document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194885>Question 1194885</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A manufacturer of automobile transmissions uses two different processes. Management ordered a study of the production costs to see if there is a difference between the two processes. A summary of the findings is shown next.\r<BR>\n" );
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document.write( "Process 1 Process 2 Total Process totals ($100s) 137 108 245\r<BR>\n" );
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document.write( "Sample size 10 10 20\r<BR>\n" );
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document.write( "Sum of squares 1,893 1,188 3,081\r<BR>\n" );
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document.write( "a. What is the critical value of F at the 5% level of significance?\r<BR>\n" );
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document.write( "b. State the null hypothesis and the alternate hypothesis\r<BR>\n" );
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document.write( "c. State the decision rule\r<BR>\n" );
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document.write( "d. Compute the value of the test statistic\r<BR>\n" );
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document.write( "e. What is your decision regarding H0 \r<BR>\n" );
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document.write( "f. Interpret the result </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848440 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=848440\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Critical Value of F at the 5% level of significance**\r<BR>\n" );
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document.write( "* **Degrees of Freedom:**<BR>\n" );
document.write( "    * Between groups (df1): k - 1 = 2 groups - 1 = 1 <BR>\n" );
document.write( "    * Within groups (df2): N - k = 20 - 2 = 18\r<BR>\n" );
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document.write( "* **F-distribution table:** <BR>\n" );
document.write( "    * Look up the F-critical value in an F-distribution table with df1 = 1 and df2 = 18, and α = 0.05. <BR>\n" );
document.write( "    * **F-critical ≈ 4.41**\r<BR>\n" );
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document.write( "**b) Null and Alternative Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** <BR>\n" );
document.write( "    * μ1 = μ2 <BR>\n" );
document.write( "    * There is no significant difference in the mean production costs between the two processes.\r<BR>\n" );
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document.write( "* **Alternative Hypothesis (H1):** <BR>\n" );
document.write( "    * μ1 ≠ μ2 <BR>\n" );
document.write( "    * There is a significant difference in the mean production costs between the two processes.\r<BR>\n" );
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document.write( "**c) Decision Rule**\r<BR>\n" );
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document.write( "* **Reject H0 if the calculated F-statistic is greater than the critical F-value (F > 4.41).**<BR>\n" );
document.write( "* **Fail to reject H0 if the calculated F-statistic is less than or equal to the critical F-value (F ≤ 4.41).**\r<BR>\n" );
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document.write( "**d) Compute the Value of the Test Statistic**\r<BR>\n" );
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document.write( "1. **Calculate Mean Squares:**<BR>\n" );
document.write( "    * **Between Groups Sum of Squares (SSB):** <BR>\n" );
document.write( "        * SSB = [Σ(n_i * (X̄_i - X̄)²)] / (k - 1) <BR>\n" );
document.write( "            * where:<BR>\n" );
document.write( "                * n_i: sample size of group i<BR>\n" );
document.write( "                * X̄_i: mean of group i<BR>\n" );
document.write( "                * X̄: overall mean <BR>\n" );
document.write( "                * k: number of groups\r<BR>\n" );
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document.write( "        * X̄_1 (Process 1) = 137 / 10 = 13.7<BR>\n" );
document.write( "        * X̄_2 (Process 2) = 108 / 10 = 10.8<BR>\n" );
document.write( "        * X̄ (Overall) = 245 / 20 = 12.25\r<BR>\n" );
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document.write( "        * SSB = [(10 * (13.7 - 12.25)²) + (10 * (10.8 - 12.25)²)] / (2 - 1) <BR>\n" );
document.write( "        * SSB = 198.05\r<BR>\n" );
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document.write( "    * **Within Groups Sum of Squares (SSW):** <BR>\n" );
document.write( "        * SSW = Total Sum of Squares (SST) - Between Groups Sum of Squares (SSB) <BR>\n" );
document.write( "        * SSW = 3081 - 198.05 = 2882.95\r<BR>\n" );
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document.write( "    * **Mean Square Between Groups (MSB):** <BR>\n" );
document.write( "        * MSB = SSB / (k - 1) = 198.05 / 1 = 198.05\r<BR>\n" );
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document.write( "    * **Mean Square Within Groups (MSW):** <BR>\n" );
document.write( "        * MSW = SSW / (N - k) = 2882.95 / 18 = 160.16\r<BR>\n" );
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document.write( "2. **Calculate F-statistic:**<BR>\n" );
document.write( "    * F = MSB / MSW = 198.05 / 160.16 = 1.236\r<BR>\n" );
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document.write( "**e) Decision**\r<BR>\n" );
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document.write( "* Since the calculated F-statistic (1.236) is less than the critical F-value (4.41), we **fail to reject the null hypothesis (H0)**.\r<BR>\n" );
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document.write( "**f) Interpretation**\r<BR>\n" );
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document.write( "* There is **not** enough evidence at the 5% level of significance to conclude that there is a significant difference in the mean production costs between the two processes.\r<BR>\n" );
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document.write( "**In summary:**\r<BR>\n" );
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document.write( "* The analysis suggests that there is no significant difference in the mean production costs between the two processes.<BR>\n" );
document.write( " </SPAN>\n" );
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