{"id":5322,"date":"2022-08-19T17:40:16","date_gmt":"2022-08-19T17:40:16","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=5322"},"modified":"2022-08-19T18:01:16","modified_gmt":"2022-08-19T18:01:16","slug":"12a-inclass","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/12a-inclass\/","title":{"raw":"12A InClass","rendered":"12A InClass"},"content":{"raw":"Airbnb is a website that connects people who are renting out rooms or homes with\u00a0 people looking for accommodations in that area. In this in-class activity, we are going to\u00a0 look at the prices of Airbnb listings under $500 per night in New York City and examine\u00a0 the behavior of sample mean prices for random samples of listings.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 1<\/h3>\r\n1) The average Airbnb listing price in New\u00a0York City is $130 per night.[footnote]Kaggle. (2019). New York City Airbnb open data. https:\/\/www.kaggle.com\/dgomonov\/new-york-city airbnb-open-data\u00a0[\/footnote] Which of the\u00a0following would be more surprising: a\u00a0single randomly selected listing price of\u00a0$200 per night or an average listing price\u00a0of $200 per night in a random sample of 25 listings? Explain.<\/div>\r\n\r\n[caption id=\"\" align=\"alignright\" width=\"494\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/26115819\/Picture26-300x200.jpg\" alt=\"A woman standing with a suitcase looking out over a river in front of a city.\" width=\"494\" height=\"329\" \/> Credit: iStock\/FillippoBacci[\/caption]\r\n\r\nIn-Class Activities 9.B and 9.C introduced you to sampling variability. A sampling\u00a0 distribution is the probability distribution of a sample statistic, such as a sample mean\u00a0 or sample proportion, as it varies from sample to sample. In-Class Activity 9.B\u00a0 introduced you to the sampling distribution of a sample proportion.\r\n\r\nIn this activity, you will explore the sampling distribution of the sample mean for varying\u00a0 sample sizes and discover the Central Limit Theorem as it applies to means.\r\n\r\nGo to the DCMP Sampling Distribution of the Sample Mean (Continuous Population) tool at https:\/\/dcmathpathways.shinyapps.io\/SampDist_cont\/.\r\n\r\nYou will use this tool to simulate random samples of different sizes from all Airbnb listings under $500 in New York City (NYC), where the sample mean price (in USD) is\u00a0 calculated in each sample. Enter the following inputs:\r\n<ul>\r\n \t<li>Select Population Distribution: Real Population Data<\/li>\r\n \t<li>Select Example: New York Airbnb Prices<\/li>\r\n<\/ul>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 2<\/h3>\r\n2) Examine the population distribution of the NYC Airbnb listing prices displayed in the\u00a0 data analysis tool. What shape is this distribution?\r\n\r\n(a) Symmetric\r\n\r\n(b) Skewed right\r\n\r\n(c) Skewed left\r\n\r\n(d) Normal\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 3<\/h3>\r\n3) Before using the tool to generate random samples, let\u2019s make a few predictions.\r\n\r\na) If you were to take 1,000 random samples of [latex]n=2[\/latex] NYC Airbnb listings and\u00a0 calculate the sample mean price for each sample, what shape would you\u00a0 expect the distribution of the sample means to have?\r\n\r\nb) If you were to take 1,000 random samples of [latex]n=10[\/latex] NYC Airbnb listings and\u00a0 calculate the sample mean price for each sample, what shape would you\u00a0 expect the distribution of the sample means to have?\r\n\r\nc) If you were to take 1,000 random samples of [latex]n=50[\/latex] NYC Airbnb listings and\u00a0 calculate the sample mean price for each sample, what shape would you\u00a0 expect the distribution of the sample means to have?\r\n\r\nd) How would you expect the variability in sample mean listing prices to change\u00a0 as the sample size increases?\r\n\r\ne) How would you expect the average of the sample mean listing prices to\u00a0 change as the sample size increases?