document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193353>Question 1193353</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me!<BR>\n" );
document.write( "In the following exercises, use this information for the overhead reach distances of adult females: mean = 205.5 cm, S.D = 8.6 cm, and overhead reach distances are normally distributed. The overhead reach distances are used in planning assembly workstations.\r<BR>\n" );
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document.write( "1. If 1 adult female is randomly selected, find the probability that her overhead reach is less than 222.7 cm.\r<BR>\n" );
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document.write( "2. If 49 adult females are randomly selected, find the probability that they have a mean overhead reach of less than 207 cm.\r<BR>\n" );
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document.write( "3. If 1 adult female is randomly selected, find the probability that her overhead reach is greater than 196.9 cm.\r<BR>\n" );
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document.write( "4. If 36 adult females are randomly selected, find the probability that they have a mean overhead reach greater than 205 cm.\r<BR>\n" );
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document.write( "5. If 1 adult female is randomly selected, find the probability that her overhead reach is between 179.7 cm and 231.3 cm.\r<BR>\n" );
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document.write( "6. If 40 adult females are randomly selected, find the probability that they have a mean overhead reach between 204.0 cm and 206.0 cm.\r<BR>\n" );
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document.write( "Thank you for your time. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #825367 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=825367\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Problem 1\r<BR>\n" );
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document.write( "mu = population mean = 205.5<BR>\n" );
document.write( "sigma = population standard deviation = 8.6\r<BR>\n" );
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document.write( "Convert the raw score x = 222.7 to its corresponding z score<BR>\n" );
document.write( "z = (x - mu)/sigma<BR>\n" );
document.write( "z = (222.7 - 205.5)/8.6<BR>\n" );
document.write( "z = 2\r<BR>\n" );
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document.write( "Now use a Z table such as this one<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.ztable.net/\">https://www.ztable.net/</a><BR>\n" );
document.write( "to find that P(Z < 2.00) = 0.97725<BR>\n" );
document.write( "You'll need to scroll down to refer to the second table on that page.<BR>\n" );
document.write( "Look at the row that starts with +2.0 and the column that starts with 0 up top. The row and column intersect at 0.97725\r<BR>\n" );
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document.write( "If you were to use a TI calculator, then you'd use the <font color=blue>normalcdf(</font> function. To access it, hit the yellow 2ND key at the top corner, then hit VARS to reach the stats distribution menu. <BR>\n" );
document.write( "The general template of the function is<BR>\n" );
document.write( "<font color=blue>normalcdf(lower, upper)</font><BR>\n" );
document.write( "so we must provide a lower and upper boundary under the curve (ie the left and right endpoints). <BR>\n" );
document.write( "The right endpoint must be z = 2. The left endpoint can be some fairly large negative value such as -99<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/cRhEWWh.png\"><BR>\n" );
document.write( "The screenshot shows that the area to the left of z = 2 is roughly 0.977249938; rounding to five decimal places gets us 0.97725 mentioned in the previous paragraph.\r<BR>\n" );
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document.write( "If you don't have a calculator, then you can use this free resource.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://davidmlane.com/normal.html\">https://davidmlane.com/normal.html</a><BR>\n" );
document.write( "Keep mean and SD as 0 and 1 respectively<BR>\n" );
document.write( "Click the \"below\" button and type 2 in the box just next to it. Then hit \"recalculate\" to have 0.9772 show up. \r<BR>\n" );
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document.write( "If you use WolframAlpha (another free calculator), you would type in <font color=blue>P(Z < 2)</font> to have the result 0.97725 show up\r<BR>\n" );
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document.write( "I'll round the result to four decimal places, though be sure to follow your teacher's instructions.\r<BR>\n" );
