document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201609>Question 1201609</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Assume that females have pulse rates that are normally distributed with a mean of = 76.0 beats per minute and a standard deviation of o = 12.5 beats per minute. Complete parts (a) through (C)<BR>\n" );
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document.write( "a. If 1 adult female is randomly selected, find the probability that her pulse rate is between 72 beats per minute and 80 beats per minute<BR>\n" );
document.write( "The probability is<BR>\n" );
document.write( "[Round to tour decimal o aces as needed.<BR>\n" );
document.write( "b. If 25 adult females are randomly selected, find the probability that they have pulse rates with a mean between 72 beats per minute and 80 beats per minute.<BR>\n" );
document.write( "The probability is<BR>\n" );
document.write( "(Round to four decimal places as needed.)<BR>\n" );
document.write( "c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30?<BR>\n" );
document.write( "A. Since the original population has a normal distribution. the distribution of sample means is a normal distribution for any sample size<BR>\n" );
document.write( "B. Since the mean pulse rate exceeds 30, the distribution of sample means is a normal distribution for any sample size<BR>\n" );
document.write( "C. Since the distribution is of individuals, not sample means, the distribution is a normal distribution for any sample size<BR>\n" );
document.write( "D. Since the distribution is of sample means, not individuals. the distribution is a normal distribution for an sample size </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #836070 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=836070\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answers: <BR>\n" );
document.write( "(a) <font color=red>0.2510</font> (approximate)<BR>\n" );
document.write( "(b) <font color=red>0.8904</font> (approximate)<BR>\n" );
document.write( "(c) <font color=red>A. Since the original population has a normal distribution. the distribution of sample means is a normal distribution for any sample size</font>\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "mu = 76 = population mean pulse rate<BR>\n" );
document.write( "sigma = 12.5 = population standard deviation of the pulse rates<BR>\n" );
document.write( "x = pulse rate of 1 individual\r<BR>\n" );
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document.write( "Each pulse rate is measured in beats per minute (bpm).\r<BR>\n" );
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document.write( "Compute the z score for the raw score x = 72<BR>\n" );
document.write( "z = (x-mu)/sigma<BR>\n" );
document.write( "z = (72-76)/12.5<BR>\n" );
document.write( "z = -4/12.5<BR>\n" );
document.write( "z = -0.32\r<BR>\n" );
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document.write( "Repeat for x = 80<BR>\n" );
document.write( "z = (x-mu)/sigma<BR>\n" );
document.write( "z = (80-76)/12.5<BR>\n" );
document.write( "z = 4/12.5<BR>\n" );
document.write( "z = 0.32\r<BR>\n" );
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document.write( "The task of finding P(72 < x < 80) is equivalent to P(-0.32 < z < 0.32)\r<BR>\n" );
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document.write( "I'll be using this Z table<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.ztable.net/\">https://www.ztable.net/</a><BR>\n" );
document.write( "a similar table can be found in the back of your stats textbook\r<BR>\n" );
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document.write( "Use such a table to find that<BR>\n" );
document.write( "P(Z < -0.32) = 0.37448<BR>\n" );
document.write( "and<BR>\n" );
document.write( "P(Z < 0.32) = 0.62552\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "P(a < Z < b) = P(Z < b) - P(Z < a)<BR>\n" );
document.write( "P(-0.32 < Z < 0.32) = P(Z < 0.32) - P(Z < -0.32)<BR>\n" );
document.write( "P(-0.32 < Z < 0.32) = 0.62552 - 0.37448<BR>\n" );
document.write( "P(-0.32 < Z < 0.32) = 0.25104<BR>\n" );
document.write( "P(-0.32 < Z < 0.32) = <font color=red>0.2510</font>\r<BR>\n" );
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document.write( "Which leads back to<BR>\n" );
document.write( "P(72 < x < 80) = <font color=red>0.2510</font>\r<BR>\n" );
