{"id":45731,"date":"2024-04-15T22:02:11","date_gmt":"2024-04-15T22:02:11","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=45731"},"modified":"2024-04-15T22:02:11","modified_gmt":"2024-04-15T22:02:11","slug":"which-of-the-following-theorem-states-that-the-distribution-of-averages-of-iid-variables-properly-normalized-becomes-that-of-a-standard-normal-as-the-sample-size-increases","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/which-of-the-following-theorem-states-that-the-distribution-of-averages-of-iid-variables-properly-normalized-becomes-that-of-a-standard-normal-as-the-sample-size-increases\/","title":{"rendered":"Which of the following theorem states that the distribution of averages of iid variables, properly normalized, becomes that of a standard normal as the sample size increases?"},"content":{"rendered":"<p>\r\n    <!-- Check if it's an AMP page -->\r\n            <!-- Non-AMP version -->\r\n        <div class=\"mcq-container\" data-quiz-id=\"quizState_6a97565945106\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"true\">\r\n                    Central Limit Theorem                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    Central Mean Theorem                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"false\">\r\n                    Centroid Limit Theorem                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"false\">\r\n                    All of the mentioned                <\/div>\r\n                            \r\n            <!-- Feedback messages for non-AMP -->\r\n            <div class=\"feedback\" data-feedback=\"wrong\">Answer is Right!<\/div>\r\n            <div class=\"feedback\" data-feedback=\"right\">Answer is Wrong!<\/div>\r\n        <\/div>\r\n\r\n        <script>\r\n        document.addEventListener('DOMContentLoaded', function () {\r\n            var containers = document.querySelectorAll('.mcq-container');\r\n\r\n            containers.forEach(function(container) {\r\n                var options = container.querySelectorAll('.option');\r\n                var feedbackSelect = container.querySelector('[data-feedback=\"select\"]');\r\n                var feedbackWrong = container.querySelector('[data-feedback=\"wrong\"]');\r\n                var feedbackRight = container.querySelector('[data-feedback=\"right\"]');\r\n\r\n                options.forEach(function(option) {\r\n                    option.addEventListener('click', function() {\r\n                        var selectedOption = option.getAttribute('data-option-key');\r\n                        var isCorrect = option.getAttribute('data-is-correct') === 'true';\r\n\r\n                        \/\/ Remove previous selections\r\n                        options.forEach(function(opt) {\r\n                            opt.classList.remove('correct', 'incorrect');\r\n                        });\r\n\r\n                        \/\/ Add the correct\/incorrect class\r\n                        if (isCorrect) {\r\n                            option.classList.add('correct');\r\n                            feedbackRight.hidden = false;\r\n                            feedbackWrong.hidden = true;\r\n                        } else {\r\n                            option.classList.add('incorrect');\r\n                            feedbackRight.hidden = true;\r\n                            feedbackWrong.hidden = false;\r\n                        }\r\n\r\n                        \/\/ Hide select feedback\r\n                        feedbackSelect.hidden = true;\r\n                    });\r\n                });\r\n            });\r\n        });\r\n        <\/script>\r\n    \r\n    <!--more--><\/p>\n<p>The correct answer is: <strong>A. Central Limit Theorem<\/strong><\/p>\n<p>The Central Limit Theorem (CLT) states that, given a sufficiently large sample size, the sampling distribution of the arithmetic mean of a variable will be approximately normally distributed, regardless of the underlying distribution of the variable.<\/p>\n<p>The CLT is a powerful tool that can be used to make inferences about a population based on <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                    <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> a sample. For example, if we know that the average height of a population is 6 feet, and we take a sample of 100 people from that population, we can use the CLT to estimate the probability that the average height of the sample is greater than 6 feet 2 inches.<\/p>\n<p>The CLT is based on the following assumptions:<\/p>\n<ul>\n<li>The population is normally distributed.<\/li>\n<li>The sample is randomly selected from the population.<\/li>\n<li>The sample <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div> size is large enough.<\/li>\n<\/ul>\n<p>The CLT is a powerful tool, but it is important to remember that it is only an approximation. The actual distribution of the sample mean may not be exactly normal, especially if the sample size is small. However, the CLT becomes more accurate as the sample size increases.<\/p>\n<p>The Central Mean Theorem (CMT) is a special case of the CLT. The CMT states that, given a sufficiently large sample size, the sampling distribution of the mean of a variable will be normally distributed, regardless of the underlying distribution of the variable, as long as the variable has a finite mean and variance.<\/p>\n<p>The Centroid Limit Theorem (CLT) is a generalization of the CLT. The CLT states that, given a sufficiently large sample size, the sampling distribution of the centroid of a variable will be normally distributed, regardless of the underlying distribution of the variable, as long as the variable has a finite mean and variance.<\/p>\n<p>The CLT, CMT, and CLT are all important theorems in statistics. They can be used to make inferences about a population based on a sample.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Join Our Telegram Channel Subscribe on YouTube<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[738],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Which of the following theorem states that the distribution of averages of iid variables, properly normalized, becomes that of a standard normal as the sample size increases?<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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