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document.write( "<font color=red>Answer: Roughly 0.9773</font>\r<BR>\n" );
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document.write( "================================================================<BR>\n" );
document.write( "Problem 2\r<BR>\n" );
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document.write( "The sample size is n = 49<BR>\n" );
document.write( "When forming the xbar distribution, it has a standard error (SE) of <BR>\n" );
document.write( "SE = sigma/sqrt(n)<BR>\n" );
document.write( "SE = 8.6/sqrt(49)<BR>\n" );
document.write( "SE = 1.22857142857142<BR>\n" );
document.write( "This will be treated as the standard deviation of the xbar distribution. The mean is the same as before\r<BR>\n" );
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document.write( "Let's convert the raw score x = 207 to its corresponding z score.<BR>\n" );
document.write( "z = (x - mu)/SE<BR>\n" );
document.write( "z = (207 - 205.5)/1.22857142857142<BR>\n" );
document.write( "z = 1.22093023255814<BR>\n" );
document.write( "z = 1.22<BR>\n" );
document.write( "When the option arises, it's standard practice to round the z values to two decimal places. Though be sure to follow your teacher's instruction if s/he wants a different level of precision.\r<BR>\n" );
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document.write( "If you were to use the table I mentioned earlier, then you should get P(Z < 1.22) = 0.88877\r<BR>\n" );
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document.write( "Using a TI calculator should produce the value 0.8887674991\r<BR>\n" );
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document.write( "Using the calculator in the link I posted earlier would give the value 0.8888\r<BR>\n" );
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document.write( "If you prefer WolframAlpha, then type in P(Z < 1.22) to get a result of 0.888768\r<BR>\n" );
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document.write( "<font color=red>Answer: Approximately 0.8888</font>\r<BR>\n" );
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document.write( "Problem 3\r<BR>\n" );
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document.write( "We will use the formula mentioned in problem 1 to convert x = 196.9 to its z score\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "z = (x - mu)/sigma<BR>\n" );
document.write( "z = (196.9 - 205.5)/8.6<BR>\n" );
document.write( "z = -1\r<BR>\n" );
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document.write( "Then use your method of choice (table or calculator) to find that P(Z < -1) is equal to...<BR>\n" );
document.write( "0.15866 when using the table<BR>\n" );
document.write( "0.1586552596 when using a TI calculator<BR>\n" );
document.write( "0.1587 when using the David M Lane calculator link<BR>\n" );
document.write( "0.158655 when using WolframAlpha\r<BR>\n" );
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document.write( "They all basically say that P(Z < -1) is equal to 0.1587 when rounding to four decimal places.\r<BR>\n" );
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document.write( "Then we can say this:<BR>\n" );
document.write( "P(Z > -1) = 1 - P(Z < -1)<BR>\n" );
document.write( "P(Z > -1) = 1 - 0.1587<BR>\n" );
document.write( "P(Z > -1) = 0.8413<BR>\n" );
document.write( "This converts to<BR>\n" );
document.write( "P(X > 196.9) = 0.8413<BR>\n" );
document.write( "when mu = 205.5 and sigma = 8.6\r<BR>\n" );
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document.write( "<font color=red>Answer: 0.8413 (approximate)</font>\r<BR>\n" );
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document.write( "================================================================<BR>\n" );
document.write( "Problem 4\r<BR>\n" );
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document.write( "Like with problem 2, we'll need to calculate the standard error (SE) first. This is because we're turning to the xbar distribution. Whenever we're asked to find the probability that the sample mean is larger than some cutoff, it's when we use the xbar distribution.\r<BR>\n" );
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document.write( "SE = sigma/sqrt(n)<BR>\n" );
document.write( "SE = 8.6/sqrt(36)<BR>\n" );
document.write( "SE = 1.43333333333333\r<BR>\n" );
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document.write( "Then we'd say<BR>\n" );
document.write( "z = (x - mu)/SE<BR>\n" );
document.write( "z = (205 - 205.5)/1.43333333333333<BR>\n" );
document.write( "z = -0.34883720930232<BR>\n" );
