document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203058>Question 1203058</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An orange juice producer buys oranges from a large orange grove that has one variety of oranges. The amount of juice squeezed from these oranges is approximately normally​ distributed, with a mean of 4.10 ounces and a standard deviation of 0.20 ounces. Suppose that you select a sample of 16 oranges.\r<BR>\n" );
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document.write( "a. What is the probability that the sample mean amount of juice will be at least 4.01 ounces?\r<BR>\n" );
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document.write( "b. The probability is 78​% that the sample mean amount of juice will be contained between what two values symmetrically distributed around the population​ mean? There is a 78​% probability that the sample mean amount of juice will be contained between ______​ounce(s) and _____ ​ounce(s) ​(Round to two decimal places as needed. Use ascending​ order.)\r<BR>\n" );
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document.write( "c. The probability is 73​% that the sample mean amount of juice will be greater than what​ value?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #838309 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=838309\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> mean = 4.1 ounces.<BR>\n" );
document.write( "standard deviation = .2 ounces.<BR>\n" );
document.write( "sample size is 16 oranges.<BR>\n" );
document.write( "standard error = standard deviation / square root of sample size = .2 / sqrt(16) = .05.\r<BR>\n" );
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document.write( "use the z-score formula of z = (x-m)/s<BR>\n" );
document.write( "z is the z-score.<BR>\n" );
document.write( "x is the sample mjean.<BR>\n" );
document.write( "m is the population mean.<BR>\n" );
document.write( "s is the standard error.\r<BR>\n" );
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document.write( "a. What is the probability that the sample mean amount of juice will be at least 4.01 ounces?\r<BR>\n" );
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document.write( "when x = 4.01, the formula becomes z = (4.01 - 4.1) / .05 = -1.8.<BR>\n" );
document.write( "area to the right of that z-score is equal to .9641.\r<BR>\n" );
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document.write( "that's the probability that the mean of the sample will be at least 4.01 ounces.\r<BR>\n" );
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document.write( "b. The probability is 78​% that the sample mean amount of juice will be contained between what two values symmetrically distributed around the population​ mean? There is a 78​% probability that the sample mean amount of juice will be contained between ______​ounce(s) and _____ ​ounce(s) ​(Round to two decimal places as needed. Use ascending​ order.)\r<BR>\n" );
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document.write( "if the area between the two z-scores is .78, then the area outside of that range will be 1 - .78 = .22 / 2 = .11.\r<BR>\n" );
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document.write( "you will need 2 z-scores.<BR>\n" );
document.write( "the high z-score will have an area to the left of it of 1 - .11 = .89<BR>\n" );
document.write( "the low z-score will have an area to the left of it equal to .11.<BR>\n" );
document.write( "using a z-score calculator, i get:<BR>\n" );
document.write( "z-score with area of .89 to the left of it = 1.2265.<BR>\n" );
document.write( "z-score with area of .11 to the left of it = -1.2265.<BR>\n" );
document.write( "area in between will be .89 - .11 = .78.<BR>\n" );
document.write( "use the z-score formula to find the raw score.<BR>\n" );
document.write( "z = (x-m)/s<BR>\n" );
document.write( "z is the z-score<BR>\n" );
document.write( "x is the raw score = the mean of the smple<BR>\n" );
document.write( "m is the mean<BR>\n" );
document.write( "s is the standard error.<BR>\n" );
document.write( "solve for x in this formula to get x = s * z + m.<BR>\n" );
document.write( "when z = -1.2265 and m = 4.1 and s = .05, the formula becomes x = .05 * -1.2265 + 4.1 = 4.038675.<BR>\n" );
document.write( "when z = .2265 and m = 4.1 and s = .05, the formula becomes x = .05 * 1.2265 + 4.1 = 4.161325.<BR>\n" );
document.write( "the area between those two sample means will be equal to.78.\r<BR>\n" );
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document.write( "c. The probability is 73​% that the sample mean amount of juice will be greater than what​ value?\r<BR>\n" );
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document.write( "z-score will have an area of 1 - .73 = .27 to the left of it.<BR>\n" );
document.write( "that z-score will be equal to -.6128.<BR>\n" );
document.write( "the sample mean will be equal to x = z * s + m which would be equal to x = -.6128 * .05 + 4.1 = 4.06936.\r<BR>\n" );
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document.write( "the calculator at <a  rel=nofollow HREF= \"https://davidmlane.com/hyperstat/z_table.html\" target = \"_blank\">https://davidmlane.com/hyperstat/z_table.html</a> can be used to confirm these results.\r<BR>\n" );
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document.write( "here are the results.\r<BR>\n" );
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document.write( "for part a.\r<BR>\n" );
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document.write( "for part b.\r<BR>\n" );
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document.write( "for part c.\r<BR>\n" );
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