document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199773>Question 1199773</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A simple random sample of size n=49 is obtained from a population with mean = 80 and standard deviation = 14. \r<BR>\n" );
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document.write( "a) Describe the sampling distribution of x̅.\r<BR>\n" );
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document.write( "b) What is P(x̅ > 83)?\r<BR>\n" );
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document.write( "c) What is P(x̅ ≤ 75.8)?\r<BR>\n" );
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document.write( "d) What is P(78.3 < x̅ < 85.1)? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848265 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=848265\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Describe the sampling distribution of x̅**\r<BR>\n" );
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document.write( "* **Mean of the sampling distribution (μ<sub>x̅</sub>):** <BR>\n" );
document.write( "    * μ<sub>x̅</sub> = μ = 80 <BR>\n" );
document.write( "    * The mean of the sampling distribution of the sample means is equal to the population mean.\r<BR>\n" );
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document.write( "* **Standard deviation of the sampling distribution (standard error):** <BR>\n" );
document.write( "    * σ<sub>x̅</sub> = σ / √n = 14 / √49 = 14 / 7 = 2 \r<BR>\n" );
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document.write( "* **Shape of the sampling distribution:** <BR>\n" );
document.write( "    * Since the sample size (n = 49) is sufficiently large (n ≥ 30), according to the Central Limit Theorem, the sampling distribution of the sample mean (x̅) will be approximately normally distributed, regardless of the shape of the original population distribution.\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "* The sampling distribution of x̅ is approximately normally distributed with:<BR>\n" );
document.write( "    * Mean (μ<sub>x̅</sub>) = 80<BR>\n" );
document.write( "    * Standard deviation (σ<sub>x̅</sub>) = 2 \r<BR>\n" );
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document.write( "**b) P(x̅ > 83)**\r<BR>\n" );
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document.write( "1. **Standardize the value:**<BR>\n" );
document.write( "    * z = (x̅ - μ<sub>x̅</sub>) / σ<sub>x̅</sub> <BR>\n" );
document.write( "    * z = (83 - 80) / 2 = 1.5\r<BR>\n" );
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document.write( "2. **Find the probability using a standard normal distribution table or calculator:**<BR>\n" );
document.write( "    * P(x̅ > 83) = P(Z > 1.5) <BR>\n" );
document.write( "    * From the z-table, P(Z > 1.5) = 0.0668\r<BR>\n" );
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document.write( "**Therefore, P(x̅ > 83) = 0.0668**\r<BR>\n" );
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document.write( "**c) P(x̅ ≤ 75.8)**\r<BR>\n" );
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document.write( "1. **Standardize the value:**<BR>\n" );
document.write( "    * z = (75.8 - 80) / 2 = -2.1\r<BR>\n" );
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document.write( "2. **Find the probability using a standard normal distribution table or calculator:**<BR>\n" );
document.write( "    * P(x̅ ≤ 75.8) = P(Z ≤ -2.1) <BR>\n" );
document.write( "    * From the z-table, P(Z ≤ -2.1) = 0.0179\r<BR>\n" );
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document.write( "**Therefore, P(x̅ ≤ 75.8) = 0.0179**\r<BR>\n" );
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document.write( "**d) P(78.3 < x̅ < 85.1)**\r<BR>\n" );
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document.write( "1. **Standardize the values:**<BR>\n" );
document.write( "    * z1 = (78.3 - 80) / 2 = -0.85<BR>\n" );
document.write( "    * z2 = (85.1 - 80) / 2 = 2.55\r<BR>\n" );
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document.write( "2. **Find the probabilities using a standard normal distribution table or calculator:**<BR>\n" );
document.write( "    * P(x̅ < 85.1) = P(Z < 2.55) <BR>\n" );
document.write( "    * P(x̅ < 78.3) = P(Z < -0.85)\r<BR>\n" );
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document.write( "3. **Calculate the desired probability:**<BR>\n" );
document.write( "    * P(78.3 < x̅ < 85.1) = P(x̅ < 85.1) - P(x̅ < 78.3) <BR>\n" );
document.write( "                           = P(Z < 2.55) - P(Z < -0.85)<BR>\n" );
document.write( "                           ≈ 0.9946 - 0.1977 <BR>\n" );
document.write( "                           ≈ 0.7969\r<BR>\n" );
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document.write( "**Therefore, P(78.3 < x̅ < 85.1) = 0.7969**<BR>\n" );
document.write( " </SPAN>\n" );
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