document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177608>Question 1177608</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The times spent by people visiting a certain dentist are independent and normally distributed with a mean of 8.2 minutes. 79% of people who visit this dentist have visits lasting less than 10 minutes.\r<BR>\n" );
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document.write( "Find the probability that, of 35 randomly chosen people, fewer than 16 have visits lasting<BR>\n" );
document.write( "less than 8.2 minutes.<BR>\n" );
document.write( "(answer = 0.250) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850438 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=850438\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **Step 1: Find the standard deviation**\r<BR>\n" );
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document.write( "We know that the visit times are normally distributed with a mean (µ) of 8.2 minutes. We also know that 79% of the visits last less than 10 minutes. This information allows us to find the standard deviation (σ) of the distribution.\r<BR>\n" );
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document.write( "* Let X be the random variable representing the time spent by a person visiting the dentist.<BR>\n" );
document.write( "* We are given P(X < 10) = 0.79.<BR>\n" );
document.write( "* We can standardize this value by finding the z-score corresponding to a cumulative probability of 0.79. Using a standard normal table or calculator, we find that the z-score is approximately 0.81.\r<BR>\n" );
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document.write( "Now, using the z-score formula:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "z = (x - µ) / σ<BR>\n" );
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document.write( "We can plug in the values:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "0.81 = (10 - 8.2) / σ<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Solving for σ:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "σ ≈ 2.22 minutes<BR>\n" );
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document.write( "**Step 2: Find the probability of a visit lasting less than 8.2 minutes**\r<BR>\n" );
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document.write( "Since the mean is 8.2 minutes, the probability of a visit lasting less than 8.2 minutes is simply 0.5 (because the normal distribution is symmetric around the mean).\r<BR>\n" );
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document.write( "**Step 3: Find the probability of fewer than 16 people out of 35 having visits less than 8.2 minutes**\r<BR>\n" );
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document.write( "Now we have a binomial distribution problem.\r<BR>\n" );
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document.write( "* n = 35 (number of trials)<BR>\n" );
document.write( "* p = 0.5 (probability of success, i.e., a visit lasting less than 8.2 minutes)<BR>\n" );
document.write( "* We want to find P(X < 16), where X is the number of people with visits less than 8.2 minutes.\r<BR>\n" );
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document.write( "We can use the binomial probability formula or a binomial calculator to find this probability. Using a calculator, we get:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "P(X < 16) ≈ 0.214<BR>\n" );
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document.write( "**Therefore, the probability that fewer than 16 out of 35 randomly chosen people have visits lasting less than 8.2 minutes is approximately 0.214.** </SPAN>\n" );
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