document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191810>Question 1191810</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You wish to test the following claim (Ha) at a significance level of α=0.005<BR>\n" );
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document.write( "Ho: p1 = p2<BR>\n" );
document.write( "Ha: p1 < p2<BR>\n" );
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document.write( "You obtain 86.4% successes in a sample of size n1=766 from the first population. You obtain 91.1% successes in a sample of size n2=649 from the second population. \r<BR>\n" );
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document.write( "What is the test statistic for this sample? (Report answer accurate to three decimal places.)<BR>\n" );
document.write( "test statistic = \r<BR>\n" );
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document.write( "What is the p-value for this sample? (Report answer accurate to four decimal places.)<BR>\n" );
document.write( "p-value = \r<BR>\n" );
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document.write( "The p-value is...\r<BR>\n" );
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document.write( "-less than (or equal to)α<BR>\n" );
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document.write( "-greater than α\r<BR>\n" );
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document.write( "This test statistic leads to a decision to...<BR>\n" );
document.write( "-reject the null<BR>\n" );
document.write( "-accept the null<BR>\n" );
document.write( "-fail to reject the null\r<BR>\n" );
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document.write( "As such, the final conclusion is that...\r<BR>\n" );
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document.write( "(a)There is sufficient evidence to warrant rejection of the claim that the first population proportion is less than the second population proportion.<BR>\n" );
document.write( "(b)There is not sufficient evidence to warrant rejection of the claim that the first population proportion is less than the second population proportion.<BR>\n" );
document.write( "(c)The sample data support the claim that the first population proportion is less than the second population proportion.<BR>\n" );
document.write( "(d)There is not sufficient sample evidence to support the claim that the first population proportion is less than the second population proportion. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #823653 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=823653\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "We're going to use a Two Proportion Z-Test.\r<BR>\n" );
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document.write( "Hypotheses:<BR>\n" );
document.write( "Ho: p1 = p2<BR>\n" );
document.write( "Ha: p1 < p2<BR>\n" );
document.write( "The claim is in the alternative hypothesis and the claim is p1 < p2<BR>\n" );
document.write( "The inequality sign in the alternative hypothesis says we have a left-tailed test.\r<BR>\n" );
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document.write( "Sample sizes<BR>\n" );
document.write( "n1 = 766<BR>\n" );
document.write( "n2 = 649\r<BR>\n" );
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document.write( "Sample proportions<BR>\n" );
document.write( "phat1 = 0.864<BR>\n" );
document.write( "phat2 = 0.911\r<BR>\n" );
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document.write( "phat = x/n<BR>\n" );
document.write( "x = n*phat\r<BR>\n" );
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document.write( "x1 = n1*phat1<BR>\n" );
document.write( "x1 = 766*0.864<BR>\n" );
document.write( "x1 = 661.824<BR>\n" );
document.write( "x1 = 662\r<BR>\n" );
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document.write( "x2 = n2*phat2<BR>\n" );
document.write( "x2 = 649*0.911<BR>\n" );
document.write( "x2 = 591.239<BR>\n" );
document.write( "x2 = 591\r<BR>\n" );
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document.write( "pbar = pooled or average sample proportion<BR>\n" );
document.write( "pbar = (x1+x2)/(n1+n2)<BR>\n" );
document.write( "pbar = (662+591)/(766+649)<BR>\n" );
document.write( "pbar = 0.885512<BR>\n" );
document.write( "Think of pbar in the same light as something like xbar<BR>\n" );
document.write( "It's the letter p with a horizontal bar over top.\r<BR>\n" );
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document.write( "SE = standard error<BR>\n" );
document.write( "SE = sqrt( pbar*(1-pbar)*(1/n1 + 1/n2) )<BR>\n" );
document.write( "SE = sqrt( 0.885512*(1-0.885512)*(1/766 + 1/649) )<BR>\n" );
document.write( "SE = 0.016987\r<BR>\n" );
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document.write( "z = test statistic<BR>\n" );
document.write( "z = (phat1 - phat2)/SE<BR>\n" );
document.write( "z = (0.864 - 0.911)/0.016987<BR>\n" );
document.write( "z = -2.767\r<BR>\n" );
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document.write( "Since the test statistic is recorded to three decimal places, we can't use a standard Z table unfortunately.<BR>\n" );
document.write( "Instead, we can only use a p-value calculator.<BR>\n" );
document.write( "There are countless many such free calculators out there, so feel free to pick your favorite.<BR>\n" );
document.write( "One such option is this really neat online calculator<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://davidmlane.com/normal.html\">https://davidmlane.com/normal.html</a><BR>\n" );
document.write( "Make sure the mean and standard deviation are 0 and 1 respectively.\r<BR>\n" );
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document.write( "The p-value you should get is roughly 0.0028<BR>\n" );
document.write( "Keep in mind this is a left-tailed test.\r<BR>\n" );
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document.write( "Comparing this to the level of significance alpha = 0.005, we see that the p-value is smaller. \r<BR>\n" );
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document.write( "Whenever the p-value is smaller than alpha, we reject the null. <BR>\n" );
document.write( "A useful saying is \"If the p-value is low, then the null must go\".\r<BR>\n" );
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document.write( "Since we rejected the null, we accept the claim that p1 < p2.\r<BR>\n" );
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document.write( "Translating back to the original problem, we have the conclusion that <BR>\n" );
document.write( "(c)The sample data support the claim that the first population proportion is less than the second population proportion.<BR>\n" );
document.write( "Once again, the claim is in the alternative hypothesis.\r<BR>\n" );
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document.write( "<font color=red>Answers:</font><BR>\n" );
document.write( "test statistic = -2.767<BR>\n" );
document.write( "p-value = 0.0028<BR>\n" );
document.write( "The p-value is <u>less</u> than alpha = 0.005<BR>\n" );
document.write( "It means we <u>reject the null</u>\r<BR>\n" );
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document.write( "Conclusion: <BR>\n" );
document.write( "(c)The sample data support the claim that the first population proportion is less than the second population proportion.\r<BR>\n" );
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