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document.write( "There's about a 25.1% chance of randomly selecting a woman who has a pulse rate between 72 bpm and 80 bpm.\r<BR>\n" );
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document.write( "You could use a specialized stats calculator such as this one<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://onlinestatbook.com/2/calculators/normal_dist.html\">https://onlinestatbook.com/2/calculators/normal_dist.html</a><BR>\n" );
document.write( "as an alternative to using a Z table.\r<BR>\n" );
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document.write( "This article goes over a few examples of how to calculate normal distribution probabilities on a TI84 calculator.<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.statology.org/normal-probabilities-ti-84-calculator/\">https://www.statology.org/normal-probabilities-ti-84-calculator/</a>\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "n = 25 = sample size<BR>\n" );
document.write( "xbar = sample mean = (add up the values)/(number of values)\r<BR>\n" );
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document.write( "Part (a) refers to selecting 1 woman, while part (b) handles selecting a random sample of 25 women.<BR>\n" );
document.write( "Then we compute xbar and want to determine P(72 < xbar < 80)<BR>\n" );
document.write( "We'll use the aptly named xbar distribution.\r<BR>\n" );
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document.write( "The xbar distribution has its center at mu = 76 and standard deviation of SE = 2.5, where SE stands for \"standard error\"<BR>\n" );
document.write( "SE = standard error<BR>\n" );
document.write( "SE  = sigma/sqrt(n) <BR>\n" );
document.write( "SE = (12.5)/(sqrt(25)) <BR>\n" );
document.write( "SE = 2.5\r<BR>\n" );
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document.write( "Compute the z scores.<BR>\n" );
document.write( "Use xbar in place of x.<BR>\n" );
document.write( "Use the SE value in place of sigma.\r<BR>\n" );
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document.write( "If xbar = 72, then,<BR>\n" );
document.write( "z = (xbar-mu)/SE<BR>\n" );
document.write( "z = (72-76)/(2.5)<BR>\n" );
document.write( "z = -4/(2.5)<BR>\n" );
document.write( "z = -1.6\r<BR>\n" );
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document.write( "If xbar = 80, then,<BR>\n" );
document.write( "z = (xbar-mu)/SE<BR>\n" );
document.write( "z = (80-76)/(2.5)<BR>\n" );
document.write( "z = 4/(2.5)<BR>\n" );
document.write( "z = 1.6\r<BR>\n" );
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document.write( "The task of finding P(72 < xbar < 80) is equivalent to P(-1.6 < z < 1.6)\r<BR>\n" );
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document.write( "Use a Z table to determine the following<BR>\n" );
document.write( "P(Z < -1.6) = 0.0548<BR>\n" );
document.write( "P(Z < 1.6) = 0.9452\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "P(a < Z < b) = P(Z < b) - P(Z < a)<BR>\n" );
document.write( "P(-1.6 < Z < 1.6) = P(Z < 1.6) - P(Z < -1.6)<BR>\n" );
document.write( "P(-1.6 < Z < 1.6) = 0.9452 - 0.0548<BR>\n" );
document.write( "P(-1.6 < Z < 1.6) = <font color=red>0.8904</font>\r<BR>\n" );
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document.write( "Which leads back to<BR>\n" );
document.write( "P(72 < xbar < 80) = <font color=red>0.8904</font>\r<BR>\n" );
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document.write( "There's about an 89.04% chance of randomly selecting a group of 25 women who have a sample mean (xbar) pulse rate between 72 bpm and 80 bpm.\r<BR>\n" );
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document.write( "--------------------------\r<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "At the top of the problem it states \"Assume that females have pulse rates that are <u><b>normally distributed</b></u>\"<BR>\n" );
document.write( "So because the population is normally distributed, it indicates the distribution of sample means (xbar distribution) is also normally distributed. This is regardless of the sample size.\r<BR>\n" );
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document.write( "Therefore, we arrive at the final answer <font color=red>A. Since the original population has a normal distribution. the distribution of sample means is a normal distribution for any sample size</font><BR>\n" );
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