document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200573>Question 1200573</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.\r<BR>\n" );
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document.write( "It is estimated that 3.5% of the general population will live past their 90th birthday. In a graduating class of 758 high school seniors, find the following probabilities. (Round your answers to four decimal places)\r<BR>\n" );
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document.write( "what is the probability that 30 or more will live beyond their 90th birthday? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848087 by </B><A HREF=/tutors/aboutme.mpl?userid=GingerAle><B>GingerAle(43)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=GingerAle><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848087\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Check for Normal Approximation**\r<BR>\n" );
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document.write( "* **Conditions:**<BR>\n" );
document.write( "    * **np ≥ 10:** (758)(0.035) = 26.53 ≥ 10<BR>\n" );
document.write( "    * **n(1-p) ≥ 10:** (758)(0.965) = 731.47 ≥ 10\r<BR>\n" );
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document.write( "* **Conclusion:** Since both conditions are met, the normal approximation to the binomial distribution is appropriate.\r<BR>\n" );
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document.write( "**2. Calculate Mean and Standard Deviation**\r<BR>\n" );
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document.write( "* **Mean (μ):** μ = n * p = 758 * 0.035 = 26.53<BR>\n" );
document.write( "* **Standard Deviation (σ):** σ = √(n * p * (1 - p)) = √(758 * 0.035 * 0.965) ≈ 5.07\r<BR>\n" );
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document.write( "**3. Calculate Z-score**\r<BR>\n" );
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document.write( "* **Continuity Correction:** For \"30 or more,\" we use 29.5 as the lower bound.<BR>\n" );
document.write( "* **Z-score:** z = (X - μ) / σ = (29.5 - 26.53) / 5.07 ≈ 0.586\r<BR>\n" );
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document.write( "**4. Find Probability**\r<BR>\n" );
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document.write( "* **Using a Standard Normal Distribution Table or Calculator:**<BR>\n" );
document.write( "    * Find the area to the right of z = 0.586. <BR>\n" );
document.write( "    * P(X ≥ 30) ≈ 1 - 0.7219 = 0.2781\r<BR>\n" );
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document.write( "**Therefore, the probability that 30 or more of the 758 high school seniors will live beyond their 90th birthday is approximately 0.2781.**<BR>\n" );
document.write( " </SPAN>\n" );
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