document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190619>Question 1190619</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Random samples of 200 men, all retired were classified according to education and number of children is as shown below <BR>\n" );
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document.write( "No of children\r<BR>\n" );
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document.write( "Education <BR>\n" );
document.write( "0 to 1 <BR>\n" );
document.write( "2 to 3<BR>\n" );
document.write( "Over 3\r<BR>\n" );
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document.write( "Elementary <BR>\n" );
document.write( "14<BR>\n" );
document.write( "37<BR>\n" );
document.write( "32\r<BR>\n" );
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document.write( "Secondary and above <BR>\n" );
document.write( "31<BR>\n" );
document.write( "59<BR>\n" );
document.write( "27\r<BR>\n" );
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document.write( "Test the hypothesis that the size of the family is independent of the level of education attained by fathers. (Use 5% level of significance) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849243 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=849243\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to perform a Chi-Square test for independence to determine if family size and education level are independent:\r<BR>\n" );
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document.write( "**1. State the Hypotheses:**\r<BR>\n" );
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document.write( "*   **Null Hypothesis (H₀):** Family size and education level are independent.<BR>\n" );
document.write( "*   **Alternative Hypothesis (H₁):** Family size and education level are not independent (they are associated).\r<BR>\n" );
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document.write( "**2. Set the Significance Level:**\r<BR>\n" );
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document.write( "α = 0.05 (given)\r<BR>\n" );
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document.write( "**3. Create the Contingency Table (Observed Frequencies):**\r<BR>\n" );
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document.write( "You already have this:\r<BR>\n" );
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document.write( "| No. of Children | 0 to 1 | 2 to 3 | Over 3 | Total (Row) |<BR>\n" );
document.write( "|---|---|---|---|---|<BR>\n" );
document.write( "| Elementary | 14 | 37 | 32 | 83 |<BR>\n" );
document.write( "| Secondary & Above | 31 | 59 | 27 | 117 |<BR>\n" );
document.write( "| Total (Column) | 45 | 96 | 59 | 200 |\r<BR>\n" );
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document.write( "**4. Calculate Expected Frequencies:**\r<BR>\n" );
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document.write( "The expected frequency for each cell is calculated as:\r<BR>\n" );
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document.write( "(Row Total * Column Total) / Grand Total\r<BR>\n" );
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document.write( "Here's the table of expected frequencies:\r<BR>\n" );
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document.write( "| No. of Children | 0 to 1 | 2 to 3 | Over 3 |<BR>\n" );
document.write( "|---|---|---|---|<BR>\n" );
document.write( "| Elementary | (83*45)/200 = 18.68 | (83*96)/200 = 39.84 | (83*59)/200 = 24.48 |<BR>\n" );
document.write( "| Secondary & Above | (117*45)/200 = 26.33 | (117*96)/200 = 56.16 | (117*59)/200 = 34.52 |\r<BR>\n" );
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document.write( "**5. Calculate the Chi-Square Statistic (χ²):**\r<BR>\n" );
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document.write( "χ² = Σ [(Observed Frequency - Expected Frequency)² / Expected Frequency]\r<BR>\n" );
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document.write( "χ² = [(14-18.68)²/18.68] + [(37-39.84)²/39.84] + [(32-24.48)²/24.48] + [(31-26.33)²/26.33] + [(59-56.16)²/56.16] + [(27-34.52)²/34.52]\r<BR>\n" );
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document.write( "χ² ≈ 1.23 + 0.20 + 2.26 + 0.85 + 0.14 + 2.09\r<BR>\n" );
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document.write( "χ² ≈ 6.77\r<BR>\n" );
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document.write( "**6. Determine Degrees of Freedom (df):**\r<BR>\n" );
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document.write( "df = (Number of Rows - 1) * (Number of Columns - 1)<BR>\n" );
document.write( "df = (2 - 1) * (3 - 1) = 1 * 2 = 2\r<BR>\n" );
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document.write( "**7. Find the Critical Value:**\r<BR>\n" );
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document.write( "Using a Chi-Square distribution table with df = 2 and α = 0.05, the critical value is approximately 5.99.\r<BR>\n" );
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document.write( "**8. Make a Decision:**\r<BR>\n" );
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document.write( "*   **Compare χ² to the critical value:** Our calculated χ² (6.77) is *greater* than the critical value (5.99).\r<BR>\n" );
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document.write( "*   **Conclusion:** Because our χ² value exceeds the critical value, we reject the null hypothesis.\r<BR>\n" );
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document.write( "**9. Interpret the Results:**\r<BR>\n" );
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document.write( "There is sufficient evidence at the 5% level of significance to conclude that family size and education level are *not* independent.  There is an association between the two variables.<BR>\n" );
document.write( " </SPAN>\n" );
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