document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196934>Question 1196934</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A computer manufacturer offers technical support that is available 24 hours a day,<BR>\n" );
document.write( "7 days a week. Timely resolution of these calls is important to the company’s image. For<BR>\n" );
document.write( "35 calls that were related to software, technicians resolved the issues in a mean time of<BR>\n" );
document.write( "18 minutes with a standard deviation of 4.2 minutes. For 45 calls related to hardware,<BR>\n" );
document.write( "technicians resolved the problems in a mean time of 15.5 minutes with a standard deviation of 3.9 minutes. At the .05 significance level, does it take longer to resolve software<BR>\n" );
document.write( "issues? What is the p-value?<BR>\n" );
document.write( "Giva Farmula for calculating when you have two sample test of mean and SD is unknown?\r<BR>\n" );
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document.write( "State what will be null hypothesis and alternate hypotheses at05 significance level. Dose it take longer to resolve software issues </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848388 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=848388\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We will conduct a two-sample t-test to determine whether it takes longer to resolve software issues compared to hardware issues. Here's the step-by-step process:\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Formula for Two-Sample t-Test (Unequal Variances)\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Where:<BR>\n" );
document.write( "- \(\bar{X}_1, \bar{X}_2\): Sample means<BR>\n" );
document.write( "- \(s_1, s_2\): Sample standard deviations<BR>\n" );
document.write( "- \(n_1, n_2\): Sample sizes\r<BR>\n" );
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document.write( "Degrees of freedom (\(df\)) are approximated as:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Hypotheses\r<BR>\n" );
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document.write( "- **Null Hypothesis (\(H_0\))**: There is no difference in the time taken to resolve software and hardware issues (\(\mu_1 = \mu_2\)).<BR>\n" );
document.write( "- **Alternative Hypothesis (\(H_a\))**: It takes longer to resolve software issues (\(\mu_1 > \mu_2\)).\r<BR>\n" );
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document.write( "This is a **one-tailed test** at the 0.05 significance level.\r<BR>\n" );
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document.write( "### Given Data\r<BR>\n" );
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document.write( "- Software calls:<BR>\n" );
document.write( "  - \(n_1 = 35\), \(\bar{X}_1 = 18\), \(s_1 = 4.2\)<BR>\n" );
document.write( "- Hardware calls:<BR>\n" );
document.write( "  - \(n_2 = 45\), \(\bar{X}_2 = 15.5\), \(s_2 = 3.9\)\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Step 1: Calculate the Test Statistic\r<BR>\n" );
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document.write( "Substitute the values into the t-test formula:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "t = \frac{18 - 15.5}{\sqrt{\frac{4.2^2}{35} + \frac{3.9^2}{45}}}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "t = \frac{2.5}{\sqrt{\frac{17.64}{35} + \frac{15.21}{45}}}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "t = \frac{2.5}{\sqrt{0.504 + 0.338}}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "t = \frac{2.5}{\sqrt{0.842}} = \frac{2.5}{0.9177} \approx 2.72<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "### Step 2: Degrees of Freedom\r<BR>\n" );
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document.write( "Calculate \(df\):\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "df = \frac{\left(0.504 + 0.338\right)^2}{\frac{0.504^2}{34} + \frac{0.338^2}{44}}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "df = \frac{0.842^2}{\frac{0.254}{34} + \frac{0.114}{44}} = \frac{0.708}{0.00747 + 0.00259} = \frac{0.708}{0.01006} \approx 70.34<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Round \(df\) to 70 for simplicity.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Step 3: Compare to Critical Value and Find p-Value\r<BR>\n" );
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document.write( "For a one-tailed test at \(\alpha = 0.05\) and \(df = 70\), the critical \(t\)-value is approximately **1.667**.\r<BR>\n" );
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document.write( "Since \(t = 2.72 > 1.667\), we reject \(H_0\).\r<BR>\n" );
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document.write( "To find the p-value, use a t-distribution table or software. The p-value for \(t = 2.72\) with \(df = 70\) is approximately **0.004**.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Conclusion\r<BR>\n" );
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document.write( "- **At the 0.05 significance level**, we conclude that it takes longer to resolve software issues.<BR>\n" );
document.write( "- **p-value**: \(0.004\) </SPAN>\n" );
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