document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168872>Question 1168872</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You receive a brochure from a large university. The brochure indicates that the mean class size for full-time faculty is fewer than 32students. You want to test this claim. You randomly select 18 classes taught by full­ time faculty and determine the class size of each. The results are listed below. At a=o.01, can you support the university's claim.<BR>\n" );
document.write( "35	28	29	33	32	40	26	25	29<BR>\n" );
document.write( "28	30	36	33	29	27	30	28	25\r<BR>\n" );
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document.write( "Research Question:<BR>\n" );
document.write( "I. Hypotheses:<BR>\n" );
document.write( "Ho: µ<BR>\n" );
document.write( "Hi:µ<BR>\n" );
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document.write( "IV. Decision:<BR>\n" );
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document.write( "II. Criteria for Decision:<BR>\n" );
document.write( "a =<BR>\n" );
document.write( "Decision Rule:<BR>\n" );
document.write( "Reject H 0 if p-value <<BR>\n" );
document.write( "Ill.Test Statistics:<BR>\n" );
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document.write( "V. Summary:\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 #851484 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=851484\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this hypothesis test step-by-step.\r<BR>\n" );
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document.write( "**I. Hypotheses:**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H₀):** The mean class size for full-time faculty is greater than or equal to 32 students.<BR>\n" );
document.write( "    * H₀: µ ≥ 32<BR>\n" );
document.write( "* **Alternative Hypothesis (H₁):** The mean class size for full-time faculty is less than 32 students.<BR>\n" );
document.write( "    * H₁: µ < 32\r<BR>\n" );
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document.write( "**II. Criteria for Decision:**\r<BR>\n" );
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document.write( "* **a (alpha):** 0.01 (given in the problem)<BR>\n" );
document.write( "* **Decision Rule:** Since the alternative hypothesis is a \"less than\" test (left-tailed), we will reject H₀ if the p-value is less than 0.01.\r<BR>\n" );
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document.write( "**III. Test Statistics:**\r<BR>\n" );
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document.write( "1.  **Calculate the Sample Mean (x̄):**<BR>\n" );
document.write( "    * Sum of class sizes: 35 + 28 + 29 + 33 + 32 + 40 + 26 + 25 + 29 + 28 + 30 + 36 + 33 + 29 + 27 + 30 + 28 + 25 = 523<BR>\n" );
document.write( "    * Sample size (n): 18<BR>\n" );
document.write( "    * x̄ = 523 / 18 ≈ 29.0556\r<BR>\n" );
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document.write( "2.  **Calculate the Sample Standard Deviation (s):**<BR>\n" );
document.write( "    * Using a calculator or statistical software, we find the sample standard deviation (s) ≈ 4.195\r<BR>\n" );
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document.write( "3.  **Calculate the Test Statistic (t):**<BR>\n" );
document.write( "    * Since the population standard deviation is unknown and the sample size is small (n < 30), we will use a t-test.<BR>\n" );
document.write( "    * t = (x̄ - µ) / (s / √n)<BR>\n" );
document.write( "    * t = (29.0556 - 32) / (4.195 / √18)<BR>\n" );
document.write( "    * t ≈ -2.94\r<BR>\n" );
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document.write( "4.  **Degrees of Freedom (df):**<BR>\n" );
document.write( "    * df = n - 1 = 18 - 1 = 17\r<BR>\n" );
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document.write( "5.  **Calculate the p-value:**<BR>\n" );
document.write( "    * Using a t-distribution table or statistical software, we find the p-value for a t-statistic of -2.94 with 17 degrees of freedom (left-tailed test).<BR>\n" );
document.write( "    * p-value ≈ 0.0046\r<BR>\n" );
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document.write( "**IV. Decision:**\r<BR>\n" );
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document.write( "* **Compare the p-value to alpha:** 0.0046 < 0.01<BR>\n" );
document.write( "* **Decision:** Reject the null hypothesis (H₀).\r<BR>\n" );
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document.write( "**V. Summary:**\r<BR>\n" );
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document.write( "* The calculated t-statistic is approximately -2.94.<BR>\n" );
document.write( "* The p-value is approximately 0.0046.<BR>\n" );
document.write( "* Since the p-value (0.0046) is less than the significance level (0.01), we reject the null hypothesis.<BR>\n" );
document.write( "* **Conclusion:** There is sufficient evidence at the α = 0.01 level to support the university's claim that the mean class size for full-time faculty is fewer than 32 students.<BR>\n" );
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