\/p>\n

    0.04 0.04<\/span><\/p>\n

    Thesamplesizeis384 for(a) and 600 for (b), showing the advantage in using the estimated \u02c6p (0.2) so long as we feel\u00a0confident that this is roughly the right guess.<\/span><\/p>\n<\/div>\n

    \n

     <\/p>\n

     <\/p>\n

    8.\u00a0 Determining the Sample Size with fpc<\/strong><\/p>\n

     <\/p>\n

    Finite population correction factor (fpc) is used to determine sample size when sampling without replacement. The use of such factor reduces the standard error by a value equal to<\/p>\n

    \u221a(\u00a0\u00a0 \u2212\u00a0\u00a0 )\/\u00a0\u00a0 \u2212 1.<\/p>\n

     <\/p>\n

    For example, in estimating the mean, the sampling error is given by<\/p>\n

    e =\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u221a\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u221a(\u00a0\u00a0 \u2212\u00a0\u00a0 )\/\u00a0\u00a0 \u2212 1<\/p>\n

     <\/p>\n

     <\/p>\n

    9.\u00a0 Summary<\/strong><\/p>\n

     <\/p>\n

    One of the most common requests that researchers get from investigating agencies are sample size calculations or sample size justifications for the proposed study. The sample size is the number of experimental units or samples included in a study. Determining the sample size to answer the research question is one of the important requirements in designing a study. In order to calculate the sample size, it is required to have some idea of the results expected in a\u00a0study. In general, the greater the variability in the outcome variable, the larger the sample size required\u00a0<\/span>to assess whether an observed effect is a true effect.<\/p>\n<\/div>\n

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

      \n
    1. Tulsian P.C. and Pandey V. (2002). Quantitative Techniques, Theory & Problems (1st<\/em> edition)<\/em>. New Delhi: Pearson India.<\/li>\n
    2. Black. K (2013) Business Statistics For Contemporary Decision Making (8th Edition) New Delhi: Wiley<\/li>\n
    3. 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
    4. https:\/\/www.unc.edu\/~rls\/s151-2010\/class23.pdf<\/li>\n<\/ol>\n
      \n

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