document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193540>Question 1193540</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (a) The regional manager of a bank wants to analyze the number of delayed repayments of <BR>\n" );
document.write( "instalments of consumer loans in two of its branch banks (Branch −X and Branch −Y ). <BR>\n" );
document.write( "The number of delayed payments of instalments in each branch bank follows normal <BR>\n" );
document.write( "distribution. The manager feels that the number of delayed payments of instalments by <BR>\n" );
document.write( "the consumers of the Branch −X is no way different from that of the Branch −Y . So, he <BR>\n" );
document.write( "selected the loan accounts of 80 different consumers from the Branch −X and found that <BR>\n" );
document.write( "the mean and variance of the number of delayed payments of instalments are 35 and 25, <BR>\n" );
document.write( "respectively. Similarly, he selected the loan accounts of 100 different consumers from the Branch −Y and found that the mean and variance of the number of delayed payments of <BR>\n" );
document.write( "instalments are 40 and 49, respectively. Test his intuition at a significance level of 0.01. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848649 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=848649\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Set up Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** The mean number of delayed payments of installments is the same for both branches. <BR>\n" );
document.write( "    * μ₁ = μ₂ <BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** The mean number of delayed payments of installments is different for the two branches.<BR>\n" );
document.write( "    * μ₁ ≠ μ₂\r<BR>\n" );
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document.write( "**2. Choose the Test Statistic**\r<BR>\n" );
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document.write( "* Since we are comparing the means of two independent samples with known (or assumed) population variances, we will use the **Z-test for the difference between two means**.\r<BR>\n" );
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document.write( "**3. Calculate the Test Statistic**\r<BR>\n" );
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document.write( "* **Given:**<BR>\n" );
document.write( "    * Sample 1 (Branch X): n₁ = 80, x̄₁ = 35, σ₁² = 25 <BR>\n" );
document.write( "    * Sample 2 (Branch Y): n₂ = 100, x̄₂ = 40, σ₂² = 49<BR>\n" );
document.write( "* **Calculate the pooled variance (not needed in this case since population variances are known):**<BR>\n" );
document.write( "    * Pooled variance (s_p²) = [(n₁ - 1)s₁² + (n₂ - 1)s₂²] / (n₁ + n₂ - 2) <BR>\n" );
document.write( "* **Calculate the standard error of the difference between means:**<BR>\n" );
document.write( "    * SE = √[(σ₁²/n₁) + (σ₂²/n₂)] <BR>\n" );
document.write( "        * SE = √[(25/80) + (49/100)] <BR>\n" );
document.write( "        * SE = √(0.3125 + 0.49) <BR>\n" );
document.write( "        * SE = √0.8025 <BR>\n" );
document.write( "        * SE ≈ 0.8958\r<BR>\n" );
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document.write( "* **Calculate the Z-score:**<BR>\n" );
document.write( "    * Z = (x̄₁ - x̄₂) / SE <BR>\n" );
document.write( "        * Z = (35 - 40) / 0.8958 <BR>\n" );
document.write( "        * Z = -5 / 0.8958 <BR>\n" );
document.write( "        * Z ≈ -5.58\r<BR>\n" );
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document.write( "**4. Determine Critical Values**\r<BR>\n" );
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document.write( "* **Significance Level:** α = 0.01<BR>\n" );
document.write( "* **Two-tailed test:** We need to find the critical values for both tails of the standard normal distribution.<BR>\n" );
document.write( "* **Using a standard normal distribution table or statistical software:**<BR>\n" );
document.write( "    * Critical values: Z_critical ≈ ±2.576\r<BR>\n" );
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document.write( "**5. Decision Rule**\r<BR>\n" );
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document.write( "* If the calculated Z-score (|Z|) is greater than the critical value (Z_critical), reject the null hypothesis.<BR>\n" );
document.write( "* If the calculated Z-score (|Z|) is less than or equal to the critical value (Z_critical), fail to reject the null hypothesis.\r<BR>\n" );
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document.write( "**6. Make a Decision**\r<BR>\n" );
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document.write( "* Our calculated Z-score (|-5.58|) is greater than the critical value (2.576).\r<BR>\n" );
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document.write( "* **Conclusion:** We reject the null hypothesis.\r<BR>\n" );
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document.write( "**Interpretation**\r<BR>\n" );
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document.write( "The evidence suggests that the mean number of delayed payments of installments is significantly different between Branch X and Branch Y at the 0.01 significance level. \r<BR>\n" );
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document.write( "**Therefore, the manager's intuition that the number of delayed payments is the same for both branches is not supported by the data.**<BR>\n" );
document.write( " </SPAN>\n" );
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