document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200718>Question 1200718</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A bank has taken a random sample of 4 customers for each set of various criteria to determine their level of customer satisfaction on a scale of 0 to 10. The data appear in the table below.\r<BR>\n" );
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document.write( "Time of day 	Automated 	Bank Representative 	Row means <BR>\n" );
document.write( "Morning 	6, 5, 8, 4 	8, 7, 9, 9 	        Row 1 <BR>\n" );
document.write( "		𝑥 = 5.75	𝑥 = 8.25                𝑥 = 7.00<BR>\n" );
document.write( "Afternoon 	3, 5, 6, 5 	9, 10, 6, 8 	          Row 2 <BR>\n" );
document.write( "                 𝑥 = 4.75	𝑥 = 8.25 		  𝑥 = 6.50<BR>\n" );
document.write( "Evening 	5, 5, 7, 5 	9, 10, 10, 9 	               Row 3 <BR>\n" );
document.write( "                 𝑥 = 5.50	𝑥 = 9.50 	        	𝑥 =7.50<BR>\n" );
document.write( "Column means 	Column 1 𝑥 = 5.33 	Column 2 𝑥 = 8.67 	Total 𝑥 =7.00 \r<BR>\n" );
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document.write( "For the two measurement problem, use an a = 0.05 level of significance. Conduct appropriate hypothesis tests and conclude whether to reject or not reject the claim that:<BR>\n" );
document.write( "a. there is no difference in population mean satisfaction depending on time of contact,<BR>\n" );
document.write( "b. there is no difference in population mean satisfaction depending on type of customer contact and<BR>\n" );
document.write( "c. there is no interaction between type of contact and time of contact.\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 #848060 by </B><A HREF=/tutors/aboutme.mpl?userid=GingerAle><B>GingerAle(43)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=GingerAle><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=848060\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Certainly, let's analyze the customer satisfaction data using a two-way ANOVA.\r<BR>\n" );
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document.write( "**a. Test for Difference in Mean Satisfaction Depending on Time of Contact**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** There is no difference in population mean satisfaction among the three time periods (Morning, Afternoon, Evening).<BR>\n" );
document.write( "* **Alternative Hypothesis (Ha):** There is a difference in population mean satisfaction among the three time periods.\r<BR>\n" );
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document.write( "**b. Test for Difference in Mean Satisfaction Depending on Type of Contact**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** There is no difference in population mean satisfaction between Automated and Bank Representative contact.<BR>\n" );
document.write( "* **Alternative Hypothesis (Ha):** There is a difference in population mean satisfaction between Automated and Bank Representative contact.\r<BR>\n" );
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document.write( "**c. Test for Interaction Between Time of Contact and Type of Contact**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** There is no interaction between time of contact and type of contact on customer satisfaction.<BR>\n" );
document.write( "* **Alternative Hypothesis (Ha):** There is an interaction between time of contact and type of contact on customer satisfaction.\r<BR>\n" );
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document.write( "**Two-Way ANOVA Procedure**\r<BR>\n" );
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document.write( "1. **Calculate Sum of Squares:**<BR>\n" );
document.write( "   * **Total Sum of Squares (SST):** Calculate the total variation in the data.<BR>\n" );
document.write( "   * **Sum of Squares Between Rows (SSR):** Calculate the variation between the row means (time of contact).<BR>\n" );
document.write( "   * **Sum of Squares Between Columns (SSC):** Calculate the variation between the column means (type of contact).<BR>\n" );
document.write( "   * **Sum of Squares of Interaction (SS(RC)):** Calculate the variation due to the interaction between rows and columns.<BR>\n" );
document.write( "   * **Sum of Squares Within Cells (SSE):** Calculate the variation within each cell (combination of time and contact).\r<BR>\n" );
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document.write( "2. **Calculate Degrees of Freedom:**<BR>\n" );
document.write( "   * **Rows (dfR):** Number of rows - 1 = 3 - 1 = 2<BR>\n" );
document.write( "   * **Columns (dfC):** Number of columns - 1 = 2 - 1 = 1<BR>\n" );
document.write( "   * **Interaction (dfRC):** dfR * dfC = 2 * 1 = 2<BR>\n" );
document.write( "   * **Error (dfE):** Total number of observations - (number of rows * number of columns) = 12 - (3 * 2) = 6<BR>\n" );
document.write( "   * **Total (dfT):** Total number of observations - 1 = 12 - 1 = 11\r<BR>\n" );
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document.write( "3. **Calculate Mean Squares:**<BR>\n" );
document.write( "   * MSR = SSR / dfR<BR>\n" );
document.write( "   * MSC = SSC / dfC<BR>\n" );
document.write( "   * MS(RC) = SS(RC) / dfRC<BR>\n" );
document.write( "   * MSE = SSE / dfE\r<BR>\n" );
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document.write( "4. **Calculate F-statistics:**<BR>\n" );
document.write( "   * F-statistic for Rows: F_R = MSR / MSE<BR>\n" );
document.write( "   * F-statistic for Columns: F_C = MSC / MSE<BR>\n" );
document.write( "   * F-statistic for Interaction: F_RC = MS(RC) / MSE\r<BR>\n" );
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document.write( "5. **Determine Critical Values:**<BR>\n" );
document.write( "   * Use an F-distribution table to find the critical F-values for each test at the 0.05 significance level.\r<BR>\n" );
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document.write( "6. **Compare F-statistics to Critical Values:**<BR>\n" );
document.write( "   * If the calculated F-statistic is greater than the critical F-value, reject the null hypothesis.\r<BR>\n" );
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document.write( "**Interpretation:**\r<BR>\n" );
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document.write( "* **If the null hypothesis for rows is rejected:** There is evidence to suggest that the mean satisfaction scores differ significantly across the different times of contact (Morning, Afternoon, Evening).<BR>\n" );
document.write( "* **If the null hypothesis for columns is rejected:** There is evidence to suggest that the mean satisfaction scores differ significantly between Automated and Bank Representative contact.<BR>\n" );
document.write( "* **If the null hypothesis for interaction is rejected:** There is evidence to suggest that the effect of time of contact on satisfaction depends on the type of contact (and vice versa).\r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* This is a general outline of the two-way ANOVA procedure. <BR>\n" );
document.write( "* The actual calculations can be quite complex and are typically performed using statistical software (such as R, Python, SPSS, or Excel).\r<BR>\n" );
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document.write( "**Disclaimer:** \r<BR>\n" );
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document.write( "* This information is for general knowledge and educational purposes only and does not constitute financial, investment, or professional advice. <BR>\n" );
document.write( "* The analysis assumes that the data meets the assumptions of ANOVA (normality, homogeneity of variances).\r<BR>\n" );
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document.write( "I hope this explanation is helpful! Let me know if you have any further questions.<BR>\n" );
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