\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 4<\/h3>\r\n4) Use the tool to generate 1,000 random samples for each of the following sample\u00a0 sizes. Select the \u201cShow Normal Approximation\u201d box to overlay a normal distribution\u00a0 on each simulated sampling distribution. For each sample size:\r\n<ul>\r\n \t<li>Sketch the graph of the resulting sampling distribution of the sample mean listing\u00a0 prices, including the x-axis label and scale (use the same x-axis scale for all\u00a0 three plots) and the overlayed normal distribution.<\/li>\r\n \t<li>Write down the mean of the sampling distribution of the sample mean listing\u00a0 prices.<\/li>\r\n \t<li>Write down the standard deviation of the sampling distribution of the sample\u00a0 mean listing prices.<\/li>\r\n<\/ul>\r\na) [latex]n= 2[\/latex]\r\n\r\nb) [latex]n= 10[\/latex]\r\n\r\nc) [latex]n= 50[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 5<\/h3>\r\n5) Now, you\u2019ll compare the results in Question 4 to your predictions from Question 3.\r\n\r\na) As the sample size increases, how does the shape of the sampling\u00a0 distribution of the mean listing prices change? Does this match the pattern in\u00a0 your predicted shapes from Question 3, Parts A, B, and C?\r\n\r\nb) As the sample size increases, how does the standard deviation of the\u00a0 sampling distribution of the mean listing prices change? Does this match\u00a0 your prediction from Question 3, Part D?\r\n\r\nc) As the sample size increases, how does the mean of the sampling distribution of the mean listing prices change? Does this match your\u00a0 prediction from Question 3, Part E?\r\n\r\n<\/div>\r\nWe can calculate the mean of sample means and standard deviation of sample means\u00a0 through simulation, as in Question 4, or through mathematical formulas. Suppose the\u00a0 mean of the population is \u00b5 and the standard deviation of the population is \u03c3. Then the\u00a0 mean of the sample means is the same as the population mean, \u00b5, but the standard\u00a0 deviation of the sample means decreases with the sample size. Specifically, the\u00a0 standard deviation of the sample means is equal to [latex]\\frac{\\sigma}{\\sqrt{n}}[\/latex].\r\n<div align=\"left\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Sampling Distribution of the Sample Mean\r\n\r\nWhen taking many random samples of size [latex]n[\/latex] from a population distribution with mean [latex]\\mu[\/latex] and standard deviation [latex]\\sigma[\/latex]:\r\n\r\nThe mean of the distribution of the sample means is [latex]\\mu[\/latex].\r\n\r\nThe standard deviation of the distribution of the sample means is [latex]\\frac{\\sigma}{\\sqrt{n}}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 6<\/h3>\r\n6) Examine again the population distribution of the NYC Airbnb listing prices displayed in the data analysis tool. What are the values of [latex]\\mu[\/latex] and [latex]\\sigma[\/latex]?<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 7<\/h3>\r\n7) Calculate the mean and standard deviation of sample mean listing prices for each of\u00a0 the following sample sizes using the mathematical formulas given previously.\r\n\r\na) [latex]n=2[\/latex]\r\n\r\nb) [latex]n=10[\/latex]\r\n\r\nc)\u00a0[latex]n=50[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 8<\/h3>\r\n<span style=\"font-size: 1rem; text-align: initial;\">8) Compare the simulated mean and standard deviation of the sample mean listing prices from Question 4 to those calculated in Question 7. Do the values seem similar? (Hint: They should!)\u00a0<\/span><\/div>\r\nIn In-Class Activity 9.C, you saw the Central Limit Theorem at work for sample\u00a0 proportions. In this activity, you just witnessed the Central Limit Theorem at work for sample means. The Central Limit Theorem states that, as the sample size gets larger,\u00a0 the distribution of the sample mean will become closer to a normal distribution.\u00a0 Combining this with the expressions for the mean and standard deviation of the sample\u00a0 means results in:\r\n<div align=\"left\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Sampling Distribution of the Sample Mean\r\n\r\nWhen taking many random samples of size [latex]n[\/latex] from a population distribution with mean [latex]\\mu[\/latex] and standard deviation [latex]\\sigma[\/latex]:\r\n\r\nThe mean of the distribution of the sample means is [latex]\\mu[\/latex].