document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163425>Question 1163425</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Assume that z-score are normally distributed with a mean of 0 and a standard deviation of 1 if p(-b<z<b)=0.2842, find b </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854377 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=854377\">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 scipy.stats as stats\r<BR>\n" );
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document.write( "# Given P(-b < Z < b) = 0.2842<BR>\n" );
document.write( "# 2 * P(Z < b) - 1 = 0.2842<BR>\n" );
document.write( "# P(Z < b) = (1 + 0.2842) / 2\r<BR>\n" );
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document.write( "prob_symmetric = 0.2842<BR>\n" );
document.write( "p_cumulative = (1 + prob_symmetric) / 2<BR>\n" );
document.write( "b = stats.norm.ppf(p_cumulative)\r<BR>\n" );
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document.write( "print(f\"{p_cumulative=}\")<BR>\n" );
document.write( "print(f\"{b=}\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "p_cumulative=0.6421<BR>\n" );
document.write( "b=0.3640776846923426\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "To find the value of $b$ for the standard normal distribution where $P(-b < Z < b) = 0.2842$, we follow these steps:\r<BR>\n" );
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document.write( "### 1. Understand the Symmetry<BR>\n" );
document.write( "The standard normal distribution is symmetric around the mean (0). Therefore, the area in the center is given by:<BR>\n" );
document.write( "$$P(-b < Z < b) = \Phi(b) - \Phi(-b)$$<BR>\n" );
document.write( "Since $\Phi(-b) = 1 - \Phi(b)$, we can rewrite this as:<BR>\n" );
document.write( "$$P(-b < Z < b) = \Phi(b) - (1 - \Phi(b)) = 2\Phi(b) - 1$$\r<BR>\n" );
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document.write( "### 2. Set Up the Equation<BR>\n" );
document.write( "Substitute the given probability ($0.2842$) into the equation:<BR>\n" );
document.write( "$$2\Phi(b) - 1 = 0.2842$$<BR>\n" );
document.write( "$$2\Phi(b) = 1.2842$$<BR>\n" );
document.write( "$$\Phi(b) = \frac{1.2842}{2}$$<BR>\n" );
document.write( "$$\Phi(b) = 0.6421$$\r<BR>\n" );
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document.write( "### 3. Find the z-score ($b$)<BR>\n" );
document.write( "We now look for the value $b$ such that the cumulative probability to its left is $0.6421$. Using a standard normal distribution table or a calculator:<BR>\n" );
document.write( "$$b = \text{invNorm}(0.6421)$$<BR>\n" );
document.write( "$$b \approx 0.3641$$\r<BR>\n" );
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document.write( "**Final Answer:**<BR>\n" );
document.write( "The value of $b$ is approximately **$0.3641$**. </SPAN>\n" );
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