document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196642>Question 1196642</A>: <I><SPAN STYLE=\"word-break: break-all;\"> HERE IS THE TABLE: https://imagizer.imageshack.com/img923/3314/zi0hjD.jpg  \r<BR>\n" );
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document.write( " A sample of 1793 homes in a neighborhood of a large city during a recent year produced the data in the table for the number of bedrooms and house price. Is the price of the house independent of whether it has 3 or more​ bedrooms? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #829587 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=829587\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "We'll be doing a ChiSquare Test of Independence.\r<BR>\n" );
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document.write( "Null Hypothesis: The two variables \"price\" and \"bedroom count\" are independent. There's no connection between them. Changing one variable does not cause the other to change.\r<BR>\n" );
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document.write( "Alternative Hypothesis: The two variables \"price\" and \"bedroom count\" are dependent. There's a connection somehow. Changing one variable causes a change in the other.\r<BR>\n" );
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document.write( "Since the level of significance isn't stated, I'll assume the standard default of alpha = 0.05\r<BR>\n" );
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document.write( "Given Data <TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD  colspan=2>3 or More Bedrooms</TD></tr><tr><TD >House Price</TD><TD >Yes</TD><TD >No</TD></tr><tr><TD >Less than $300K</TD><TD >344</TD><TD >499</TD></tr><tr><TD >$300-450K</TD><TD >190</TD><TD >334</TD></tr><tr><TD >$450-600K</TD><TD >133</TD><TD >293</TD></tr>\n" );
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document.write( "Compute the row and column totals by adding the values.<BR>\n" );
document.write( "Eg: 344+499 = 843 total houses are priced less than $300K <TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD  colspan=2>3 or More Bedrooms</TD><TD ></TD></tr><tr><TD >House Price</TD><TD >Yes</TD><TD >No</TD><TD >Total</TD></tr><tr><TD >Less than $300K</TD><TD >344</TD><TD >499</TD><TD >843</TD></tr><tr><TD >$300-450K</TD><TD >190</TD><TD >334</TD><TD >524</TD></tr><tr><TD >$450-600K</TD><TD >133</TD><TD >293</TD><TD >426</TD></tr><tr><TD >Total</TD><TD >667</TD><TD >1126</TD><TD >1793</TD></tr>\n" );
document.write( "</TABLE>The fact we get 1793 as the grand total helps act as a checksum of sorts.\r<BR>\n" );
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document.write( "The given data table represents the set of observed values. With ChiSquare problems, the letter O represents \"observed values\". <BR>\n" );
document.write( "This is unfortunate since this letter looks a lot like the number zero.\r<BR>\n" );
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document.write( "The letter E represents expected values. To compute each expected value, we multiply the row and column totals, and then divide by the grand total N = 1793\r<BR>\n" );
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document.write( "For example, we expect the value 313.597881 in row1,column1 since:<BR>\n" );
document.write( "E = (rowTotal*columnTotal)/N<BR>\n" );
document.write( "E = (843*667)/1793<BR>\n" );
document.write( "E = 313.59788064696<BR>\n" );
document.write( "E = 313.597881<BR>\n" );
document.write( "For more information and further examples, check out this page<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://online.stat.psu.edu/stat200/book/export/html/230\">https://online.stat.psu.edu/stat200/book/export/html/230</a>\r<BR>\n" );
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document.write( "We'll follow that idea for the remaining other cells to get this set of expected values<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD  colspan=2>3 or More Bedrooms</TD></tr><tr><TD >House Price</TD><TD >Yes</TD><TD >No</TD></tr><tr><TD >Less than $300K</TD><TD >313.597881</TD><TD >529.402119</TD></tr><tr><TD >$300-450K</TD><TD >194.929169</TD><TD >329.070831</TD></tr><tr><TD >$450-600K</TD><TD >158.47295</TD><TD >267.52705</TD></tr>\n" );
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document.write( "Once again:<BR>\n" );
document.write( "O = observed value<BR>\n" );
document.write( "E = expected value<BR>\n" );
document.write( "We'll compute the value <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28O-E%29%5E2%29%2FE ALIGN=MIDDLE  ALT=\"%28%28O-E%29%5E2%29%2FE\"   DISABLED_style= \"max-width:90%; height:auto;\" > for each of the 6 cells\r<BR>\n" );
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document.write( "Here's an example calculation for the item in row1,column1.<BR>\n" );
document.write( "O = 344<BR>\n" );
document.write( "E = 313.597881<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28O-E%29%5E2%29%2FE=%28%28344-313.597881%29%5E2%29%2F313.597881+=2.94737 ALIGN=MIDDLE  ALT=\"%28%28O-E%29%5E2%29%2FE=%28%28344-313.597881%29%5E2%29%2F313.597881+=2.94737\"   DISABLED_style= \"max-width:90%; height:auto;\" > approximately.<BR>\n" );
document.write( "Similar calculations will apply to the remaining five other cells\r<BR>\n" );
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document.write( "This is what that table of values looks like<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD  colspan=2>3 or More Bedrooms</TD></tr><tr><TD >House Price</TD><TD >Yes</TD><TD >No</TD></tr><tr><TD >Less than $300K</TD><TD >2.94737</TD><TD >1.745911</TD></tr><tr><TD >$300-450K</TD><TD >0.124644</TD><TD >0.073834</TD></tr><tr><TD >$450-600K</TD><TD >4.094523</TD><TD >2.425441</TD></tr>\n" );
