document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1162307>Question 1162307</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A mechanical engineer is analyzing tensile strength of steel ( API 5L X65). A sample of 7 specimens showed a sample mean of 530 MPa. The standard deviation is known to be LaTeX: MPaσ = 5 . A 95% upper confidence interval for the true mean tensile strength is: </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854407 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854407\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To find the **95% upper confidence interval** (a one-sided confidence interval) for the true mean tensile strength, we follow these steps:\r<BR>\n" );
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document.write( "### 1. Identify the given parameters<BR>\n" );
document.write( "* **Sample Mean ($\bar{x}$):** 530 MPa<BR>\n" );
document.write( "* **Known Population Standard Deviation ($\sigma$):** 5 MPa<BR>\n" );
document.write( "* **Sample Size ($n$):** 7<BR>\n" );
document.write( "* **Confidence Level:** 95% (one-sided)\r<BR>\n" );
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document.write( "### 2. Determine the Critical Value ($z_{\alpha}$)<BR>\n" );
document.write( "Because the population standard deviation ($\sigma$) is known, we use the standard normal distribution ($z$-distribution). For a **one-sided** 95% upper confidence interval, the entire 5% error ($\alpha = 0.05$) is placed in the upper tail. <BR>\n" );
document.write( "* The $z$-score that leaves 5% in the upper tail (or corresponds to a cumulative area of 0.95) is **$1.645$**.\r<BR>\n" );
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document.write( "### 3. Calculate the Standard Error ($SE$)<BR>\n" );
document.write( "The standard error of the mean is calculated as:<BR>\n" );
document.write( "$$SE = \frac{\sigma}{\sqrt{n}} = \frac{5}{\sqrt{7}} \approx \frac{5}{2.6458} \approx 1.8898 \text{ MPa}$$\r<BR>\n" );
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document.write( "### 4. Calculate the Upper Bound<BR>\n" );
document.write( "The formula for the 95% upper confidence interval is:<BR>\n" );
document.write( "$$\text{Upper Limit} = \bar{x} + z_{\alpha} \cdot \left( \frac{\sigma}{\sqrt{n}} \right)$$<BR>\n" );
document.write( "$$\text{Upper Limit} = 530 + (1.645 \cdot 1.8898)$$<BR>\n" );
document.write( "$$\text{Upper Limit} = 530 + 3.1087$$<BR>\n" );
document.write( "$$\text{Upper Limit} \approx 533.11 \text{ MPa}$$\r<BR>\n" );
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document.write( "### Final Result<BR>\n" );
document.write( "The 95% upper confidence interval for the true mean tensile strength is:<BR>\n" );
document.write( "$$(-\infty, 533.11 \text{ MPa}]$$\r<BR>\n" );
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document.write( "**Answer:** The true mean tensile strength is less than or equal to **533.11 MPa** with 95% confidence. </SPAN>\n" );
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