document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171027>Question 1171027</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The local bakery bakes more than a thousand 1-pound loaves of bread daily, and the weights of these loaves varies. The mean weight is 1.9 lb. and 4 oz., or 975 grams. Assume the standard deviation of the weights is 26 grams and a sample of 35 loaves is to be randomly selected.\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 965 and 985? (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 less than 967? (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 grams of the mean? (Give your answer correct to four decimal places.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850999 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=850999\">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) = 975 grams<BR>\n" );
document.write( "* Population standard deviation (σ) = 26 grams<BR>\n" );
document.write( "* Sample size (n) = 35\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 = 975\r<BR>\n" );
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document.write( "Therefore, the mean of the sampling distribution is 975.\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> = 26 / √35<BR>\n" );
document.write( "* σ<sub>x̄</sub> ≈ 26 / 5.9161<BR>\n" );
document.write( "* σ<sub>x̄</sub> ≈ 4.3946\r<BR>\n" );
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document.write( "Rounded to two decimal places, the standard error is 4.39.\r<BR>\n" );
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document.write( "**(d) Probability that the Sample Mean is Between 965 and 985**\r<BR>\n" );
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document.write( "We need to find P(965 < x̄ < 985). First, we need to convert the sample means to z-scores:\r<BR>\n" );
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document.write( "* z₁ = (965 - 975) / 4.39 = -10 / 4.39 ≈ -2.28<BR>\n" );
document.write( "* z₂ = (985 - 975) / 4.39 = 10 / 4.39 ≈ 2.28\r<BR>\n" );
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document.write( "Now, we need to find P(-2.28 < Z < 2.28).\r<BR>\n" );
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document.write( "* P(Z < 2.28) ≈ 0.9887<BR>\n" );
document.write( "* P(Z < -2.28) ≈ 0.0113\r<BR>\n" );
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document.write( "Therefore, P(-2.28 < Z < 2.28) = P(Z < 2.28) - P(Z < -2.28) ≈ 0.9887 - 0.0113 = 0.9774\r<BR>\n" );
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document.write( "**(e) Probability that the Sample Mean is Less Than 967**\r<BR>\n" );
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document.write( "We need to find P(x̄ < 967). First, we need to convert 967 to a z-score:\r<BR>\n" );
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document.write( "* z = (967 - 975) / 4.39 = -8 / 4.39 ≈ -1.82\r<BR>\n" );
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document.write( "Now, we need to find P(Z < -1.82).\r<BR>\n" );
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document.write( "* P(Z < -1.82) ≈ 0.0344\r<BR>\n" );
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document.write( "**(f) Probability that the Sample Mean is Within 3 Grams of the Mean**\r<BR>\n" );
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document.write( "We need to find P(975 - 3 < x̄ < 975 + 3), which is P(972 < x̄ < 978).\r<BR>\n" );
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document.write( "* z₁ = (972 - 975) / 4.39 = -3 / 4.39 ≈ -0.68<BR>\n" );
document.write( "* z₂ = (978 - 975) / 4.39 = 3 / 4.39 ≈ 0.68\r<BR>\n" );
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document.write( "We need to find P(-0.68 < Z < 0.68).\r<BR>\n" );
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document.write( "* P(Z < 0.68) ≈ 0.7517<BR>\n" );
document.write( "* P(Z < -0.68) ≈ 0.2483\r<BR>\n" );
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document.write( "Therefore, P(-0.68 < Z < 0.68) = P(Z < 0.68) - P(Z < -0.68) ≈ 0.7517 - 0.2483 = 0.5034\r<BR>\n" );
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document.write( "**Answers:**\r<BR>\n" );
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document.write( "(b) 975<BR>\n" );
document.write( "(c) 4.39<BR>\n" );
document.write( "(d) 0.9774<BR>\n" );
document.write( "(e) 0.0344<BR>\n" );
document.write( "(f) 0.5034<BR>\n" );
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