document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199774>Question 1199774</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A simple random sample of size n = 200 is obtained from a population whose size is N = 25,000 and whose population proportion with a specified characteristic is = = 0.65.\r<BR>\n" );
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document.write( "a) Describe the sampling distribution of 𝑝\r<BR>\n" );
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document.write( "b) What is the probability of obtaining x = 136 or fewer individuals with the characteristic? That is, what is P(𝑝 >= 0.68)?\r<BR>\n" );
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document.write( "c) What is the probability of obtaining x = 118 or fewer individuals with the characteristic? That is, what is P (p = 0.59)?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848165 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848165\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Describe the sampling distribution of p**\r<BR>\n" );
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document.write( "* **Shape:** Since the sample size (n = 200) is large and the population size (N = 25,000) is much larger, the sampling distribution of the sample proportion (p̂) can be approximated by a normal distribution. This is due to the Central Limit Theorem.\r<BR>\n" );
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document.write( "* **Mean:** The mean of the sampling distribution of p̂ is equal to the population proportion (p): <BR>\n" );
document.write( "   * μ<sub>p̂</sub> = p = 0.65\r<BR>\n" );
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document.write( "* **Standard Deviation:** The standard deviation of the sampling distribution of p̂ is given by:<BR>\n" );
document.write( "   * σ<sub>p̂</sub> = √[p * (1 - p) / n] <BR>\n" );
document.write( "   * σ<sub>p̂</sub> = √[0.65 * (1 - 0.65) / 200] <BR>\n" );
document.write( "   * σ<sub>p̂</sub> = √[0.2275 / 200] <BR>\n" );
document.write( "   * σ<sub>p̂</sub> ≈ 0.0337\r<BR>\n" );
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document.write( "**Therefore, the sampling distribution of p̂ is approximately normal with mean 0.65 and standard deviation 0.0337.**\r<BR>\n" );
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document.write( "**b) Probability of obtaining x = 136 or fewer individuals with the characteristic (P(p̂ >= 0.68))**\r<BR>\n" );
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document.write( "1. **Calculate the z-score for p̂ = 0.68:**<BR>\n" );
document.write( "   * z = (p̂ - μ<sub>p̂</sub>) / σ<sub>p̂</sub> <BR>\n" );
document.write( "   * z = (0.68 - 0.65) / 0.0337 <BR>\n" );
document.write( "   * z ≈ 0.89\r<BR>\n" );
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document.write( "2. **Find the probability:**<BR>\n" );
document.write( "   * P(p̂ >= 0.68) = P(Z >= 0.89) <BR>\n" );
document.write( "   * Using a standard normal table or calculator, find the area to the right of z = 0.89. <BR>\n" );
document.write( "   * P(Z >= 0.89) ≈ 0.1867\r<BR>\n" );
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document.write( "**Therefore, the probability of obtaining 136 or fewer individuals with the characteristic is approximately 0.1867.**\r<BR>\n" );
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document.write( "**c) Probability of obtaining x = 118 or fewer individuals with the characteristic (P(p̂ <= 0.59))**\r<BR>\n" );
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document.write( "1. **Calculate the z-score for p̂ = 0.59:**<BR>\n" );
document.write( "   * z = (0.59 - 0.65) / 0.0337 <BR>\n" );
document.write( "   * z ≈ -1.78\r<BR>\n" );
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document.write( "2. **Find the probability:**<BR>\n" );
document.write( "   * P(p̂ <= 0.59) = P(Z <= -1.78)<BR>\n" );
document.write( "   * Using a standard normal table or calculator, find the area to the left of z = -1.78. <BR>\n" );
document.write( "   * P(Z <= -1.78) ≈ 0.0375\r<BR>\n" );
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document.write( "**Therefore, the probability of obtaining 118 or fewer individuals with the characteristic is approximately 0.0375.**<BR>\n" );
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