document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1010631>Question 1010631</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 67.5% of the us population were born in their state of residence. in a random sample of 200 americans find the probability that:<BR>\n" );
document.write( "a) at least 175 were born in their state of residence<BR>\n" );
document.write( "b) exactly 150 were born in their state of residence </SPAN>\n" );
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document.write( "Question:<BR>\n" );
document.write( "67.5% of the us population were born in their state of residence. in a random sample of 200 americans find the probability that:<BR>\n" );
document.write( "a) at least 175 were born in their state of residence<BR>\n" );
document.write( "b) exactly 150 were born in their state of residence<BR>\n" );
document.write( " <BR>\n" );
document.write( "Solution:<BR>\n" );
document.write( "This question qualifies for the binomial distribution, assuming all samples are independent, and population size is large compared to sample.<BR>\n" );
document.write( "given: <BR>\n" );
document.write( "n=200<BR>\n" );
document.write( "p=0.675<BR>\n" );
document.write( " <BR>\n" );
document.write( "Using Binomial Distribution<BR>\n" );
document.write( "A. P(X>=175)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X%3E=175%29=sum%28C%28n%2Cx%29%2Ap%5Ex%2A%281-p%29%5E%28n-x%29%29+for+x=175+to+200 ALIGN=MIDDLE  ALT=\"P%28X%3E=175%29=sum%28C%28n%2Cx%29%2Ap%5Ex%2A%281-p%29%5E%28n-x%29%29+for+x=175+to+200\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where C(n,x)=n!/(x!(n-x)!) is the number of combination of n choose x.<BR>\n" );
document.write( "so after summing the terms (laborious job), we get<BR>\n" );
document.write( "P(X>=175)=5.36*10^(-11)<BR>\n" );
document.write( " <BR>\n" );
document.write( "B. P(X=150)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X=150%29=C%28n%2C150%29%2Ap%5E150%2A%281-p%29%5E%2850%29=0.004432 ALIGN=MIDDLE  ALT=\"P%28X=150%29=C%28n%2C150%29%2Ap%5E150%2A%281-p%29%5E%2850%29=0.004432\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Using normal approximation:<BR>\n" );
document.write( "mean=mu=np=135<BR>\n" );
document.write( "variance=np(1-p)=43.875<BR>\n" );
document.write( "standard deviation=sigma=sqrt(variance)=6.62382<BR>\n" );
document.write( "Use continuity correction<BR>\n" );
document.write( "A. <BR>\n" );
document.write( "z=(174.5-135)/6.62382=5.9423<BR>\n" );
document.write( "upper tail = 1.355*10^(-9)<BR>\n" );
document.write( " <BR>\n" );
document.write( "B. it is more accurate and simpler to work with the binomial distribution.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );