document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168824>Question 1168824</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Based on historical data, your manager believes that 30% of the company's orders come from first-time customers. A random sample of 97 orders will be used to estimate the proportion of first-time-customers.\r<BR>\n" );
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document.write( "What is the probability that the sample proportion is greater than than 0.2?\r<BR>\n" );
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document.write( "Answer = (Enter your answer as a number accurate to 4 decimal places.)  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851512 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=851512\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Define the Parameters**\r<BR>\n" );
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document.write( "* Population proportion (p) = 0.30<BR>\n" );
document.write( "* Sample size (n) = 97<BR>\n" );
document.write( "* Sample proportion (p̂)<BR>\n" );
document.write( "* We want to find P(p̂ > 0.2)\r<BR>\n" );
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document.write( "**2. Check the Conditions for a Normal Approximation**\r<BR>\n" );
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document.write( "To use the normal approximation for the sampling distribution of p̂, we need to check if:\r<BR>\n" );
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document.write( "* np ≥ 10<BR>\n" );
document.write( "* n(1 - p) ≥ 10\r<BR>\n" );
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document.write( "Let's check:\r<BR>\n" );
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document.write( "* np = 97 * 0.30 = 29.1 ≥ 10 (condition met)<BR>\n" );
document.write( "* n(1 - p) = 97 * 0.70 = 67.9 ≥ 10 (condition met)\r<BR>\n" );
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document.write( "Since both conditions are met, we can use the normal approximation.\r<BR>\n" );
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document.write( "**3. Calculate the Mean and Standard Deviation of the Sampling Distribution**\r<BR>\n" );
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document.write( "* Mean of p̂ (μ_p̂) = p = 0.30<BR>\n" );
document.write( "* Standard deviation of p̂ (σ_p̂) = √[p(1 - p) / n]<BR>\n" );
document.write( "    * σ_p̂ = √[(0.30 * 0.70) / 97]<BR>\n" );
document.write( "    * σ_p̂ = √(0.21 / 97)<BR>\n" );
document.write( "    * σ_p̂ = √0.00216494845<BR>\n" );
document.write( "    * σ_p̂ ≈ 0.04652\r<BR>\n" );
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document.write( "**4. Calculate the Z-score**\r<BR>\n" );
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document.write( "We want to find P(p̂ > 0.2). To do this, we need to convert 0.2 to a z-score:\r<BR>\n" );
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document.write( "* z = (p̂ - μ_p̂) / σ_p̂<BR>\n" );
document.write( "* z = (0.2 - 0.30) / 0.04652<BR>\n" );
document.write( "* z = -0.10 / 0.04652<BR>\n" );
document.write( "* z ≈ -2.15\r<BR>\n" );
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document.write( "**5. Find the Probability**\r<BR>\n" );
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document.write( "We want to find P(p̂ > 0.2), which is equivalent to P(Z > -2.15).\r<BR>\n" );
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document.write( "* Using a standard normal distribution table or calculator, we find:<BR>\n" );
document.write( "    * P(Z < -2.15) ≈ 0.0158<BR>\n" );
document.write( "* Since P(Z > -2.15) = 1-P(Z < -2.15)<BR>\n" );
document.write( "    * P(Z > -2.15) = 1-0.0158 = 0.9842\r<BR>\n" );
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document.write( "**6. Final Answer**\r<BR>\n" );
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document.write( "The probability that the sample proportion is greater than 0.2 is approximately 0.9842.\r<BR>\n" );
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document.write( "Answer = 0.9842<BR>\n" );
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