document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177956>Question 1177956</A>: <I><SPAN STYLE=\"word-break: break-all;\"> CNNBC recently reported that the mean annual cost of auto insurance is 1025 dollars. Assume the standard deviation is 252 dollars. You take a simple random sample of 57 auto insurance policies.\r<BR>\n" );
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document.write( "Find the probability that a single randomly selected value is less than 989 dollars.<BR>\n" );
document.write( "P(X < 989) = \r<BR>\n" );
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document.write( "Find the probability that a sample of size  <BR>\n" );
document.write( "n<BR>\n" );
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document.write( "57<BR>\n" );
document.write( "  is randomly selected with a mean less than 989 dollars.<BR>\n" );
document.write( "P(M < 989) = \r<BR>\n" );
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document.write( "Enter your answers as numbers accurate to 4 decimal places. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850393 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.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=850393\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**Given Information**\r<BR>\n" );
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document.write( "* Population Mean (μ): $1025<BR>\n" );
document.write( "* Population Standard Deviation (σ): $252<BR>\n" );
document.write( "* Sample Size (n): 57\r<BR>\n" );
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document.write( "**1. Probability for a Single Randomly Selected Value (X < 989)**\r<BR>\n" );
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document.write( "* We need to find P(X < 989) where X is a normally distributed random variable.\r<BR>\n" );
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document.write( "1.  **Calculate the Z-score:**<BR>\n" );
document.write( "    * Z = (X - μ) / σ<BR>\n" );
document.write( "    * Z = (989 - 1025) / 252<BR>\n" );
document.write( "    * Z = -36 / 252 ≈ -0.1429\r<BR>\n" );
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document.write( "2.  **Find the Probability using the Z-table or Calculator:**<BR>\n" );
document.write( "    * P(Z < -0.1429) ≈ 0.4431\r<BR>\n" );
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document.write( "    Therefore, P(X < 989) ≈ 0.4431\r<BR>\n" );
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document.write( "**2. Probability for a Sample Mean (M < 989)**\r<BR>\n" );
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document.write( "* We need to find P(M < 989) where M is the sample mean.\r<BR>\n" );
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document.write( "1.  **Calculate the Standard Error (SE):**<BR>\n" );
document.write( "    * SE = σ / √n<BR>\n" );
document.write( "    * SE = 252 / √57<BR>\n" );
document.write( "    * SE ≈ 252 / 7.5498 ≈ 33.377\r<BR>\n" );
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document.write( "2.  **Calculate the Z-score for the Sample Mean:**<BR>\n" );
document.write( "    * Z = (M - μ) / SE<BR>\n" );
document.write( "    * Z = (989 - 1025) / 33.377<BR>\n" );
document.write( "    * Z = -36 / 33.377 ≈ -1.0786\r<BR>\n" );
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document.write( "3.  **Find the Probability using the Z-table or Calculator:**<BR>\n" );
document.write( "    * P(Z < -1.0786) ≈ 0.1403\r<BR>\n" );
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document.write( "    Therefore, P(M < 989) ≈ 0.1403\r<BR>\n" );
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document.write( "**Answers**\r<BR>\n" );
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document.write( "* P(X < 989) ≈ 0.4431<BR>\n" );
document.write( "* P(M < 989) ≈ 0.1403<BR>\n" );
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