\r\n\r\nThe standard deviation of the distribution of the sample means is [latex]\\frac{\\sigma}{\\sqrt{n}}[\/latex].\r\n\r\nIf the population distribution is not normal, the Central Limit Theorem states that the\u00a0 distribution of the sample means still follows an approximate normal distribution as long\u00a0 as the sample size is large (e.g., [latex]n\u226530[\/latex]) and the population distribution is not strongly\u00a0 skewed.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 9<\/h3>\r\n9) Suppose you are planning a vacation to Los Angeles (LA), and you would like to\u00a0 learn more about the distribution of the Airbnb listing prices in LA. You take a random sample of 50 LA Airbnb listings. The mean listing price in your sample is\u00a0 $152.\r\n\r\nAssume the population of all LA Airbnb listing prices has the same mean and\u00a0 standard deviation as that for NYC. (You found these in Question 6.)\r\n\r\na) Use the mean and standard deviation you found in Question 7, Part C to\u00a0 calculate the z-score for a sample mean of $152. Write a sentence\u00a0 interpreting this value in the context of the problem.\r\n\r\nb) Using the normal approximation, find the probability of observing a sample\u00a0 mean listing price of $152 or higher from a random sample of 50 listings.\r\n\r\nc) Based on your answers in Parts A and B, do these data provide evidence that the mean Airbnb listing price in LA is higher than the mean Airbnb listing\u00a0 price in NYC? Explain.\r\n\r\n<\/div>","rendered":"<p>Airbnb is a website that connects people who are renting out rooms or homes with\u00a0 people looking for accommodations in that area. In this in-class activity, we are going to\u00a0 look at the prices of Airbnb listings under $500 per night in New York City and examine\u00a0 the behavior of sample mean prices for random samples of listings.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 1<\/h3>\n<p>1) The average Airbnb listing price in New\u00a0York City is $130 per night.<a class=\"footnote\" title=\"Kaggle. (2019). New York City Airbnb open data. https:\/\/www.kaggle.com\/dgomonov\/new-york-city airbnb-open-data\u00a0\" id=\"return-footnote-5322-1\" href=\"#footnote-5322-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> Which of the\u00a0following would be more surprising: a\u00a0single randomly selected listing price of\u00a0$200 per night or an average listing price\u00a0of $200 per night in a random sample of 25 listings? Explain.<\/div>\n<div style=\"width: 504px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/26115819\/Picture26-300x200.jpg\" alt=\"A woman standing with a suitcase looking out over a river in front of a city.\" width=\"494\" height=\"329\" \/><\/p>\n<p class=\"wp-caption-text\">Credit: iStock\/FillippoBacci<\/p>\n<\/div>\n<p>In-Class Activities 9.B and 9.C introduced you to sampling variability. A sampling\u00a0 distribution is the probability distribution of a sample statistic, such as a sample mean\u00a0 or sample proportion, as it varies from sample to sample. In-Class Activity 9.B\u00a0 introduced you to the sampling distribution of a sample proportion.<\/p>\n<p>In this activity, you will explore the sampling distribution of the sample mean for varying\u00a0 sample sizes and discover the Central Limit Theorem as it applies to means.<\/p>\n<p>Go to the DCMP Sampling Distribution of the Sample Mean (Continuous Population) tool at https:\/\/dcmathpathways.shinyapps.io\/SampDist_cont\/.<\/p>\n<p>You will use this tool to simulate random samples of different sizes from all Airbnb listings under $500 in New York City (NYC), where the sample mean price (in USD) is\u00a0 calculated in each sample. Enter the following inputs:<\/p>\n<ul>\n<li>Select Population Distribution: Real Population Data<\/li>\n<li>Select Example: New York Airbnb Prices<\/li>\n<\/ul>\n<div class=\"textbox key-takeaways\">\n<h3>Question 2<\/h3>\n<p>2) Examine the population distribution of the NYC Airbnb listing prices displayed in the\u00a0 data analysis tool. What shape is this distribution?