{"id":"cb_question_bank:d73839e9-ec9e-430f-a071-d84cc5413145","source_id":"cb_question_bank","source_item_id":"d73839e9-ec9e-430f-a071-d84cc5413145","external_id":null,"ibn":"14941-DC","question_id":"79137c1b","program":"SAT","module":"math","domain_code":"Q","domain":"Problem-Solving and Data Analysis","skill_code":"Q.D.","skill":"Two-variable data: Models and scatterplots","difficulty":"H","score_band":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The scatterplot above shows the size x and the sale price y of 25 houses for sale in Town H. Which of the following could be an equation for a line of best fit for the data?","stimulus":"The figure presents a scatterplot titled “Size and Sale Price of Houses in Town H.” The x axis is labeled “Size, in thousands of square feet,” and the numbers 0 through 3, in increments of 1, are indicated. The y axis is labeled “Sale price, in thousands of dollars,” and the numbers 0 through 400, in increments of 50, are indicated. There are 25 data points on the scatterplot. The data points begin in the middle left part of the graph, at the point with coordinates zero point 9 comma one hundred ninety. They trend upward and to the right in clusters, ending at the point with coordinates 2.55 comma 350. A line of best fit for the data is not shown in the graph. However, it appears that if a line of best fit were drawn, it would pass through the point with coordinates zero comma one hundred and the point with coordinates 2 point 5 comma three hundred fifty.","stimulus_html":"<div class=\"stimulus_reference \">\n        <div class=\"standalone_image \">\n          <img src=\"/api/assets/1b0cb776f6921226ab73fc4e23944335b35684bc7ffbe1029629f049cc664f9e\" alt=\"The figure presents a scatterplot titled “Size and Sale Price of Houses in Town H.” The x axis is labeled “Size, in thousands of square feet,” and the numbers 0 through 3, in increments of 1, are indicated. The y axis is labeled “Sale price, in thousands of dollars,” and the numbers 0 through 400, in increments of 50, are indicated. \n\nThere are 25 data points on the scatterplot. The data points begin in the middle left part of the graph, at the point with coordinates zero point 9 comma one hundred ninety. They trend upward and to the right in clusters, ending at the point with coordinates 2.55 comma 350.\n\nA line of best fit for the data is not shown in the graph. However, it appears that if a line of best fit were drawn, it would pass through the point with coordinates zero comma one hundred and the point with coordinates 2 point 5 comma three hundred fifty.   \n\" width=\"189\" height=\"199\"></div>\n      </div>\n","stem_html":"<p class=\"stem_paragraph \">The scatterplot above shows the size <span class=\"italic\">x</span> and the sale price <span class=\"italic\">y</span> of 25 houses for sale in Town H. Which of the following could be an equation for a line of best fit for the data?</p>\n","rationale_html":"<p>Choice B is correct. From the shape of the cluster of points, the line of best fit should pass roughly through the points <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b295bdaf30b8b92e81f537fbe4983d96eb95734907ec9cec69c4f62dda0e08d5\" alt=\"with coordinates 1 comma 200\"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2b911c60c9e1e47496f03ed4931bd1f49336328cd2f606570342e1b8ed83eca8\" alt=\"2 point 5 comma 350\"></span>. Therefore, these two points can be used to find an approximate equation for the line of best fit. The slope of this line of best fit is therefore <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/57e28d88c827e371d1026bd851f1a5cb5018b9759aa3e38ddc312f313c3d6296\" alt=\"the fraction with numerator y sub 2, minus y sub 1, and denominator x sub 2, minus x sub 1, end fraction, equals, the fraction with numerator 350 minus 200, and denominator 2 point 5 minus 1, end fraction\"></span>, or 100. The equation for the line of best fit, in slope-intercept form, is <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f3900f9cdb2e2d492adbaf03e1a1308b0822ddb6acfc7c27d7f6fb8475025318\" alt=\"y equals, 100 x plus b\"></span> for some value of <span class=\"italic\">b</span>. Using the point <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b295bdaf30b8b92e81f537fbe4983d96eb95734907ec9cec69c4f62dda0e08d5\" alt=\"with coordinates 1 comma 200\"></span>, 1 can be substituted for <span class=\"italic\">x</span> and 200 can be substituted for <span class=\"italic\">y</span>: <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2743ba7f9e0a757d95bf59af8553cf073f654e4922a4f3d2910a858d3291a9dd\" alt=\"200 equals, 100 times 1, plus b\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/383fc5c9321f5846114ce5e1873518ac2858cd58370ab61e85e33923f2d12b0e\" alt=\"b equals 100\"></span>. Substituting this value into the slope-intercept form of the equation gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5b6dd3a99a676dd4c1d8a77f4695b0641fa17476e947ba6097d0e0f3419723d4\" alt=\"y equals, 100 x plus 100\"></span>.</p><p>Choice A is incorrect. The line defined by <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/008e2439a37b4f2665285134a8452d5c35d80c7af5e3696f3b7784b04568323c\" alt=\"y equals, 200 x plus 100\"></span> passes through the points <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/aa2af017db9bb9fca9dd5669fffa5f49144c105f5b1f082424f6f7b31c455169\" alt=\"with coordinates 1 comma 300\"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f60ad776a9fa0f214c23d3ad0c755e85ce10bdf6ad1780659eadf4c7efa89a22\" alt=\"2 comma 500\"></span>, both of which are well above the cluster of points, so it cannot be a line of best fit. Choice C is incorrect. The line defined by <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c943b4ec90525c5df0962b76d853dfdc79b22474d462f7a63cd4cd042700e6d5\" alt=\"y equals, 50 x plus 100\"></span> passes through the points <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/8482eb208894e10c4b816e6d24788fa5934cb5bcfe835656f37254212f118972\" alt=\"with coordinates 1 comma 150\"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c98327ae1ff359c7374bd6d4de096b60cf90b27ffb6d58d7b03d4d2b733fd198\" alt=\"2 comma 200\"></span>, both of which lie at the bottom of the cluster of points, so it cannot be a line of best fit. Choice D is incorrect and may result from correctly calculating the slope of a line of best fit but incorrectly assuming the <span class=\"italic\">y</span>-intercept is at <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d025d5d28279edb5c1e9ce6dbf942cee3cc3caaee927a6a6932dfe130c7aac9b\" alt=\"the point with coordinates 0 comma 0\"></span>.</p><p></p>\n","correct_answer":["B"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.639000+00:00","source_updated_at":"2023-08-02T20:25:59.639000+00:00","options":[{"label":"A","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/008e2439a37b4f2665285134a8452d5c35d80c7af5e3696f3b7784b04568323c\" alt=\"y equals, 200 x plus 100\"></span></span>\n          </p>\n","ord":0},{"label":"B","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5b6dd3a99a676dd4c1d8a77f4695b0641fa17476e947ba6097d0e0f3419723d4\" alt=\"y equals, 100 x plus 100\"></span></span>\n          </p>\n","ord":1},{"label":"C","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c943b4ec90525c5df0962b76d853dfdc79b22474d462f7a63cd4cd042700e6d5\" alt=\"y equals, 50 x plus 100\"></span></span>\n          </p>\n","ord":2},{"label":"D","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f9f071bad9fdcd12c7759542568f27896fe72a67f4f52e7adc20bb71a56a3f53\" alt=\"y equals, 100 x\"></span></span>\n          </p>\n","ord":3}],"attribution":{"source_id":"cb_question_bank","source_name":"College Board digital SAT question bank (community dump)","rights_holder":"College Board","canonical_url":"https://satsuitequestionbank.collegeboard.org/","retrieved_from":"https://raw.githubusercontent.com/mdn522/sat-question-bank/main/data/cb-digital-questions.json","attribution":"SAT practice questions © College Board, from the official SAT Suite Question Bank. Retrieved via the community dump at github.com/mdn522/sat-question-bank.","disclaimer":"SAT® and PSAT/NMSQT® are trademarks registered by the College Board, which is not affiliated with, does not sponsor, and does not endorse this project.","license":"College Board copyright; source repository carries no licence. Local use only.","redistributable":false,"item_count":2017}}