document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186329>Question 1186329</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A random variable is normally distributed.  It has a mean of 245 and a standard deviation of 21.  <BR>\n" );
document.write( "a.)	If you take a sample of size 10, can you say what the shape of the distribution for the sample mean is?  Why?<BR>\n" );
document.write( "b.)	For a sample of size 10, state the mean of the sample mean and the standard deviation of the sample mean. <BR>\n" );
document.write( "c.)	For a sample of size 10, find the probability that the sample mean is more than 241.<BR>\n" );
document.write( "d.)	If you take a sample of size 35, can you say what the shape of the distribution of the sample mean is?  Why?<BR>\n" );
document.write( "e.)	For a sample of size 35, state the mean of the sample mean and the standard deviation of the sample mean. <BR>\n" );
document.write( "f.)	For a sample of size 35, find the probability that the sample mean is more than 241.<BR>\n" );
document.write( "g.)	Compare your answers in part c and f.  Why is one smaller than the other?<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 #849453 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=849453\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's the solution:\r<BR>\n" );
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document.write( "**a.) Shape of the distribution for the sample mean (n=10):**\r<BR>\n" );
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document.write( "Yes. Because the original population is normally distributed, the sampling distribution of the sample mean will *also* be normally distributed, regardless of the sample size.\r<BR>\n" );
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document.write( "**b.) Mean and standard deviation of the sample mean (n=10):**\r<BR>\n" );
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document.write( "*   Mean of the sample mean (μₓ̄) = Population mean (μ) = 245<BR>\n" );
document.write( "*   Standard deviation of the sample mean (σₓ̄) = Population standard deviation (σ) / √n = 21 / √10 ≈ 6.64\r<BR>\n" );
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document.write( "**c.) Probability that the sample mean is more than 241 (n=10):**\r<BR>\n" );
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document.write( "1.  **Calculate the z-score:**<BR>\n" );
document.write( "    z = (x̄ - μ) / σₓ̄<BR>\n" );
document.write( "    z = (241 - 245) / 6.64<BR>\n" );
document.write( "    z ≈ -0.60\r<BR>\n" );
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document.write( "2.  **Find the probability:**<BR>\n" );
document.write( "    Using a z-table or calculator, find the probability of z being *greater* than -0.60.<BR>\n" );
document.write( "    P(z > -0.60) ≈ 0.7257\r<BR>\n" );
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document.write( "**d.) Shape of the distribution of the sample mean (n=35):**\r<BR>\n" );
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document.write( "Yes.  Again, since the original population is normally distributed, the sampling distribution of the sample mean will *also* be normally distributed, even with a larger sample size.\r<BR>\n" );
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document.write( "**e.) Mean and standard deviation of the sample mean (n=35):**\r<BR>\n" );
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document.write( "*   Mean of the sample mean (μₓ̄) = Population mean (μ) = 245<BR>\n" );
document.write( "*   Standard deviation of the sample mean (σₓ̄) = Population standard deviation (σ) / √n = 21 / √35 ≈ 3.56\r<BR>\n" );
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document.write( "**f.) Probability that the sample mean is more than 241 (n=35):**\r<BR>\n" );
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document.write( "1.  **Calculate the z-score:**<BR>\n" );
document.write( "    z = (x̄ - μ) / σₓ̄<BR>\n" );
document.write( "    z = (241 - 245) / 3.56<BR>\n" );
document.write( "    z ≈ -1.12\r<BR>\n" );
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document.write( "2.  **Find the probability:**<BR>\n" );
document.write( "    Using a z-table or calculator, find the probability of z being *greater* than -1.12.<BR>\n" );
document.write( "    P(z > -1.12) ≈ 0.8686\r<BR>\n" );
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document.write( "**g.) Comparison of probabilities and explanation:**\r<BR>\n" );
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document.write( "The probability in part f (n=35) is larger than the probability in part c (n=10). This is because the standard deviation of the sample mean is *smaller* for the larger sample size.  A smaller standard deviation means the sample means are more tightly clustered around the population mean.  Therefore, it's more likely that a sample mean from a larger sample will be closer to the population mean (and thus more likely to be above 241 in this case).  In simpler terms, larger samples provide more precise estimates.<BR>\n" );
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