<\/p>\n<p>(a) Symmetric<\/p>\n<p>(b) Skewed right<\/p>\n<p>(c) Skewed left<\/p>\n<p>(d) Normal<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 3<\/h3>\n<p>3) Before using the tool to generate random samples, let\u2019s make a few predictions.<\/p>\n<p>a) If you were to take 1,000 random samples of [latex]n=2[\/latex] NYC Airbnb listings and\u00a0 calculate the sample mean price for each sample, what shape would you\u00a0 expect the distribution of the sample means to have?<\/p>\n<p>b) If you were to take 1,000 random samples of [latex]n=10[\/latex] NYC Airbnb listings and\u00a0 calculate the sample mean price for each sample, what shape would you\u00a0 expect the distribution of the sample means to have?<\/p>\n<p>c) If you were to take 1,000 random samples of [latex]n=50[\/latex] NYC Airbnb listings and\u00a0 calculate the sample mean price for each sample, what shape would you\u00a0 expect the distribution of the sample means to have?<\/p>\n<p>d) How would you expect the variability in sample mean listing prices to change\u00a0 as the sample size increases?<\/p>\n<p>e) How would you expect the average of the sample mean listing prices to\u00a0 change as the sample size increases?<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 4<\/h3>\n<p>4) Use the tool to generate 1,000 random samples for each of the following sample\u00a0 sizes. Select the \u201cShow Normal Approximation\u201d box to overlay a normal distribution\u00a0 on each simulated sampling distribution. For each sample size:<\/p>\n<ul>\n<li>Sketch the graph of the resulting sampling distribution of the sample mean listing\u00a0 prices, including the x-axis label and scale (use the same x-axis scale for all\u00a0 three plots) and the overlayed normal distribution.<\/li>\n<li>Write down the mean of the sampling distribution of the sample mean listing\u00a0 prices.<\/li>\n<li>Write down the standard deviation of the sampling distribution of the sample\u00a0 mean listing prices.<\/li>\n<\/ul>\n<p>a) [latex]n= 2[\/latex]<\/p>\n<p>b) [latex]n= 10[\/latex]<\/p>\n<p>c) [latex]n= 50[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 5<\/h3>\n<p>5) Now, you\u2019ll compare the results in Question 4 to your predictions from Question 3.<\/p>\n<p>a) As the sample size increases, how does the shape of the sampling\u00a0 distribution of the mean listing prices change? Does this match the pattern in\u00a0 your predicted shapes from Question 3, Parts A, B, and C?<\/p>\n<p>b) As the sample size increases, how does the standard deviation of the\u00a0 sampling distribution of the mean listing prices change? Does this match\u00a0 your prediction from Question 3, Part D?<\/p>\n<p>c) As the sample size increases, how does the mean of the sampling distribution of the mean listing prices change? Does this match your\u00a0 prediction from Question 3, Part E?<\/p>\n<\/div>\n<p>We can calculate the mean of sample means and standard deviation of sample means\u00a0 through simulation, as in Question 4, or through mathematical formulas. Suppose the\u00a0 mean of the population is \u00b5 and the standard deviation of the population is \u03c3. Then the\u00a0 mean of the sample means is the same as the population mean, \u00b5, but the standard\u00a0 deviation of the sample means decreases with the sample size. Specifically, the\u00a0 standard deviation of the sample means is equal to [latex]\\frac{\\sigma}{\\sqrt{n}}[\/latex].<\/p>\n<div style=\"text-align: left;\">\n<table>\n<tbody>\n<tr>\n<td>Sampling Distribution of the Sample Mean<\/p>\n<p>When taking many random samples of size [latex]n[\/latex] from a population distribution with mean [latex]\\mu[\/latex] and standard deviation [latex]\\sigma[\/latex]:<\/p>\n<p>The mean of the distribution of the sample means is [latex]\\mu[\/latex].<\/p>\n<p>The standard deviation of the distribution of the sample means is [latex]\\frac{\\sigma}{\\sqrt{n}}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 6<\/h3>\n<p>6) Examine again the population distribution of the NYC Airbnb listing prices displayed in the data analysis tool. What are the values of [latex]\\mu[\/latex] and [latex]\\sigma[\/latex]?