document.write( "</TABLE>At this point we add up all of those values<BR>\n" );
document.write( "2.94737+1.745911+0.124644+0.073834+4.094523+2.425441 = 11.411723\r<BR>\n" );
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document.write( "Therefore,<BR>\n" );
document.write( "ChiSquare = 11.411723<BR>\n" );
document.write( "This is the test statistic in which we'll compare to the critical value.\r<BR>\n" );
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document.write( "Now you'll refer to a ChiSquare table which is found in the appendix section of your stats textbook.\r<BR>\n" );
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document.write( "Check out this page for more info<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.statology.org/how-to-read-chi-square-distribution-table/\">https://www.statology.org/how-to-read-chi-square-distribution-table/</a><BR>\n" );
document.write( "That page explains how to read a ChiSquare table. \r<BR>\n" );
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document.write( "In this case we have<BR>\n" );
document.write( "r = 3 rows<BR>\n" );
document.write( "c = 2 columns<BR>\n" );
document.write( "df = degrees of freedom for a ChiSquare independence test<BR>\n" );
document.write( "df = (r-1)*(c-1)<BR>\n" );
document.write( "df = (3-1)*(2-1)<BR>\n" );
document.write( "df = 2\r<BR>\n" );
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document.write( "Locate the df = 2 row<BR>\n" );
document.write( "Then locate the column with header 0.05 to represent the alpha value<BR>\n" );
document.write( "The value 5.991 is found here<BR>\n" );
document.write( "What this means is that P(ChiSquare > 5.991) = 0.05 approximately when df = 2.<BR>\n" );
document.write( "About 5% of the area is to the right of the critical value 5.991 when df = 2.\r<BR>\n" );
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document.write( "To recap we found that:<BR>\n" );
document.write( "ChiSquare Test Statistic = 11.411723<BR>\n" );
document.write( "ChiSquare Critical Value = 5.991\r<BR>\n" );
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document.write( "Keep in mind that ChiSquare tests are always right-tailed. This fact is mentioned in the first link I posted.\r<BR>\n" );
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document.write( "So this fact means that the rejection region is to the right of the critical value 5.991<BR>\n" );
document.write( "The test statistic (11.411723) is greater than the critical value (5.991); therefore we are in the rejection region and we reject the null.\r<BR>\n" );
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document.write( "We reject the idea the variables are independent, so we conclude the <font color=red>price is <u>dependent</u> on the number of bedrooms</font>. This makes sense. A bigger house should cost more. Of course there are other factors to consider such as location and the condition of the house, among other things. \r<BR>\n" );
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document.write( "Here are a few handy calculators to check your work:<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.mathsisfun.com/data/chi-square-calculator.html\">https://www.mathsisfun.com/data/chi-square-calculator.html</a><BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.statology.org/chi-square-test-of-independence-calculator/\">https://www.statology.org/chi-square-test-of-independence-calculator/</a><BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.gigacalculator.com/calculators/chi-square-calculator.php\">https://www.gigacalculator.com/calculators/chi-square-calculator.php</a>\r<BR>\n" );
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document.write( "-------------------------\r<BR>\n" );
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document.write( "Note: if the p-value is less than alpha = 0.05, then reject the null. Otherwise, you would fail to reject it (i.e. \"accept\" the null). In this case, the p-value is roughly 0.003326 which helps confirm why we reject the null.<BR>\n" );
document.write( "Here's what the command looks like to compute the ChiSquare p-value on a TI83<BR>\n" );
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document.write( "To get to the command, press the key labeled \"2nd\" then press the VARS key. Scroll down to item 7.<BR>\n" );
document.write( "If you don't have a TI83 or TI84 type calculator, then you can use one of the previous links mentioned earlier. Many more can be found by searching out \"chi square calculator\". \r<BR>\n" );
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document.write( "The command CHIDIST is used to compute ChiSquare p-values in spreadsheet programs like Excel. \r<BR>\n" );
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document.write( "Edit: Here's what it would look like using a spreadsheet. You don't need a Google account to access this Google Sheets document.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://docs.google.com/spreadsheets/d/1CFeiCWPokyyataD6hwabsFZnCSMSxfTw08qwJvGq_R8/edit?usp=sharing\">https://docs.google.com/spreadsheets/d/1CFeiCWPokyyataD6hwabsFZnCSMSxfTw08qwJvGq_R8/edit?usp=sharing</a><BR>\n" );
document.write( "You can click on any cell to see what formulas were used.<BR>\n" );
document.write( "</font> </SPAN>\n" );
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