document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165658>Question 1165658</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A random sample of 30 students was selected from a large group of students taking a certain calculus test. The mean score for the students in the sample was 88. Assume that σ = 7.4. Construct a 99% confidence interval for the mean score, μ, of all students taking the test.\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 #852953 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</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=852953\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You can construct the $99\%$ confidence interval using the formula for a population mean when the population standard deviation ($\sigma$) is known (the $Z$-interval).\r<BR>\n" );
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document.write( "The resulting $99\%$ confidence interval for the mean score ($\mu$) of all students is **$84.52$ to $91.48$**.\r<BR>\n" );
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document.write( "## 1. Identify the Key Values\r<BR>\n" );
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document.write( "| Symbol | Description | Value |<BR>\n" );
document.write( "| :---: | :--- | :--- |<BR>\n" );
document.write( "| $\bar{x}$ | Sample Mean | 88 |<BR>\n" );
document.write( "| $\sigma$ | Population Standard Deviation | 7.4 |<BR>\n" );
document.write( "| $n$ | Sample Size | 30 |<BR>\n" );
document.write( "| C.L. | Confidence Level | 99% |<BR>\n" );
document.write( "| $Z_{\alpha/2}$ | Critical Z-score for 99% CI | 2.576 |\r<BR>\n" );
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document.write( "### Finding the Critical Z-score<BR>\n" );
document.write( "For a $99\%$ confidence level, the area in the two tails is $1 - 0.99 = 0.01$, and the area in one tail ($\alpha/2$) is $0.005$. The corresponding $Z$-score is $\mathbf{2.576}$. \r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 2. Calculate the Margin of Error ($ME$)\r<BR>\n" );
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document.write( "The margin of error is calculated as:<BR>\n" );
document.write( "$$ME = Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$$\r<BR>\n" );
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document.write( "$$ME = 2.576 \cdot \frac{7.4}{\sqrt{30}}$$\r<BR>\n" );
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document.write( "$$ME \approx 2.576 \cdot \frac{7.4}{5.4772}$$\r<BR>\n" );
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document.write( "$$ME \approx 2.576 \cdot 1.3510$$\r<BR>\n" );
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document.write( "$$\mathbf{ME \approx 3.48}$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 3. Construct the Confidence Interval\r<BR>\n" );
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document.write( "The confidence interval is calculated as:<BR>\n" );
document.write( "$$\text{Confidence Interval} = \bar{x} \pm ME$$\r<BR>\n" );
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document.write( "$$\text{C.I.} = 88 \pm 3.48$$\r<BR>\n" );
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document.write( "$$\text{Lower Bound} = 88 - 3.48 = \mathbf{84.52}$$<BR>\n" );
document.write( "$$\text{Upper Bound} = 88 + 3.48 = \mathbf{91.48}$$\r<BR>\n" );
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document.write( "The $99\%$ confidence interval for the mean score of all students is **$(84.52, 91.48)$**. </SPAN>\n" );
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