document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192318>Question 1192318</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 40% of the employees at a large corporation are female a sample of 50 employees is taken and gender is recorded what is the probability of sample proportion of females in sample between 0.35 and 0.45? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849048 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=849048\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Define Variables**\r<BR>\n" );
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document.write( "* **Population Proportion (p):** 0.40 (40% of employees are female)<BR>\n" );
document.write( "* **Sample Size (n):** 50<BR>\n" );
document.write( "* **Sample Proportion (p-hat):** Proportion of females in the sample\r<BR>\n" );
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document.write( "**2. Check Conditions for Normal Approximation**\r<BR>\n" );
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document.write( "* **Independence:** We assume that the employees are randomly selected and that the sample size is less than 10% of the total population.<BR>\n" );
document.write( "* **Success-Failure Condition:**<BR>\n" );
document.write( "    * np = 0.40 * 50 = 20 (Number of female employees in the sample)<BR>\n" );
document.write( "    * n(1-p) = 50 * 0.60 = 30 (Number of male employees in the sample)<BR>\n" );
document.write( "    * Both np and n(1-p) are greater than 10, so the condition is met.\r<BR>\n" );
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document.write( "Since the conditions are met, we can use the normal approximation to the binomial distribution.\r<BR>\n" );
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document.write( "**3. Calculate Standard Error**\r<BR>\n" );
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document.write( "* Standard Error (SE) = √[p * (1-p) / n] <BR>\n" );
document.write( "* SE = √[0.40 * 0.60 / 50] <BR>\n" );
document.write( "* SE = √[0.24 / 50] <BR>\n" );
document.write( "* SE = √0.0048 <BR>\n" );
document.write( "* SE ≈ 0.0693\r<BR>\n" );
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document.write( "**4. Standardize the Values**\r<BR>\n" );
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document.write( "* **For p-hat = 0.35:**<BR>\n" );
document.write( "    * z = (0.35 - 0.40) / 0.0693 ≈ -0.72<BR>\n" );
document.write( "* **For p-hat = 0.45:**<BR>\n" );
document.write( "    * z = (0.45 - 0.40) / 0.0693 ≈ 0.72\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(0.35 < p-hat < 0.45) <BR>\n" );
document.write( "* This is equivalent to finding P(-0.72 < z < 0.72) <BR>\n" );
document.write( "* Using a standard normal table or calculator, we find:<BR>\n" );
document.write( "    * P(z < 0.72) ≈ 0.7642<BR>\n" );
document.write( "    * P(z < -0.72) ≈ 0.2358<BR>\n" );
document.write( "* P(-0.72 < z < 0.72) = P(z < 0.72) - P(z < -0.72) <BR>\n" );
document.write( "* P(-0.72 < z < 0.72) ≈ 0.7642 - 0.2358 <BR>\n" );
document.write( "* P(-0.72 < z < 0.72) ≈ 0.5284\r<BR>\n" );
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document.write( "**Therefore, the probability that the sample proportion of females in the sample is between 0.35 and 0.45 is approximately 0.5284.**<BR>\n" );
document.write( " </SPAN>\n" );
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