document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1150934>Question 1150934</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A nationwide sample of influential Republicans and Democrats was asked as a part of a comprehensive survey whether they favored lowering environmental standards so that high-sulfur coal could be burned in coal-fired power plants. The results were:\r<BR>\n" );
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document.write( "Number sampled : 1,000 (republican) , 800 (democrats)<BR>\n" );
document.write( "Number in favor: 200 (republican) , 168 (democrats)\r<BR>\n" );
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document.write( "At the .02 level of significance, can we conclude that there is a larger proportion of Democrats in favor of lowering the standards? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #772880 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=772880\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" color=\"black\" size=\"3\"><BR>\n" );
document.write( "Let group 1 be the sample of Republicans. Group 2 will be the sample of Democrats.\r<BR>\n" );
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document.write( "H0: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p_{1} = p_{2}\"> (equivalent to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?p_{1}-p_{2} = 0\">) is the null hypothesis<BR>\n" );
document.write( "H1: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p_{1} < p_{2}\"> (equivalent to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?p_{1}-p_{2} < 0\">) is the alternative hypothesis<BR>\n" );
document.write( "where,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p_{1}\"> = population proportion of Republicans in favor of lowering the standards<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p_{2}\"> = population proportion of Democrats in favor of lowering the standards\r<BR>\n" );
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document.write( "We are conducting a left tailed test because of the \"less than\" sign in the alternative hypothesis.  \r<BR>\n" );
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document.write( "We are given<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x_{1} = 200\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x_{2} = 168\"><BR>\n" );
document.write( "as the two counts of successes (ie the number of people who responded in favor for each group). And also, we have these sample sizes<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n_{1} = 1000\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n_{2} = 800\">\r<BR>\n" );
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document.write( "Which helps us find the sample proportions for each group<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \hat{p}_{1} = \frac{x_1}{n_1} = \frac{200}{1000} = 0.2\"> is the sample proportion of Republicans in favor of lowering the standards<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \hat{p}_{2} = \frac{x_2}{n_2} = \frac{168}{800} = 0.21\"> is the sample proportion of Democrats in favor of lowering the standards<BR>\n" );
document.write( "Based on these two p-hat values, it does look like there is a higher proportion of Democrats in favor of lowering the standards. However, let's continue with the hypothesis test.\r<BR>\n" );
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document.write( "Compute the pooled proportion. The bar over the 'p' indicates an average.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \overline{p} = \frac{x_1+x_2}{n_1+n_2} = \frac{200+168}{1000+800} \approx 0.204444444\"> \r<BR>\n" );
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document.write( "Now onto the Standard Error. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large SE = \sqrt{\overline{p}*(1-\overline{p})*\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large SE \approx \sqrt{0.204 444 444*(1-0.204444444)*\left(\frac{1}{1000}+\frac{1}{800}\right)}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large SE \approx 0.019129965\"> is the approximate standard error\r<BR>\n" );
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document.write( "Compute the test statistic<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large z = \frac{ (\hat{p}_{1}-\hat{p}_{2}) - (p_{1}-p_{2}) }{SE}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large z \approx \frac{ (0.2 - 0.21) - (0) }{0.019129965}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large z \approx -0.522740110\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large z \approx -0.52\">\r<BR>\n" );
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document.write( "Use a Z table similar to this one<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.z-table.com/\">http://www.z-table.com/</a><BR>\n" );
document.write( "To find that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P(Z < -0.52) \approx 0.3015\">\r<BR>\n" );
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document.write( "The P-value is approximately 0.3015\r<BR>\n" );
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document.write( "This is much larger than alpha = 0.02, so we fail to reject the null hypothesis.<BR>\n" );
document.write( "We do not have enough statistically significant evidence to overturn or reject the null. <BR>\n" );
document.write( "Therefore, we must accept that the population proportions p1 and p2 are the same. <BR>\n" );
document.write( "In other words, the proportion of people in favor of lowering the standards are the same for each group. \r<BR>\n" );
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document.write( "Going back to the question \"<font color=blue>can we conclude that there is a larger proportion of Democrats in favor of lowering the standards?</font>\", the answer is \"no, we cannot conclude there is a larger proportion of Democrats in favor of lowering the standards\".<BR>\n" );
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