document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171028>Question 1171028</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A random sample of size 43 is to be selected from a population that has a mean 𝜇 = 55 and a standard deviation 𝜎 of 15.\r<BR>\n" );
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document.write( "(b) Find the mean of this sampling distribution. (Give your answer correct to nearest whole number.)\r<BR>\n" );
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document.write( "(c) Find the standard error of this sampling distribution. (Give your answer correct to two decimal places.)\r<BR>\n" );
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document.write( "(d) What is the probability that this sample mean will be between 43 and 51? (Give your answer correct to four decimal places.)\r<BR>\n" );
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document.write( "(e) What is the probability that the sample mean will have a value greater than 51? (Give your answer correct to four decimal places.)\r<BR>\n" );
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document.write( "(f) What is the probability that the sample mean will be within 3 units of the mean? (Give your answer correct to four decimal places.)<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 #850998 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=850998\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**Given Information:**\r<BR>\n" );
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document.write( "* Population mean (μ) = 55<BR>\n" );
document.write( "* Population standard deviation (σ) = 15<BR>\n" );
document.write( "* Sample size (n) = 43\r<BR>\n" );
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document.write( "**(b) Find the Mean of the Sampling Distribution**\r<BR>\n" );
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document.write( "The mean of the sampling distribution of the sample mean (μ<sub>x̄</sub>) is equal to the population mean (μ).\r<BR>\n" );
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document.write( "* μ<sub>x̄</sub> = μ = 55\r<BR>\n" );
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document.write( "Therefore, the mean of the sampling distribution is 55.\r<BR>\n" );
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document.write( "**(c) Find the Standard Error of the Sampling Distribution**\r<BR>\n" );
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document.write( "The standard error (σ<sub>x̄</sub>) is the standard deviation of the sampling distribution of the sample mean. It's calculated as:\r<BR>\n" );
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document.write( "* σ<sub>x̄</sub> = σ / √n<BR>\n" );
document.write( "* σ<sub>x̄</sub> = 15 / √43<BR>\n" );
document.write( "* σ<sub>x̄</sub> ≈ 15 / 6.5574<BR>\n" );
document.write( "* σ<sub>x̄</sub> ≈ 2.2875\r<BR>\n" );
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document.write( "Rounded to two decimal places, the standard error is 2.29.\r<BR>\n" );
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document.write( "**(d) Probability that the Sample Mean is Between 43 and 51**\r<BR>\n" );
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document.write( "We need to find P(43 < x̄ < 51). First, we need to convert the sample means to z-scores:\r<BR>\n" );
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document.write( "* z₁ = (43 - 55) / 2.29 = -12 / 2.29 ≈ -5.24<BR>\n" );
document.write( "* z₂ = (51 - 55) / 2.29 = -4 / 2.29 ≈ -1.75\r<BR>\n" );
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document.write( "Now, we need to find P(-5.24 < Z < -1.75).\r<BR>\n" );
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document.write( "* P(Z < -1.75) ≈ 0.0401<BR>\n" );
document.write( "* P(Z < -5.24) ≈ 0 (very close to 0)\r<BR>\n" );
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document.write( "Therefore, P(-5.24 < Z < -1.75) = P(Z < -1.75) - P(Z < -5.24) ≈ 0.0401 - 0 ≈ 0.0401\r<BR>\n" );
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document.write( "**(e) Probability that the Sample Mean is Greater Than 51**\r<BR>\n" );
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document.write( "We need to find P(x̄ > 51). We already calculated the z-score for 51: z = -1.75.\r<BR>\n" );
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document.write( "* P(Z > -1.75) = 1 - P(Z < -1.75)<BR>\n" );
document.write( "* P(Z > -1.75) ≈ 1 - 0.0401 = 0.9599\r<BR>\n" );
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document.write( "**(f) Probability that the Sample Mean is Within 3 Units of the Mean**\r<BR>\n" );
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document.write( "We need to find P(55 - 3 < x̄ < 55 + 3), which is P(52 < x̄ < 58).\r<BR>\n" );
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document.write( "* z₁ = (52 - 55) / 2.29 = -3 / 2.29 ≈ -1.31<BR>\n" );
document.write( "* z₂ = (58 - 55) / 2.29 = 3 / 2.29 ≈ 1.31\r<BR>\n" );
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document.write( "We need to find P(-1.31 < Z < 1.31).\r<BR>\n" );
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document.write( "* P(Z < 1.31) ≈ 0.9049<BR>\n" );
document.write( "* P(Z < -1.31) ≈ 0.0951\r<BR>\n" );
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document.write( "Therefore, P(-1.31 < Z < 1.31) = P(Z < 1.31) - P(Z < -1.31) ≈ 0.9049 - 0.0951 = 0.8098\r<BR>\n" );
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document.write( "**Answers:**\r<BR>\n" );
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document.write( "(b) 55<BR>\n" );
document.write( "(c) 2.29<BR>\n" );
document.write( "(d) 0.0401<BR>\n" );
document.write( "(e) 0.9599<BR>\n" );
document.write( "(f) 0.8098<BR>\n" );
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