document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169897>Question 1169897</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A random sample of n=1000 registered voters and found that 520 would vote for the Republican candidate in a state senate race. Let p represent the proportion of registered voters who would vote for the Republican candidate.\r<BR>\n" );
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document.write( "H0:p=.50<BR>\n" );
document.write( "Ha:p>.50\r<BR>\n" );
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document.write( "(a) The test statistic is z = \r<BR>\n" );
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document.write( "(b) Regardless of what you acutally computed, suppose your answer to part (a) was z = 1.28. Using this z, p-value = </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851283 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851283\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely, let's solve this hypothesis testing problem.\r<BR>\n" );
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document.write( "**Understanding the Problem**\r<BR>\n" );
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document.write( "We're conducting a one-tailed z-test for proportions to determine if there's enough evidence to suggest that more than 50% of registered voters would vote for the Republican candidate.\r<BR>\n" );
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document.write( "**Given Information**\r<BR>\n" );
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document.write( "* Sample size (n) = 1000<BR>\n" );
document.write( "* Number of voters favoring the Republican candidate (x) = 520<BR>\n" );
document.write( "* Sample proportion (p̂) = x/n = 520/1000 = 0.52<BR>\n" );
document.write( "* Null hypothesis (H0): p = 0.50<BR>\n" );
document.write( "* Alternative hypothesis (Ha): p > 0.50\r<BR>\n" );
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document.write( "**(a) Calculating the Test Statistic (z)**\r<BR>\n" );
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document.write( "The formula for the z-test statistic for proportions is:\r<BR>\n" );
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document.write( "$$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$$\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* p̂ is the sample proportion (0.52)<BR>\n" );
document.write( "* p0 is the hypothesized population proportion (0.50)<BR>\n" );
document.write( "* n is the sample size (1000)\r<BR>\n" );
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document.write( "Let's plug in the values:\r<BR>\n" );
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document.write( "$$z = \frac{0.52 - 0.50}{\sqrt{\frac{0.50(1 - 0.50)}{1000}}}$$\r<BR>\n" );
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document.write( "$$z = \frac{0.02}{\sqrt{\frac{0.25}{1000}}}$$\r<BR>\n" );
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document.write( "$$z = \frac{0.02}{\sqrt{0.00025}}$$\r<BR>\n" );
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document.write( "$$z = \frac{0.02}{0.015811388}$$\r<BR>\n" );
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document.write( "$$z \approx 1.265$$\r<BR>\n" );
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document.write( "So, the test statistic is approximately z = 1.265.\r<BR>\n" );
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document.write( "**(b) Calculating the p-value (using z = 1.28)**\r<BR>\n" );
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document.write( "We are given that z = 1.28. We need to find the p-value for a right-tailed test (since Ha: p > 0.50).\r<BR>\n" );
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document.write( "Using a standard normal distribution table or a calculator, we find the area to the right of z = 1.28.\r<BR>\n" );
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document.write( "* The area to the left of z = 1.28 is approximately 0.8997.<BR>\n" );
document.write( "* The area to the right of z = 1.28 is 1 - 0.8997 = 0.1003.\r<BR>\n" );
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document.write( "Therefore, the p-value is approximately 0.1003.\r<BR>\n" );
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document.write( "**Answers**\r<BR>\n" );
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document.write( "(a) The test statistic is z ≈ 1.265.\r<BR>\n" );
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document.write( "(b) If z = 1.28, the p-value ≈ 0.1003.<BR>\n" );
document.write( " </SPAN>\n" );
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