document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182545>Question 1182545</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the 1992 presidential election, Alaska's 40 election districts averaged 2150 votes per district for President Clinton.<BR>\n" );
document.write( "The standard deviation was 593.<BR>\n" );
document.write( "(There are only 40 election districts in Alaska).<BR>\n" );
document.write( "The distribution of the votes per district for President Clinton was bell-shaped.<BR>\n" );
document.write( "Let X = number of votes for President Clinton for an election district.<BR>\n" );
document.write( "Round all answers except part e. to 4 decimals.\r<BR>\n" );
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document.write( "a. What is the distribution of X? <BR>\n" );
document.write( "   X - N = 2150,593\r<BR>\n" );
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document.write( "b. Is 2150 a population mean or sample mean? Mean\r<BR>\n" );
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document.write( "c. Find the probability that a randomly selected district had fewer than 2142 votes for President Clinton.\r<BR>\n" );
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document.write( "d. Find the probability that a randomly selected district had between 2291 and 2561 votes for President Clinton.\r<BR>\n" );
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document.write( "e. Find the third quartile for votes President Clinton.<BR>\n" );
document.write( "Round to the nearest whole number </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849924 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=849924\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem:\r<BR>\n" );
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document.write( "**a. What is the distribution of X?**\r<BR>\n" );
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document.write( "X ~ N(2150, 593)\r<BR>\n" );
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document.write( "This means X follows a normal distribution with a mean (μ) of 2150 and a standard deviation (σ) of 593.\r<BR>\n" );
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document.write( "**b. Is 2150 a population mean or sample mean?**\r<BR>\n" );
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document.write( "Population mean.  The problem states that 2150 is the *average* of *all* 40 election districts in Alaska.  Since it pertains to the entire population, it's the population mean.\r<BR>\n" );
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document.write( "**c. Find the probability that a randomly selected district had fewer than 2142 votes for President Clinton.**\r<BR>\n" );
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document.write( "To find this probability, we need to calculate the z-score and then use a standard normal distribution table (or calculator) to find the corresponding probability.\r<BR>\n" );
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document.write( "*   z = (x - μ) / σ<BR>\n" );
document.write( "*   z = (2142 - 2150) / 593<BR>\n" );
document.write( "*   z ≈ -0.0135\r<BR>\n" );
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document.write( "Now, we look up the probability associated with z = -0.0135.  P(X < 2142) = P(Z < -0.0135) ≈ 0.4946\r<BR>\n" );
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document.write( "**d. Find the probability that a randomly selected district had between 2291 and 2561 votes for President Clinton.**\r<BR>\n" );
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document.write( "We need to calculate two z-scores and find the area under the normal curve between them.\r<BR>\n" );
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document.write( "*   z₁ = (2291 - 2150) / 593 ≈ 0.2378<BR>\n" );
document.write( "*   z₂ = (2561 - 2150) / 593 ≈ 0.7099\r<BR>\n" );
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document.write( "Now, find the probabilities associated with these z-scores: P(Z < 0.7099) and P(Z < 0.2378) and subtract to find the probability between the two Z-scores:<BR>\n" );
document.write( "P(2291 < X < 2561) = P(0.2378 < Z < 0.7099) = P(Z < 0.7099) - P(Z < 0.2378) ≈ 0.7611-0.5937 ≈ 0.1674\r<BR>\n" );
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document.write( "**e. Find the third quartile for votes President Clinton. Round to the nearest whole number.**\r<BR>\n" );
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document.write( "The third quartile (Q3) is the value below which 75% of the data falls.  We need to find the z-score corresponding to 0.75 and then convert it to the corresponding number of votes.\r<BR>\n" );
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document.write( "*   Find z such that P(Z < z) = 0.75.  z ≈ 0.674\r<BR>\n" );
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document.write( "*   Now, convert the z-score to the corresponding value of X:<BR>\n" );
document.write( "    X = μ + zσ<BR>\n" );
document.write( "    X = 2150 + (0.674 * 593)<BR>\n" );
document.write( "    X ≈ 2549.982\r<BR>\n" );
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document.write( "Rounding to the nearest whole number, the third quartile is approximately 2550 votes.<BR>\n" );
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