<\/p><\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 7<\/h3>\n<p>7) Calculate the mean and standard deviation of sample mean listing prices for each of\u00a0 the following sample sizes using the mathematical formulas given previously.<\/p>\n<p>a) [latex]n=2[\/latex]<\/p>\n<p>b) [latex]n=10[\/latex]<\/p>\n<p>c)\u00a0[latex]n=50[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 8<\/h3>\n<p><span style=\"font-size: 1rem; text-align: initial;\">8) Compare the simulated mean and standard deviation of the sample mean listing prices from Question 4 to those calculated in Question 7. Do the values seem similar? (Hint: They should!)\u00a0<\/span><\/div>\n<p>In In-Class Activity 9.C, you saw the Central Limit Theorem at work for sample\u00a0 proportions. In this activity, you just witnessed the Central Limit Theorem at work for sample means. The Central Limit Theorem states that, as the sample size gets larger,\u00a0 the distribution of the sample mean will become closer to a normal distribution.\u00a0 Combining this with the expressions for the mean and standard deviation of the sample\u00a0 means results in:<\/p>\n<div style=\"text-align: left;\">\n<table>\n<tbody>\n<tr>\n<td>Sampling Distribution of the Sample Mean<\/p>\n<p>When taking many random samples of size [latex]n[\/latex] from a population distribution with mean [latex]\\mu[\/latex] and standard deviation [latex]\\sigma[\/latex]:<\/p>\n<p>The mean of the distribution of the sample means is [latex]\\mu[\/latex].<\/p>\n<p>The standard deviation of the distribution of the sample means is [latex]\\frac{\\sigma}{\\sqrt{n}}[\/latex].<\/p>\n<p>If the population distribution is not normal, the Central Limit Theorem states that the\u00a0 distribution of the sample means still follows an approximate normal distribution as long\u00a0 as the sample size is large (e.g., [latex]n\u226530[\/latex]) and the population distribution is not strongly\u00a0 skewed.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 9<\/h3>\n<p>9) Suppose you are planning a vacation to Los Angeles (LA), and you would like to\u00a0 learn more about the distribution of the Airbnb listing prices in LA. You take a random sample of 50 LA Airbnb listings. The mean listing price in your sample is\u00a0 $152.<\/p>\n<p>Assume the population of all LA Airbnb listing prices has the same mean and\u00a0 standard deviation as that for NYC. (You found these in Question 6.)<\/p>\n<p>a) Use the mean and standard deviation you found in Question 7, Part C to\u00a0 calculate the z-score for a sample mean of $152. Write a sentence\u00a0 interpreting this value in the context of the problem.<\/p>\n<p>b) Using the normal approximation, find the probability of observing a sample\u00a0 mean listing price of $152 or higher from a random sample of 50 listings.<\/p>\n<p>c) Based on your answers in Parts A and B, do these data provide evidence that the mean Airbnb listing price in LA is higher than the mean Airbnb listing\u00a0 price in NYC? Explain.<\/p>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-5322-1\">Kaggle. (2019). New York City Airbnb open data. https:\/\/www.kaggle.com\/dgomonov\/new-york-city airbnb-open-data\u00a0 <a href=\"#return-footnote-5322-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":23592,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5322","chapter","type-chapter","status-publish","hentry"],"part":5315,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5322","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/23592"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5322\/revisions"}],"predecessor-version":[{"id":5328,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5322\/revisions\/5328"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/5315"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5322\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5322"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5322"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5322"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5322"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}