document.write( "z = -0.35\r<BR>\n" );
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document.write( "Using whatever method of choice (table or calculator), you should find that P(Z < -0.35) = 0.3632 when rounding to four decimal places\r<BR>\n" );
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document.write( "Therefore,<BR>\n" );
document.write( "P(Z > -0.35) = 1 - P(Z < -0.35)<BR>\n" );
document.write( "P(Z > -0.35) = 1 - 0.3632<BR>\n" );
document.write( "P(Z > -0.35) = 0.6368<BR>\n" );
document.write( "which converts over to<BR>\n" );
document.write( "P(xbar > 205) = 0.6368<BR>\n" );
document.write( "There's roughly a 63.68% chance we have a sample mean (xbar) larger than 205\r<BR>\n" );
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document.write( "<font color=red>Answer: 0.6368 (approximate)</font>\r<BR>\n" );
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document.write( "================================================================<BR>\n" );
document.write( "Problem 5\r<BR>\n" );
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document.write( "Convert x = 179.7 to its z score<BR>\n" );
document.write( "z = (x-mu)/sigma<BR>\n" );
document.write( "z = (179.7-205.5)/8.6<BR>\n" );
document.write( "z = -3\r<BR>\n" );
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document.write( "Do the same for x = 231.3<BR>\n" );
document.write( "z = (x - mu)/sigma<BR>\n" );
document.write( "z = (231.3 - 205.5)/8.6<BR>\n" );
document.write( "z = 3\r<BR>\n" );
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document.write( "Determining P(179.7 < X < 231.3) is exactly equivalent to P(-3 < Z < 3)\r<BR>\n" );
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document.write( "Use your method of choice to find that<BR>\n" );
document.write( "P(Z < -3) = 0.0014<BR>\n" );
document.write( "P(Z < 3) = 0.9987<BR>\n" );
document.write( "So,<BR>\n" );
document.write( "P(a < Z < b) = P(Z < b) - P(Z < a)<BR>\n" );
document.write( "P(-3 < Z < 3) = P(Z < 3) - P(Z < -3)<BR>\n" );
document.write( "P(-3 < Z < 3) = 0.9987 - 0.0014<BR>\n" );
document.write( "P(-3 < Z < 3) = 0.9973\r<BR>\n" );
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document.write( "The nice thing about the WolframAlpha calculator is that you can type in P(-3 < Z < 3) as is, and it will produce roughly 0.9973 as a way to check your work.<BR>\n" );
document.write( "The David M Lane calculator link also provides a direct route to finding the area between two given endpoints. Click on the \"between\" option.<BR>\n" );
document.write( "With the TI calculator, you'll of course have the \"lower\" and \"upper\" as -3 and 3 respectively.\r<BR>\n" );
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document.write( "<font color=red>Answer: 0.9973 (approximate)</font>\r<BR>\n" );
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document.write( "================================================================<BR>\n" );
document.write( "Problem 6\r<BR>\n" );
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document.write( "sample size = n = 40<BR>\n" );
document.write( "SE = standard error<BR>\n" );
document.write( "SE = sigma/sqrt(n)<BR>\n" );
document.write( "SE = 8.6/sqrt(40)<BR>\n" );
document.write( "SE = 1.3597793938724\r<BR>\n" );
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document.write( "Convert xbar = 204.0 to its z score<BR>\n" );
document.write( "z = (xbar - mu)/SE<BR>\n" );
document.write( "z = (204.0 - 205.5)/1.3597793938724<BR>\n" );
document.write( "z = -1.10312011401222<BR>\n" );
document.write( "z = -1.10\r<BR>\n" );
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document.write( "Repeat for xbar = 206.0 <BR>\n" );
document.write( "z = (xbar - mu)/SE<BR>\n" );
document.write( "z = (206.0 - 205.5)/1.3597793938724<BR>\n" );
document.write( "z = 0.36770670467074<BR>\n" );
document.write( "z = 0.37\r<BR>\n" );
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document.write( "Then we can say<BR>\n" );
document.write( "P(Z < -1.10) = 0.1357<BR>\n" );
document.write( "P(Z < 0.37) = 0.6443<BR>\n" );
document.write( "so,<BR>\n" );
document.write( "P(a < Z < b) = P(Z < b) - P(Z < a)<BR>\n" );
document.write( "P(-1.10 < Z < 0.37) = P(Z < 0.37) - P(Z < -1.10)<BR>\n" );
document.write( "P(-1.10 < Z < 0.37) = 0.6443 - 0.1357<BR>\n" );
document.write( "P(-1.10 < Z < 0.37) = 0.5086\r<BR>\n" );
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document.write( "<font color=red>Answer: 0.5086</font>\r<BR>\n" );
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document.write( "As with all of the other answers, this result is approximate to four decimal places.<BR>\n" );
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