document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193448>Question 1193448</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Randomly selected 130 student cars have ages with a mean of 7.9 years and a standard deviation of 3.4 years, while randomly selected 95 faculty cars have ages with a mean of 5.4 years and a standard deviation of 3.3 years.\r<BR>\n" );
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document.write( "1.  Use a 0.05 significance level to test the claim that student cars are older than faculty cars. <BR>\n" );
document.write( "- The test statistic is ______. (5 sig. figs)<BR>\n" );
document.write( "- The critical value is ______. (5 sig. figs)\r<BR>\n" );
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document.write( "2. Construct a 95% confidence interval estimate of the difference μ1−μ2, where μ1 is the mean age of student cars and μ2 is the mean age of faculty cars. \r<BR>\n" );
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document.write( "- ________ <(μ1−μ2)< _________ (5 sig. figs) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848670 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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=848670\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Hypothesis Testing**\r<BR>\n" );
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document.write( "* **Hypotheses:**<BR>\n" );
document.write( "    * H0: μ1 - μ2 ≤ 0 (Null Hypothesis: Student cars are not older than faculty cars)<BR>\n" );
document.write( "    * H1: μ1 - μ2 > 0 (Alternative Hypothesis: Student cars are older than faculty cars)\r<BR>\n" );
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document.write( "* **Sample Data:**<BR>\n" );
document.write( "    * Sample 1 (Student Cars): n1 = 130, x̄1 = 7.9 years, s1 = 3.4 years<BR>\n" );
document.write( "    * Sample 2 (Faculty Cars): n2 = 95, x̄2 = 5.4 years, s2 = 3.3 years\r<BR>\n" );
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document.write( "* **Calculate the Pooled Variance (assuming equal variances):**<BR>\n" );
document.write( "    * s_p² = [((n1 - 1) * s1²) + ((n2 - 1) * s2²)] / (n1 + n2 - 2)<BR>\n" );
document.write( "    * s_p² = [((130 - 1) * 3.4²) + ((95 - 1) * 3.3²)] / (130 + 95 - 2)<BR>\n" );
document.write( "    * s_p² ≈ 11.1225\r<BR>\n" );
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document.write( "* **Calculate the Standard Error of the Difference:**<BR>\n" );
document.write( "    * SE = s_p * √(1/n1 + 1/n2)<BR>\n" );
document.write( "    * SE = √11.1225 * √(1/130 + 1/95)<BR>\n" );
document.write( "    * SE ≈ 0.5069\r<BR>\n" );
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document.write( "* **Calculate the t-statistic:**<BR>\n" );
document.write( "    * t = (x̄1 - x̄2) / SE<BR>\n" );
document.write( "    * t = (7.9 - 5.4) / 0.5069<BR>\n" );
document.write( "    * t ≈ 4.9177\r<BR>\n" );
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document.write( "* **Determine Degrees of Freedom:**<BR>\n" );
document.write( "    * df = n1 + n2 - 2 = 130 + 95 - 2 = 223\r<BR>\n" );
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document.write( "* **Find the Critical Value (t-critical) using a t-distribution table or statistical software:**<BR>\n" );
document.write( "    * For a one-tailed test with α = 0.05 and df = 223, t-critical ≈ 1.6525\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "* **Test Statistic:** t ≈ 4.9177<BR>\n" );
document.write( "* **Critical Value:** t-critical ≈ 1.6525\r<BR>\n" );
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document.write( "**2. Construct a 95% Confidence Interval**\r<BR>\n" );
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document.write( "* **Calculate the Margin of Error:**<BR>\n" );
document.write( "    * Margin of Error = t-critical * SE<BR>\n" );
document.write( "    * Margin of Error = 1.96 * 0.5069 <BR>\n" );
document.write( "    * Margin of Error ≈ 0.9935\r<BR>\n" );
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document.write( "* **Calculate the Confidence Interval:**<BR>\n" );
document.write( "    * (x̄1 - x̄2) - Margin of Error < (μ1 - μ2) < (x̄1 - x̄2) + Margin of Error<BR>\n" );
document.write( "    * (7.9 - 5.4) - 0.9935 < (μ1 - μ2) < (7.9 - 5.4) + 0.9935<BR>\n" );
document.write( "    * 1.5065 < (μ1 - μ2) < 3.4935\r<BR>\n" );
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document.write( "**Therefore, the 95% confidence interval estimate of the difference (μ1 - μ2) is 1.5065 < (μ1 - μ2) < 3.4935.**\r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "* The calculated t-statistic (4.9177) is greater than the critical value (1.6525), so we **reject the null hypothesis**. <BR>\n" );
document.write( "* There is sufficient evidence to support the claim that student cars are older than faculty cars at the 0.05 significance level.<BR>\n" );
document.write( "* The 95% confidence interval for the difference in mean ages of student and faculty cars does not include zero, further supporting the conclusion that student cars are older.<BR>\n" );
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