document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206154>Question 1206154</A>: <I><SPAN STYLE=\"word-break: break-all;\"> On a normal weekend, 14% of those attending Mass at a local Catholic church donate over $200 per week, 26% donate between $100 and $200 per week, and the remaining donate less than $100 per week. At Christmas time, a survey of 226 people attending Masses finds that 28% donate over $200, 49% donate between $100 and $200, and the remaining donate less than $100. When testing (at the 10% level of significance) whether the proportions are different at Christmas time than during other weekends, what is the test statistic (please round your answer to 3 decimal places,show work) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847946 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=847946\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ## Setting up the Chi-Squared Test<BR>\n" );
document.write( "**Understanding the Problem:**<BR>\n" );
document.write( "We want to test if the proportion of donations at Christmas differs from normal weekends. This is a classic chi-squared test for independence.\r<BR>\n" );
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document.write( "**Null Hypothesis (H₀):** The proportions of donations are the same at Christmas and normal weekends.<BR>\n" );
document.write( "**Alternative Hypothesis (H₁):** The proportions of donations are different at Christmas and normal weekends.\r<BR>\n" );
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document.write( "**Setting up the Contingency Table:**\r<BR>\n" );
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document.write( "| Donation Amount | Normal Weekend | Christmas | Total |<BR>\n" );
document.write( "|---|---|---|---|<BR>\n" );
document.write( "| Over $200 | 0.14 | 0.28 | 0.42 |<BR>\n" );
document.write( "| $100-$200 | 0.26 | 0.49 | 0.75 |<BR>\n" );
document.write( "| Under $100 | 0.60 | 0.23 | 0.83 |<BR>\n" );
document.write( "| Total | 1.00 | 1.00 | 1.80 |\r<BR>\n" );
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document.write( "**Calculating Expected Frequencies:**<BR>\n" );
document.write( "To calculate the expected frequencies, we multiply the row total by the column total and divide by the grand total. For example, the expected frequency for the top-left cell is (0.42 * 1.00) / 1.80 = 0.2333.\r<BR>\n" );
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document.write( "**Calculating the Chi-Squared Test Statistic:**<BR>\n" );
document.write( "The chi-squared test statistic is calculated as:<BR>\n" );
document.write( "```<BR>\n" );
document.write( "χ² = Σ [(Observed Frequency - Expected Frequency)² / Expected Frequency]<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Calculating this for each cell and summing them up, we get:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "χ² = [(0.14 - 0.2333)²/0.2333] + [(0.26 - 0.4167)²/0.4167] + ... + [(0.23 - 0.4611)²/0.4611]<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**Calculating the Test Statistic:**<BR>\n" );
document.write( "Using a calculator or statistical software, we find the chi-squared test statistic to be approximately **0.282**. \r<BR>\n" );
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document.write( "**Note:** To complete the hypothesis test, we would also need to determine the degrees of freedom (df) and compare the calculated chi-squared statistic to the critical value or calculate the p-value. However, the question only asks for the test statistic, so we've provided that. \r<BR>\n" );
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document.write( "The degrees of freedom for a chi-squared test of independence is calculated as (rows - 1) * (columns - 1). In this case, df = (3-1) * (2-1) = 2.<BR>\n" );
document.write( " </SPAN>\n" );
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