document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193979>Question 1193979</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A researcher is investigating the relationship between academic performance and self-esteem. A sample of 150 ten-year old<BR>\n" );
document.write( "children is obtained, and each child is classified by level of academic performance and level of self-esteem. Test at 0.05 level of<BR>\n" );
document.write( "significance.<BR>\n" );
document.write( "Level of acad <BR>\n" );
document.write( "performance/ level <BR>\n" );
document.write( "of self esteem.      High.  Medium. Low<BR>\n" );
document.write( "      High            17.     32.    11<BR>\n" );
document.write( "      Low.            13.     43.    34  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848542 by </B><A HREF=/tutors/aboutme.mpl?userid=parmen><B>parmen(42)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=parmen><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=848542\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Set up Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** There is no association between academic performance and self-esteem in the population of ten-year-old children. <BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** There is an association between academic performance and self-esteem in the population of ten-year-old children.\r<BR>\n" );
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document.write( "**2. Calculate Expected Frequencies**\r<BR>\n" );
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document.write( "* **Create a table of expected frequencies:**<BR>\n" );
document.write( "    | Level of Academic Performance | High Self-Esteem | Medium Self-Esteem | Low Self-Esteem | Row Total |<BR>\n" );
document.write( "    |---|---|---|---|---|<BR>\n" );
document.write( "    | High | (60*60)/150 = 24 | (60*75)/150 = 30 | (60*15)/150 = 18 | 60 |<BR>\n" );
document.write( "    | Low | (90*60)/150 = 36 | (90*75)/150 = 45 | (90*15)/150 = 27 | 90 |<BR>\n" );
document.write( "    | Column Total | 60 | 75 | 15 | 150 |\r<BR>\n" );
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document.write( "* **Where:**<BR>\n" );
document.write( "    * Row Total: Total number of children in each academic performance level.<BR>\n" );
document.write( "    * Column Total: Total number of children in each self-esteem level.<BR>\n" );
document.write( "    * Grand Total: Total number of children in the sample (150).\r<BR>\n" );
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document.write( "**3. Calculate the Chi-Square Test Statistic**\r<BR>\n" );
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document.write( "* **Formula:** χ² = Σ [(O - E)² / E] <BR>\n" );
document.write( "    * Where:<BR>\n" );
document.write( "        * O = Observed frequency in each cell<BR>\n" );
document.write( "        * E = Expected frequency in each cell\r<BR>\n" );
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document.write( "* **Calculate for each cell:**<BR>\n" );
document.write( "    * (17-24)²/24 = 2.04<BR>\n" );
document.write( "    * (32-30)²/30 = 0.13<BR>\n" );
document.write( "    * (11-18)²/18 = 2.72<BR>\n" );
document.write( "    * (13-36)²/36 = 12.5<BR>\n" );
document.write( "    * (43-45)²/45 = 0.09<BR>\n" );
document.write( "    * (34-27)²/27 = 1.70\r<BR>\n" );
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document.write( "* **Sum the values:** χ² = 2.04 + 0.13 + 2.72 + 12.5 + 0.09 + 1.70 = 19.18\r<BR>\n" );
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document.write( "**4. Determine the Degrees of Freedom**\r<BR>\n" );
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document.write( "* Degrees of Freedom (df) = (Number of rows - 1) * (Number of columns - 1) = (2 - 1) * (3 - 1) = 2\r<BR>\n" );
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document.write( "**5. Find the Critical Value**\r<BR>\n" );
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document.write( "* Using a chi-square distribution table, find the critical value for α = 0.05 and df = 2. <BR>\n" );
document.write( "* The critical value is approximately 5.991.\r<BR>\n" );
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document.write( "**6. Make a Decision**\r<BR>\n" );
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document.write( "* **Compare the calculated chi-square statistic to the critical value:**<BR>\n" );
document.write( "    * 19.18 > 5.991\r<BR>\n" );
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document.write( "* **Decision:** Since the calculated chi-square statistic (19.18) is greater than the critical value (5.991), we reject the null hypothesis.\r<BR>\n" );
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document.write( "**7. Conclusion**\r<BR>\n" );
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document.write( "* There is sufficient evidence at the 0.05 level of significance to conclude that there is an association between academic performance and self-esteem in the population of ten-year-old children.\r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* This analysis assumes that the expected frequencies in each cell are greater than 5. <BR>\n" );
document.write( "* You can use statistical software (like R or SPSS) to perform the chi-square test for independence more easily and accurately.\r<BR>\n" );
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document.write( "This analysis provides a basic framework for conducting a chi-square test for independence. <BR>\n" );
document.write( " </SPAN>\n" );
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