document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164884>Question 1164884</A>: <I><SPAN STYLE=\"word-break: break-all;\"> CNNBC recently reported that the mean annual cost of auto insurance is 1036 dollars. Assume the standard deviation is 268 dollars, and the cost is normally distributed. You take a simple random sample of 35 auto insurance policies. Round your answers to 4 decimal places.\r<BR>\n" );
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document.write( "What is the distribution of X?X~ N(________,_______)\r<BR>\n" );
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document.write( "What is the distribution of ¯x?¯x~ N(________,__________)\r<BR>\n" );
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document.write( "What is the probability that one randomly selected auto insurance is less than $996? \r<BR>\n" );
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document.write( "a simple random sample of 35 auto insurance policies, find the probability that the average cost is less than $996. \r<BR>\n" );
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document.write( "For part d), is the assumption of normal necessary? Yes or No </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853506 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=853506\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To solve this, we will use the parameters of the population and the **Central Limit Theorem** to find the probabilities for both an individual policy and a sample mean.\r<BR>\n" );
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document.write( "### a) What is the distribution of ?\r<BR>\n" );
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document.write( "The distribution of individual auto insurance costs () is given as normally distributed with the population mean and standard deviation.\r<BR>\n" );
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document.write( "### b) What is the distribution of ?\r<BR>\n" );
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document.write( "For a sample of size , the sampling distribution of the mean () is also normal.\r<BR>\n" );
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document.write( "* **Mean ():** Equal to the population mean, **1036**.<BR>\n" );
document.write( "* **Standard Error ():** Calculated as .\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### c) What is the probability that one randomly selected auto insurance is less than $996?\r<BR>\n" );
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document.write( "We need to find . First, we calculate the -score:\r<BR>\n" );
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document.write( "Using a standard normal distribution table or calculator for :<BR>\n" );
document.write( "**Probability = 0.4407**\r<BR>\n" );
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document.write( "### d) For a simple random sample of 35 auto insurance policies, find the probability that the average cost is less than $996.\r<BR>\n" );
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document.write( "We need to find . We use the standard error () to find the new -score:\r<BR>\n" );
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document.write( "Using a standard normal distribution table or calculator for :<BR>\n" );
document.write( "**Probability = 0.1886**\r<BR>\n" );
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document.write( "### e) For part d), is the assumption of normal necessary?\r<BR>\n" );
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document.write( "**No**.<BR>\n" );
document.write( "Even if the original population were not normally distributed, the **Central Limit Theorem** states that the sampling distribution of the mean will be approximately normal because the sample size () is greater than 30.\r<BR>\n" );
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document.write( "Would you like me to show you how the probability changes if we increase the sample size to 100? </SPAN>\n" );
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