document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1132222>Question 1132222</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The city of Providence plans to conduct a study on population expenditures related to the use of the city�s public transit such as bus and subway services. Data will be obtained by the mean of a random sample of the families living in Providence city. To determine the sample size, it was hypothesized that on average, families spend annually 2000 QAR with a standard deviation of 1200 QAR.\r<BR>\n" );
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document.write( "1. Calculate the sample size allowing Providence city to have an estimation of average family�s expenditures (sample mean) on public transit that falls within an interval of 200 QAR above or below average population expenditures with a 95% level of confidence? Repeat the same analysis with a 90% level of confidence instead of 95%.\r<BR>\n" );
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document.write( "2. Assuming the precision of the estimation was relaxed from 200 QAR to 400 QAR, what is the required sample size at a 95% level of confidence?\r<BR>\n" );
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document.write( "3. Considering the result obtained in (1) and (2), what are the impacts of levels of confidence and precision on sample size?<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 #749219 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=749219\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The formula for sample size(n) is<BR>\n" );
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document.write( "n = ((critical value(CV) * standard deviation)/margin of error(ME))^2<BR>\n" );
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document.write( "CV is the z-value associated with the given confidence level<BR>\n" );
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document.write( "Note we assume the mean family income is Normally distributed<BR>\n" );
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document.write( "CV for 95% confidence level is the z-score associated with critical probability(p*) of 0.975 which is 1.96<BR>\n" );
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document.write( "CV for 90% confidence level is z-score associated with p* of 0.95 which is 1.645<BR>\n" );
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document.write( "1.a 95% confidence level, ME is 200<BR>\n" );
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document.write( "n = (1.96 * 1200 / 200)^2 = 138.2976 is approximately 138<BR>\n" );
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document.write( "1.b 90% confidence level, ME is 200<BR>\n" );
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document.write( "n = (1.645 * 1200 /200)^2 = 97.4169 is approximately 97<BR>\n" );
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document.write( "2.  95% confidence level, ME is 400<BR>\n" );
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document.write( "n = (1.96 * 1200 / 400)^2 = 34.5744 is approximately 35<BR>\n" );
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document.write( "3.  A greater degree of confidence with the same margin of error requires more samples while relaxing the precision of the margin of error requires fewer  samples.<BR>\n" );
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