document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181367>Question 1181367</A>: <I><SPAN STYLE=\"word-break: break-all;\"> - Solve for the t  -  computed and determine the t  -  tabular\r<BR>\n" );
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document.write( "A course in mathematics is taught to 15 students by explicit method. Another<BR>\n" );
document.write( "group of 17 students was given the same course by means of another method. At the<BR>\n" );
document.write( "end of the semester, the same test was administered to each group. The 15 students<BR>\n" );
document.write( "under method A made an average of 85 with a standard deviation of 4, while the 17<BR>\n" );
document.write( "students under method B made an average of 81 with a standard deviation of 5. Test<BR>\n" );
document.write( "the null hypothesis of no significant difference in the performance of the two groups of<BR>\n" );
document.write( "students at 5% level of significance.\r<BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "1. H0:<BR>\n" );
document.write( "H1:<BR>\n" );
document.write( "2. Level of Significance:<BR>\n" );
document.write( "3.Test:<BR>\n" );
document.write( "4. Critical Region:<BR>\n" );
document.write( "5. Solution:<BR>\n" );
document.write( "6. Decision:<BR>\n" );
document.write( "7. Conclusion: </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850054 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=850054\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's the solution to the hypothesis test:\r<BR>\n" );
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document.write( "1. **Hypotheses:**\r<BR>\n" );
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document.write( "* H₀: There is no significant difference in the performance of the two groups. (μ₁ = μ₂)<BR>\n" );
document.write( "* H₁: There is a significant difference in the performance of the two groups. (μ₁ ≠ μ₂)\r<BR>\n" );
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document.write( "2. **Level of Significance:** α = 0.05\r<BR>\n" );
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document.write( "3. **Test:** Since we have two independent samples, small sample sizes (both less than 30), and unknown population standard deviations, we use a two-sample t-test.  Because the sample standard deviations are different, we'll use the version of the t-test that *does not* assume equal variances.\r<BR>\n" );
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document.write( "4. **Critical Region:** This is a two-tailed test (because H₁ is μ₁ ≠ μ₂).  We need to find the degrees of freedom (df). For unequal variances, the calculation for df is complex. A conservative approach is to use the smaller of n₁-1 and n₂-1 which is min(15-1, 17-1)=14.  For df = 14 and α = 0.05 (two-tailed), the critical t-value from the t-table is approximately ±2.145.  Our critical region is t < -2.145 or t > 2.145.\r<BR>\n" );
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document.write( "5. **Solution (Calculations):**\r<BR>\n" );
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document.write( "* **Calculate the pooled variance (Sp²):**  Since we're not assuming equal variances, we *don't* pool the variances.\r<BR>\n" );
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document.write( "* **Calculate the test statistic (t):**\r<BR>\n" );
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document.write( "   t = (x̄₁ - x̄₂) / sqrt(s₁²/n₁ + s₂²/n₂)<BR>\n" );
document.write( "   t = (85 - 81) / sqrt(4²/15 + 5²/17)<BR>\n" );
document.write( "   t = 4 / sqrt(1.067 + 1.471)<BR>\n" );
document.write( "   t = 4 / sqrt(2.538)<BR>\n" );
document.write( "   t ≈ 4 / 1.593<BR>\n" );
document.write( "   t ≈ 2.51\r<BR>\n" );
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document.write( "6. **Decision:**  Our calculated t-value (2.51) falls *within* the critical region (t > 2.145).  Therefore, we *reject* the null hypothesis.\r<BR>\n" );
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document.write( "7. **Conclusion:** There *is* sufficient evidence at the 5% level of significance to conclude that there is a significant difference in the performance of the two groups of students. Method A appears to be more effective than Method B.\r<BR>\n" );
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document.write( "**Summary:**\r<BR>\n" );
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document.write( "* **t-computed:** ≈ 2.51<BR>\n" );
document.write( "* **t-tabular (critical value):** ≈ ±2.145<BR>\n" );
document.write( "* **Decision:** Reject H₀<BR>\n" );
document.write( "* **Conclusion:**  There is a statistically significant difference between the two methods.<BR>\n" );
document.write( " </SPAN>\n" );
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