document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1180440>Question 1180440</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.Suppose a population hold the ages of five family members,as follows: 1,13,32,34, and 59.Calculate the population mean and the sample means of the random variable of size n=2. Illustrate the probability distributions of the sample means of the ages of the five family members.\r<BR>\n" );
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document.write( "2.Referring to the situation above, construct a sampling distribution of the sample variances and sample standard deviations. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850110 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=850110\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Population Mean and Sample Means:**\r<BR>\n" );
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document.write( "*   **Population Mean:**<BR>\n" );
document.write( "    The population mean (μ) is the average of all the ages:<BR>\n" );
document.write( "    μ = (1 + 13 + 32 + 34 + 59) / 5 = 139 / 5 = 27.8\r<BR>\n" );
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document.write( "*   **Sample Means:**<BR>\n" );
document.write( "    Since the sample size is 2, we need to find all possible combinations of 2 ages from the population. There are 10 such combinations:<BR>\n" );
document.write( "    (1, 13), (1, 32), (1, 34), (1, 59), (13, 32), (13, 34), (13, 59), (32, 34), (32, 59), (34, 59)<BR>\n" );
document.write( "    Now, calculate the mean of each sample:<BR>\n" );
document.write( "    7, 16.5, 17.5, 30, 22.5, 23.5, 36, 33, 45.5, 46.5\r<BR>\n" );
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document.write( "*   **Probability Distribution of Sample Means:**<BR>\n" );
document.write( "    Each of the 10 sample means has an equal probability of occurring (1/10). The probability distribution is:\r<BR>\n" );
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document.write( "    | Sample Mean | Probability |<BR>\n" );
document.write( "    |---|---|<BR>\n" );
document.write( "    | 7 | 1/10 |<BR>\n" );
document.write( "    | 16.5 | 1/10 |<BR>\n" );
document.write( "    | 17.5 | 1/10 |<BR>\n" );
document.write( "    | 30 | 1/10 |<BR>\n" );
document.write( "    | 22.5 | 1/10 |<BR>\n" );
document.write( "    | 23.5 | 1/10 |<BR>\n" );
document.write( "    | 36 | 1/10 |<BR>\n" );
document.write( "    | 33 | 1/10 |<BR>\n" );
document.write( "    | 45.5 | 1/10 |<BR>\n" );
document.write( "    | 46.5 | 1/10 |\r<BR>\n" );
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document.write( "**2. Sampling Distribution of Sample Variances and Standard Deviations:**\r<BR>\n" );
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document.write( "*   **Sample Variances:**<BR>\n" );
document.write( "    For each sample, calculate the variance using the formula:<BR>\n" );
document.write( "    s² = Σ(xᵢ - x̄)² / (n - 1)<BR>\n" );
document.write( "    where xᵢ are the ages in the sample, x̄ is the sample mean, and n is the sample size (2).<BR>\n" );
document.write( "    The sample variances are:<BR>\n" );
document.write( "    72, 480.5, 544.5, 1682, 180.5, 220.5, 1058, 2, 364.5, 312.5\r<BR>\n" );
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document.write( "*   **Sample Standard Deviations:**<BR>\n" );
document.write( "    For each sample, take the square root of the variance:<BR>\n" );
document.write( "    s = √s²<BR>\n" );
document.write( "    The sample standard deviations are:<BR>\n" );
document.write( "    8.49, 21.92, 23.33, 41.01, 13.44, 14.85, 32.53, 1.41, 19.09, 17.68\r<BR>\n" );
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document.write( "*   **Sampling Distribution:**<BR>\n" );
document.write( "    Each sample variance and standard deviation has a probability of 1/10 of occurring. The sampling distributions are:\r<BR>\n" );
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document.write( "    **Sample Variances:**\r<BR>\n" );
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document.write( "    | Sample Variance | Probability |<BR>\n" );
document.write( "    |---|---|<BR>\n" );
document.write( "    | 72 | 1/10 |<BR>\n" );
document.write( "    | 480.5 | 1/10 |<BR>\n" );
document.write( "    | 544.5 | 1/10 |<BR>\n" );
document.write( "    | 1682 | 1/10 |<BR>\n" );
document.write( "    | 180.5 | 1/10 |<BR>\n" );
document.write( "    | 220.5 | 1/10 |<BR>\n" );
document.write( "    | 1058 | 1/10 |<BR>\n" );
document.write( "    | 2 | 1/10 |<BR>\n" );
document.write( "    | 364.5 | 1/10 |<BR>\n" );
document.write( "    | 312.5 | 1/10 |\r<BR>\n" );
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document.write( "    **Sample Standard Deviations:**\r<BR>\n" );
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document.write( "    | Sample Standard Deviation | Probability |<BR>\n" );
document.write( "    |---|---|<BR>\n" );
document.write( "    | 8.49 | 1/10 |<BR>\n" );
document.write( "    | 21.92 | 1/10 |<BR>\n" );
document.write( "    | 23.33 | 1/10 |<BR>\n" );
document.write( "    | 41.01 | 1/10 |<BR>\n" );
document.write( "    | 13.44 | 1/10 |<BR>\n" );
document.write( "    | 14.85 | 1/10 |<BR>\n" );
document.write( "    | 32.53 | 1/10 |<BR>\n" );
document.write( "    | 1.41 | 1/10 |<BR>\n" );
document.write( "    | 19.09 | 1/10 |<BR>\n" );
document.write( "    | 17.68 | 1/10 | </SPAN>\n" );
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