document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192206>Question 1192206</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. A Science teacher used two different languages in teaching her two classes. Thirty students from<BR>\n" );
document.write( "class A, which used Filipino as medium of instruction, showed a mean performance of 98 and a<BR>\n" );
document.write( "standard deviation of 9.2. While 35 students from class B, which used the English medium,<BR>\n" );
document.write( "revealed an average performance of 93 with standard deviation of 7.3. Is there a difference in<BR>\n" );
document.write( "performance between the two samples? Use α = 0.01.\r<BR>\n" );
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document.write( "2. A language proficiency test was given to 55 Liberal Arts and 49 Business Administration<BR>\n" );
document.write( "freshmen. The Liberal Arts students got a mean score of 78 with a standard deviation of 11. The<BR>\n" );
document.write( "Business Administration students got a mean score of 82 with a standard deviation of 12. Is there<BR>\n" );
document.write( "a reason to believe that there is significant difference in the performance of the two groups of<BR>\n" );
document.write( "students? Use α = 0.0\r<BR>\n" );
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document.write( "3.Records of 40 used passenger cars and 40 used pickup trucks (none used commercially) were<BR>\n" );
document.write( "randomly selected to investigate whether there was any difference in the mean time in years that<BR>\n" );
document.write( "they were kept by the original owner before being sold. For cars the mean was 5.3 years with<BR>\n" );
document.write( "standard deviation 2.2 years. For pickup trucks the mean was 7.1 years with standard deviation<BR>\n" );
document.write( "3.0 years.<BR>\n" );
document.write( "a. Construct the 95% confidence interval for the difference in the means based on these data.<BR>\n" );
document.write( "b. Test the hypothesis that there is a difference in the means against the null hypothesis that<BR>\n" );
document.write( "there is no difference. Use the 1% level of significance.<BR>\n" );
document.write( "c. Compute the observed significance of the test in part (b).\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849071 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=849071\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve these hypothesis testing problems:\r<BR>\n" );
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document.write( "**Problem 1: Science Teacher's Classes**\r<BR>\n" );
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document.write( "* **Hypotheses:**<BR>\n" );
document.write( "    * H₀: There is no difference in performance between the two classes (μ₁ = μ₂)<BR>\n" );
document.write( "    * H₁: There is a difference in performance between the two classes (μ₁ ≠ μ₂)  (Two-tailed test)\r<BR>\n" );
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document.write( "* **Significance Level:** α = 0.01\r<BR>\n" );
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document.write( "* **Test Statistic (z-score):**  Since the sample sizes are large, we use a z-test.\r<BR>\n" );
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document.write( "   z = (x̄₁ - x̄₂) / √((s₁²/n₁) + (s₂²/n₂))<BR>\n" );
document.write( "   z = (98 - 93) / √((9.2²/30) + (7.3²/35))<BR>\n" );
document.write( "   z = 5 / √(2.821 + 1.526)<BR>\n" );
document.write( "   z = 5 / √4.347<BR>\n" );
document.write( "   z ≈ 2.39\r<BR>\n" );
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document.write( "* **Critical Values:** For a two-tailed test at α = 0.01, the critical values are ±2.576.\r<BR>\n" );
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document.write( "* **Decision:** Since |z| = 2.39 < 2.576, we *fail to reject* the null hypothesis.\r<BR>\n" );
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document.write( "* **Conclusion:** There is not enough evidence at the 0.01 level of significance to conclude that there is a difference in performance between the two classes.\r<BR>\n" );
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document.write( "**Problem 2: Language Proficiency Test**\r<BR>\n" );
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document.write( "* **Hypotheses:**<BR>\n" );
document.write( "    * H₀: There is no significant difference in performance (μ₁ = μ₂)<BR>\n" );
document.write( "    * H₁: There is a significant difference in performance (μ₁ ≠ μ₂) (Two-tailed test)\r<BR>\n" );
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document.write( "* **Significance Level:** α = 0.0 (This is unusual.  A significance level of 0 means you will *never* reject the null hypothesis unless the sample means are *exactly* equal.  I'll proceed with α = 0.05, which is more common. If you truly meant α = 0, the answer is that you would never reject the null hypothesis.)\r<BR>\n" );
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document.write( "* **Test Statistic (z-score):**  Again, we use a z-test due to the large sample sizes.\r<BR>\n" );
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document.write( "   z = (78 - 82) / √((11²/55) + (12²/49))<BR>\n" );
document.write( "   z = -4 / √(2.2 + 2.939)<BR>\n" );
document.write( "   z = -4 / √5.139<BR>\n" );
document.write( "   z ≈ -1.77\r<BR>\n" );
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document.write( "* **Critical Values (using α = 0.05):** For a two-tailed test at α = 0.05, the critical values are ±1.96.\r<BR>\n" );
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document.write( "* **Decision (using α = 0.05):** Since |z| = 1.77 < 1.96, we *fail to reject* the null hypothesis.\r<BR>\n" );
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document.write( "* **Conclusion (using α = 0.05):**  There is not sufficient evidence at the 0.05 level of significance to believe that there is a significant difference in the performance of the two groups of students.\r<BR>\n" );
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document.write( "**Problem 3: Used Cars and Trucks**\r<BR>\n" );
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document.write( "This problem was already addressed in your previous question.  I'll summarize the answers:\r<BR>\n" );
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document.write( "* **a. 95% Confidence Interval:** (-2.953, -0.647)<BR>\n" );
document.write( "* **b. Hypothesis Test:**<BR>\n" );
document.write( "    * H₀: μ₁ = μ₂<BR>\n" );
document.write( "    * H₁: μ₁ ≠ μ₂<BR>\n" );
document.write( "    * α = 0.01<BR>\n" );
document.write( "    * z ≈ -3.06<BR>\n" );
document.write( "    * Critical values: ±2.576<BR>\n" );
document.write( "    * Decision: Reject the null hypothesis.<BR>\n" );
document.write( "    * Conclusion: There is sufficient evidence at the 1% level of significance to conclude that there is a difference in the mean time that cars and pickup trucks are kept by their original owners.<BR>\n" );
document.write( "* **c. Observed Significance (p-value):** ≈ 0.0022<BR>\n" );
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