document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169896>Question 1169896</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An article published in the Washington Post claims that 45 percent of all Americans have brown eyes. A random sample of n=76 college students students found 28 who had brown eyes.\r<BR>\n" );
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document.write( "H0:p=.45<BR>\n" );
document.write( "Ha:p≠.45\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) 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 #851284 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.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=851284\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this hypothesis testing problem step by step.\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 two-tailed z-test for proportions to determine if the proportion of college students with brown eyes is significantly different from the claimed 45% of all Americans.\r<BR>\n" );
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document.write( "**Given Information**\r<BR>\n" );
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document.write( "* Claimed population proportion (p0) = 0.45<BR>\n" );
document.write( "* Sample size (n) = 76<BR>\n" );
document.write( "* Number of college students with brown eyes (x) = 28<BR>\n" );
document.write( "* Sample proportion (p̂) = x/n = 28/76 ≈ 0.3684<BR>\n" );
document.write( "* Null hypothesis (H0): p = 0.45<BR>\n" );
document.write( "* Alternative hypothesis (Ha): p ≠ 0.45\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 (0.3684)<BR>\n" );
document.write( "* p0 is the hypothesized population proportion (0.45)<BR>\n" );
document.write( "* n is the sample size (76)\r<BR>\n" );
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document.write( "Let's plug in the values:\r<BR>\n" );
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document.write( "$$z = \frac{0.3684 - 0.45}{\sqrt{\frac{0.45(1 - 0.45)}{76}}}$$\r<BR>\n" );
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document.write( "$$z = \frac{-0.0816}{\sqrt{\frac{0.45(0.55)}{76}}}$$\r<BR>\n" );
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document.write( "$$z = \frac{-0.0816}{\sqrt{\frac{0.2475}{76}}}$$\r<BR>\n" );
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document.write( "$$z = \frac{-0.0816}{\sqrt{0.0032565789}}$$\r<BR>\n" );
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document.write( "$$z = \frac{-0.0816}{0.057066442}$$\r<BR>\n" );
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document.write( "$$z \approx -1.4299$$\r<BR>\n" );
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document.write( "Therefore, the test statistic is approximately z = -1.43.\r<BR>\n" );
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document.write( "**(b) Calculating the p-value**\r<BR>\n" );
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document.write( "Since this is a two-tailed test (Ha: p ≠ 0.45), we need to find the area in both tails of the standard normal distribution.\r<BR>\n" );
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document.write( "1.  **Find the area to the left of z = -1.43:**<BR>\n" );
document.write( "    * Using a standard normal distribution table or a calculator, the area to the left of z = -1.43 is approximately 0.0764.\r<BR>\n" );
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document.write( "2.  **Double the area for the two-tailed p-value:**<BR>\n" );
document.write( "    * Since it's a two-tailed test, we multiply the area by 2:<BR>\n" );
document.write( "        * p-value = 2 * 0.0764 = 0.1528.\r<BR>\n" );
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document.write( "Therefore, the p-value is approximately 0.1528.\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 ≈ -1.43.\r<BR>\n" );
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document.write( "(b) The p-value ≈ 0.1528.<BR>\n" );
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