document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1081803>Question 1081803</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Do “A” students tend to sit in a particular part of the classroom?  An instructor recorded the locations of the students who received grades of A, with these results: 14 sat in the front, 12 sat in the middle, and 5 sat in the back of the classroom. Is there sufficient evidence to support the claim that the “A” students are not evenly distributed throughout the classroom? [Construct an appropriate frequency table.]  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #696024 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=696024\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Note: The teacher didn't state it but I'm assuming alpha = 0.05 which is often the default significance level.\r<BR>\n" );
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document.write( "First generate the proper frequency table shown below\r<BR>\n" );
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document.write( "Note: Table generated by Excel, which is part of Microsoft Office. If you cannot afford to get this program, then there are free alternatives out there such as Open Office, Libre Office, Google Doc Spreadsheets, etc which pretty much can do the same thing. I'm sure there are tons more free programs out there. \r<BR>\n" );
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document.write( "Explanation of the table:\r<BR>\n" );
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document.write( "The first column represented with an \"A\" at the top is simply a column for the row labels \"Front\", \"Middle\", and \"Back\". Also, there's a \"Total\" label to indicate sums for any given column of values. \r<BR>\n" );
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document.write( "The second column, represented with \"B\" up top\", represents the number of observed entries for the front, middle, and back (rows 2 through 4). For instance, the 14 in cell B2 represents the idea that the teacher observed 14 students in the front. I'm coding \"Observed\" as \"OBS\" for short so I can use it later in the table.\r<BR>\n" );
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document.write( "The third column, represented with \"C\" up top, is the expected results we should get if students were evenly distributed. If 31 students are evenly divided by 3 different sections (front,middle,back) then we'd expect roughly 31/3 = 10.33333333 students per section. This is an average value. Do NOT round this value. We'll use this later in our calculations.\r<BR>\n" );
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document.write( "The fourth column represents the difference between columns B and C. You can notate this as B-C or as OBS-EXP like I have done in the table. Each entry is simply the result of subtracting the expected from the observed value. For example, the value 3.666666667 in cell D2 is the result of the subtraction 14-10.33333333, basically D2 = B2-C2. The rest of the values are calculated the same way. \r<BR>\n" );
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document.write( "The \"E\" column, which is the fifth column, is the result of squaring what we got in column D. Put one way, E = (D)^2. I've notated it as (OBS-EXP)^2. For example, (D2)^2 = (3.666666667)^2 = 13.44444444\r<BR>\n" );
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document.write( "The \"F\" column, the last column, is the result of dividing what we got in column E over the column C. In other words, divide the expression (OBS-EXP)^2 over EXP. I've notated this as [ (OBS-EXP)^2 ]/EXP\r<BR>\n" );
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document.write( "Once we have our F column set up, add up everything in it. You can add by typing the individual values into the calculator or you can use the <font color=green>SUM</font> Excel function (OpenOffice and LibreOffice and others have this function too)\r<BR>\n" );
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document.write( "The portion <font color=blue>Chi-Square Critical Value 5.991464547</font>, specifically the numeric entry, is found by typing in <font color=green>=CHIINV(0.05,2)</font> into the spreadsheet. The equal sign up front tells the spreadsheet \"this is a formula to be calculated\" instead of some text that is displayed. The first argument 0.05 is the alpha value. This is the area in the right tail. The '2' refers to the degrees of freedom df = k-1 = 3-1 = 2. The value k = 3 indicates the number of groups (front,middle,back).\r<BR>\n" );
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document.write( "After you're comfortable setting up the table and reading the values, the only values that we really care about are marked in <font color=red>red</font>, which is <font color=red>4.322580645</font> (<font color=red>Chi-Square test statistic</font>), and the value marked in <font color=blue>blue</font>, which is <font color=blue>5.991464547</font> (<font color=blue>Chi-Square critical value</font>).\r<BR>\n" );
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document.write( "The Chi-Square test statistic and critical value will help us do a <a  rel=nofollow HREF=\"http://stattrek.com/chi-square-test/goodness-of-fit.aspx\">chi-square goodness of fit test</a>. This is test that determines if one group of values fit a specified distribution. In this case, our given group of values is the set of observed values {front=14,middle=12,back=5} and we want to test if this observed group fits with the expected distribution {front=10.33333333,middle=10.33333333,back=10.33333333}\r<BR>\n" );
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document.write( "The null hypothesis would be <BR>\n" );
document.write( "H0: The students are evenly distributed\r<BR>\n" );
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document.write( "The alternative hypothesis would be <BR>\n" );
document.write( "H1: The students are not evenly distributed\r<BR>\n" );
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document.write( "The claim is in the alternative hypothesis. If we reject the null, then we accept the claim. Take note how the null and alternative are the only two options possible. This binary nature is important when setting up hypotheses.\r<BR>\n" );
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document.write( "Using the <font color=blue>Chi-Square critical value</font> of <font color=blue>5.991464547</font> and the <font color=red>Chi-Square test statistic</font> of <font color=red>4.322580645</font>, we simply compare to see which is larger. If the test-statistic is larger, then we are in the rejection region and we reject the null. Otherwise, we fail to reject the null and we have no choice but to accept the null.\r<BR>\n" );
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document.write( "In this case, the test statistic is smaller so we are in the acceptance region. We have no choice but to accept the null. We do not have enough statistically significant evidence to overturn the null. The difference is simply due to randomness or statistical error or other unknown factors. \r<BR>\n" );
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document.write( "We can see this on a graph\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/hG3Yfc1.png\"><BR>\n" );
document.write( "Image generated by <a  rel=nofollow HREF= \"https://www.geogebra.org/home?ggbLang=en\">GeoGebra</a> (free graphing software).\r<BR>\n" );
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document.write( "The red point B represents the location of the test statistic which is NOT in the rejection region (shown in green). The point A represents the critical value. This is the boundary between the rejection region and the acceptance region. \r<BR>\n" );
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document.write( "Conclusion: We have accepted the null meaning that the students are evenly distributed. The seating arrangement doesn't seem to have an affect on grades.\r<BR>\n" );
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document.write( "Fitting it in the framework of a Q&A format, we'd have\r<BR>\n" );
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document.write( "Question: \"Is there sufficient evidence to support the claim that the A students are not evenly distributed throughout the classroom?\"\r<BR>\n" );
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document.write( "Answer: \"No, there isn't sufficient evidence to support the claim that A students are not evenly distributed throughout the classroom\" </SPAN>\n" );
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