document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1162467>Question 1162467</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An industrial engineer intends to use the mean of a random sample of size n = 150 to estimate the average mechanical aptitude (as measured by a certain test) of assembly line workers in a large industry. If on the b basis of experience, the engineer can assume that alpha = 6.2 for such data, what can be assert with probability 0.99 about the maximum size of his error?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854402 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2261)</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=854402\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import math<BR>\n" );
document.write( "from scipy.stats import norm\r<BR>\n" );
document.write( "\n" );
document.write( "# Given values<BR>\n" );
document.write( "n = 150<BR>\n" );
document.write( "sigma = 6.2<BR>\n" );
document.write( "confidence_level = 0.99\r<BR>\n" );
document.write( "\n" );
document.write( "# Calculate z-score for 0.99 probability (two-tailed)<BR>\n" );
document.write( "# Alpha is 1 - confidence_level<BR>\n" );
document.write( "alpha_val = 1 - confidence_level<BR>\n" );
document.write( "z_score = norm.ppf(1 - alpha_val / 2)\r<BR>\n" );
document.write( "\n" );
document.write( "# Calculate Margin of Error (E)<BR>\n" );
document.write( "error = z_score * (sigma / math.sqrt(n))\r<BR>\n" );
document.write( "\n" );
document.write( "print(f\"{z_score=}\")<BR>\n" );
document.write( "print(f\"{error=}\")\r<BR>\n" );
document.write( "\n" );
document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "z_score=2.5758293035489004<BR>\n" );
document.write( "error=1.3039566080286962\r<BR>\n" );
document.write( "\n" );
document.write( "```<BR>\n" );
document.write( "To determine the maximum size of the error for estimating the population mean with a specific probability, we use the formula for the **Margin of Error ($E$)**:\r<BR>\n" );
document.write( "\n" );
document.write( "$$E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$$\r<BR>\n" );
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document.write( "### 1. Identify the given information<BR>\n" );
document.write( "* **Sample size ($n$):** 150<BR>\n" );
document.write( "* **Population standard deviation ($\sigma$):** 6.2 (The \"alpha\" mentioned in the prompt refers to the population standard deviation for the data)<BR>\n" );
document.write( "* **Confidence level (Probability):** 0.99\r<BR>\n" );
document.write( "\n" );
document.write( "### 2. Find the critical value ($z$)<BR>\n" );
document.write( "For a probability of 0.99, the significance level ($\alpha$) is $1 - 0.99 = 0.01$. Since we are estimating a range around the mean, we look for the $z$-score corresponding to $\alpha/2 = 0.005$ in the tails.<BR>\n" );
document.write( "* The $z$-score for a 0.99 confidence level is approximately **2.576**.\r<BR>\n" );
document.write( "\n" );
document.write( "### 3. Calculate the maximum error<BR>\n" );
document.write( "Substitute the values into the formula:<BR>\n" );
document.write( "$$E = 2.576 \cdot \frac{6.2}{\sqrt{150}}$$<BR>\n" );
document.write( "$$E = 2.576 \cdot \frac{6.2}{12.2474}$$<BR>\n" );
document.write( "$$E = 2.576 \cdot 0.5062$$<BR>\n" );
document.write( "$$E \approx 1.304$$\r<BR>\n" );
document.write( "\n" );
document.write( "### Final Assertion<BR>\n" );
document.write( "With a probability of 0.99, the industrial engineer can assert that the maximum size of his error in estimating the average mechanical aptitude is **1.304**. \r<BR>\n" );
document.write( "\n" );
document.write( "This means the engineer is 99% confident that the sample mean will not differ from the true population mean by more than **1.304 units**. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );