document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186293>Question 1186293</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Based on historical data, your manager believes that 38% of the company's orders come from first-time customers. A random sample of 229 orders will be used to estimate the proportion of first-time-customers. What is the probability that the sample proportion is between 0.3 and 0.4?\r<BR>\n" );
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document.write( "Note: You should carefully round any z-values you calculate to 4 decimal places to match wamap's approach and calculations. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849456 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=849456\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem using the normal approximation to the binomial distribution:\r<BR>\n" );
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document.write( "1.  **Check if the normal approximation is appropriate:**\r<BR>\n" );
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document.write( "    *   n * p = 229 * 0.38 = 86.82 >= 10<BR>\n" );
document.write( "    *   n * (1-p) = 229 * (1 - 0.38) = 229 * 0.62 = 142.18 >= 10\r<BR>\n" );
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document.write( "    Since both conditions are met, the normal approximation is reasonable.\r<BR>\n" );
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document.write( "2.  **Calculate the mean (μ) and standard deviation (σ) of the sample proportion:**\r<BR>\n" );
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document.write( "    *   μ = p = 0.38<BR>\n" );
document.write( "    *   σ = sqrt[p * (1-p) / n] = sqrt[0.38 * 0.62 / 229] ≈ 0.0322\r<BR>\n" );
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document.write( "3.  **Calculate the z-scores:**\r<BR>\n" );
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document.write( "    The z-score formula for proportions is: z = (p̂ - p) / σ\r<BR>\n" );
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document.write( "    *   For p̂ = 0.3: z = (0.3 - 0.38) / 0.0322 ≈ -2.4845<BR>\n" );
document.write( "    *   For p̂ = 0.4: z = (0.4 - 0.38) / 0.0322 ≈ 0.6211\r<BR>\n" );
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document.write( "4.  **Find the probabilities using the z-table or calculator:**\r<BR>\n" );
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document.write( "    *   P(z < -2.4845) ≈ 0.0065 (This is the area to the left of -2.4845)<BR>\n" );
document.write( "    *   P(z < 0.6211) ≈ 0.7327 (This is the area to the left of 0.6211)\r<BR>\n" );
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document.write( "5.  **Calculate the probability between 0.3 and 0.4:**\r<BR>\n" );
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document.write( "    P(0.3 < p̂ < 0.4) = P(z < 0.6211) - P(z < -2.4845)<BR>\n" );
document.write( "    P(0.3 < p̂ < 0.4) ≈ 0.7327 - 0.0065<BR>\n" );
document.write( "    P(0.3 < p̂ < 0.4) ≈ 0.7262\r<BR>\n" );
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document.write( "Therefore, the probability that the sample proportion is between 0.3 and 0.4 is approximately **0.7262**.<BR>\n" );
document.write( " </SPAN>\n" );
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