document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190594>Question 1190594</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A population consists of (3,8,10,15). Consider all possible sample size 2 that can be drawn with replacement.\r<BR>\n" );
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document.write( "-what is the standard deviation of the sampling distribution of means? =\r<BR>\n" );
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document.write( "-what is the population mean? =\r<BR>\n" );
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document.write( "-what is the population standard deviation? =\r<BR>\n" );
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document.write( "-what is the population variance? =\r<BR>\n" );
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document.write( "-what is the variance of the sampling distribution of means? =\r<BR>\n" );
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document.write( "-what is the mean of the sampling distribution of means? =\r<BR>\n" );
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document.write( "A population consists of (3,8,10,15). Consider all possible sample size<BR>\n" );
document.write( " 2 that can be drawn without replacement.\r<BR>\n" );
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document.write( "-what is the variance of the sampling distribution of means? =\r<BR>\n" );
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document.write( "-what is the standard deviation of the sampling distribution of means? =\r<BR>\n" );
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document.write( "-what is the mean of the sampling distribution of means? =<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 #849246 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=849246\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to calculate the requested values for both scenarios (with and without replacement):\r<BR>\n" );
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document.write( "**Scenario 1: With Replacement**\r<BR>\n" );
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document.write( "1.  **Population Mean (μ):**<BR>\n" );
document.write( "    μ = (3 + 8 + 10 + 15) / 4 = 36 / 4 = 9\r<BR>\n" );
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document.write( "2.  **Population Variance (σ²):**<BR>\n" );
document.write( "    σ² = [(3-9)² + (8-9)² + (10-9)² + (15-9)²] / 4<BR>\n" );
document.write( "    σ² = [36 + 1 + 1 + 36] / 4 = 74 / 4 = 18.5\r<BR>\n" );
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document.write( "3.  **Population Standard Deviation (σ):**<BR>\n" );
document.write( "    σ = √σ² = √18.5 ≈ 4.3\r<BR>\n" );
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document.write( "4.  **Variance of the Sampling Distribution of Means:**<BR>\n" );
document.write( "    When sampling with replacement, the variance of the sampling distribution of means (σₓ̄²) is given by:<BR>\n" );
document.write( "    σₓ̄² = σ² / n  where n is the sample size.<BR>\n" );
document.write( "    σₓ̄² = 18.5 / 2 = 9.25\r<BR>\n" );
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document.write( "5.  **Standard Deviation of the Sampling Distribution of Means (Standard Error):**<BR>\n" );
document.write( "    This is the square root of the variance of the sampling distribution:<BR>\n" );
document.write( "    σₓ̄ = √σₓ̄² = √9.25 ≈ 3.04\r<BR>\n" );
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document.write( "6.  **Mean of the Sampling Distribution of Means (μₓ̄):**<BR>\n" );
document.write( "    When sampling with replacement, the mean of the sampling distribution of means is equal to the population mean:<BR>\n" );
document.write( "    μₓ̄ = μ = 9\r<BR>\n" );
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document.write( "**Scenario 2: Without Replacement**\r<BR>\n" );
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document.write( "1.  **Population Mean (μ):**  (Same as before)<BR>\n" );
document.write( "    μ = 9\r<BR>\n" );
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document.write( "2.  **Population Variance (σ²):** (Same as before)<BR>\n" );
document.write( "    σ² = 18.5\r<BR>\n" );
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document.write( "3.  **Population Standard Deviation (σ):** (Same as before)<BR>\n" );
document.write( "    σ ≈ 4.3\r<BR>\n" );
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document.write( "4.  **Variance of the Sampling Distribution of Means:**<BR>\n" );
document.write( "    When sampling *without* replacement, the variance of the sampling distribution is adjusted by a finite population correction factor:<BR>\n" );
document.write( "    σₓ̄² = (σ² / n) * [(N - n) / (N - 1)]<BR>\n" );
document.write( "    Where N is the population size.<BR>\n" );
document.write( "    σₓ̄² = (18.5 / 2) * [(4 - 2) / (4 - 1)]<BR>\n" );
document.write( "    σₓ̄² = 9.25 * (2/3) = 6.17 (approximately)\r<BR>\n" );
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document.write( "5.  **Standard Deviation of the Sampling Distribution of Means (Standard Error):**<BR>\n" );
document.write( "    σₓ̄ = √σₓ̄² = √6.17 ≈ 2.48\r<BR>\n" );
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document.write( "6.  **Mean of the Sampling Distribution of Means (μₓ̄):**<BR>\n" );
document.write( "    Even when sampling without replacement, the mean of the sampling distribution of means is still equal to the population mean:<BR>\n" );
document.write( "    μₓ̄ = μ = 9\r<BR>\n" );
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document.write( "**Summary Table:**\r<BR>\n" );
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document.write( "| Statistic | With Replacement | Without Replacement |<BR>\n" );
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document.write( "| Population Mean (μ) | 9 | 9 |<BR>\n" );
document.write( "| Population Variance (σ²) | 18.5 | 18.5 |<BR>\n" );
document.write( "| Population Standard Deviation (σ) | 4.3 | 4.3 |<BR>\n" );
document.write( "| Variance of Sampling Distribution (σₓ̄²) | 9.25 | 6.17 |<BR>\n" );
document.write( "| Standard Error of the Mean (σₓ̄) | 3.04 | 2.48 |<BR>\n" );
document.write( "| Mean of Sampling Distribution (μₓ̄) | 9 | 9 |<BR>\n" );
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