document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208010>Question 1208010</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A random sample of n = 36 observations has a mean x = 28.7 and a standard deviation s = 3.8.\r<BR>\n" );
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document.write( "(a) Find a 90% lower confidence bound for the population mean 𝜇. (Round your answer to two decimal places.)\r<BR>\n" );
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document.write( "(b) How many observations do you need to estimate 𝜇 to within 0.6, with probability equal to 0.95? (Round your answer up to the nearest whole number.)<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 #846199 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=846199\">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>27.66</font><BR>\n" );
document.write( "(b) <font color=red>155</font>\r<BR>\n" );
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document.write( "Explanation for Part (a)\r<BR>\n" );
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document.write( "n = 36 = sample size<BR>\n" );
document.write( "xbar = 28.7 = sample mean<BR>\n" );
document.write( "s = 3.8 = sample standard deviation\r<BR>\n" );
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document.write( "mu = population mean = unknown<BR>\n" );
document.write( "sigma = population standard deviation = unknown\r<BR>\n" );
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document.write( "Since n = 36 fits the criteria n > 30,  we can use the Z distribution even if we don't know the value of sigma.\r<BR>\n" );
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document.write( "At a 90% confidence level, the critical z value is roughly z = 1.645<BR>\n" );
document.write( "Use a reference table or stats calculator to determine this.<BR>\n" );
document.write( "I'm using this table<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf\">https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf</a><BR>\n" );
document.write( "A table like this should be somewhere in the back of your stats textbook. <BR>\n" );
document.write( "Your professor will likely hand out such a table during exams if s/he expects you to use them.<BR>\n" );
document.write( "Refer to the bottom row of the table where it lists the confidence levels. Just above \"90%\" is the value 1.645\r<BR>\n" );
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document.write( "E = margin of error for population mean<BR>\n" );
document.write( "E = z*s/sqrt(n)<BR>\n" );
document.write( "E = 1.645*3.8/sqrt(36)<BR>\n" );
document.write( "E = 1.041833333333 approximately\r<BR>\n" );
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document.write( "L = lower boundary of confidence interval<BR>\n" );
document.write( "L = xbar - E<BR>\n" );
document.write( "L = 28.7 - 1.041833333333<BR>\n" );
document.write( "L = 27.658166666667<BR>\n" );
document.write( "L = <font color=red>27.66</font> which is the answer to part (a).\r<BR>\n" );
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document.write( "Explanation for Part (b)\r<BR>\n" );
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document.write( "E = margin of error for population mean<BR>\n" );
document.write( "E = z*s/sqrt(n)<BR>\n" );
document.write( "E*sqrt(n) = z*s<BR>\n" );
document.write( "sqrt(n) = z*s/E<BR>\n" );
document.write( "n = (z*s/E)^2\r<BR>\n" );
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document.write( "This formula gives us the minimum sample size needed when we specify a desired margin of error.<BR>\n" );
document.write( "In this case we have E = 0.6 <BR>\n" );
document.write( "The portion that says \"with probability equal to 0.95\" refers to a 95% confidence interval.<BR>\n" );
document.write( "At 95% confidence, the z critical value is roughly z = 1.960 (use a table or stats calculator).\r<BR>\n" );
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document.write( "Here are the inputs we'll need<BR>\n" );
document.write( "z = 1.960 (approximate)<BR>\n" );
document.write( "s = 3.8<BR>\n" );
document.write( "E = 0.6\r<BR>\n" );
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document.write( "Let's calculate the minimum sample size.<BR>\n" );
document.write( "n = (z*s/E)^2<BR>\n" );
document.write( "n = (1.960*3.8/0.6)^2<BR>\n" );
document.write( "n = 154.090844444444<BR>\n" );
document.write( "n = <font color=red>155</font> .... rounding up to nearest integer.<BR>\n" );
document.write( "Despite n being much closer to 154, we must round up to 155 to clear the hurdle.\r<BR>\n" );
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document.write( "Let's see what happens when we try n = 154.<BR>\n" );
document.write( "E = z*s/sqrt(n)<BR>\n" );
document.write( "E = 1.960*3.8/sqrt(154)<BR>\n" );
document.write( "E = 0.60018 approximately<BR>\n" );
document.write( "This is slightly over the 0.6 threshold we want.<BR>\n" );
document.write( "The goal is to get E = 0.6 exactly or E < 0.6\r<BR>\n" );
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document.write( "Now try n = 155.<BR>\n" );
document.write( "E = z*s/sqrt(n)<BR>\n" );
document.write( "E = 1.960*3.8/sqrt(155)<BR>\n" );
document.write( "E = 0.59824 approximately<BR>\n" );
document.write( "We are now under the 0.6 threshold. \r<BR>\n" );
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document.write( "Answer to part (b) is <font color=red>155</font><BR>\n" );
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