document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170718>Question 1170718</A>: <I><SPAN STYLE=\"word-break: break-all;\"> It is known that the weights of apples from a farm are normally distributed. In order to estimate the mean weight, a random sample of 150 apples is considered and the sample mean and population standard deviation are 6 kg and 0.8 kg respectively.\r<BR>\n" );
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document.write( "(a)	Construct a 95% confidence interval estimate for the population mean weight of apples.<BR>\n" );
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document.write( "(b)	The researcher suggests doing the study again so that 98% confidence interval estimate for the population mean weight of apples is (5.8835,6.1165) kg. How large should the sample size be?<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 #851105 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=851105\">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:**\r<BR>\n" );
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document.write( "* Sample size (n) = 150<BR>\n" );
document.write( "* Sample mean (x̄) = 6 kg<BR>\n" );
document.write( "* Population standard deviation (σ) = 0.8 kg\r<BR>\n" );
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document.write( "**(a) Construct a 95% Confidence Interval**\r<BR>\n" );
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document.write( "1.  **Find the critical z-value (z_c):**<BR>\n" );
document.write( "    * For a 95% confidence interval, z_c = 1.96 (from a z-table or calculator).\r<BR>\n" );
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document.write( "2.  **Calculate the margin of error (E):**<BR>\n" );
document.write( "    * E = z_c * (σ / √n)<BR>\n" );
document.write( "    * E = 1.96 * (0.8 / √150)<BR>\n" );
document.write( "    * E ≈ 1.96 * (0.8 / 12.247)<BR>\n" );
document.write( "    * E ≈ 1.96 * 0.0653<BR>\n" );
document.write( "    * E ≈ 0.128 kg\r<BR>\n" );
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document.write( "3.  **Construct the confidence interval:**<BR>\n" );
document.write( "    * Confidence Interval = x̄ ± E<BR>\n" );
document.write( "    * Confidence Interval = 6 ± 0.128<BR>\n" );
document.write( "    * Confidence Interval = (6 - 0.128, 6 + 0.128)<BR>\n" );
document.write( "    * Confidence Interval = (5.872, 6.128) kg\r<BR>\n" );
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document.write( "4.  **Answer:**<BR>\n" );
document.write( "    * The 95% confidence interval for the population mean weight of apples is (5.872, 6.128) kg.\r<BR>\n" );
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document.write( "**(b) Find the sample size for a 98% Confidence Interval**\r<BR>\n" );
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document.write( "1.  **Given Confidence Interval:**<BR>\n" );
document.write( "    * (5.8835, 6.1165) kg\r<BR>\n" );
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document.write( "2.  **Calculate the margin of error (E):**<BR>\n" );
document.write( "    * E = (Upper Bound - Lower Bound) / 2<BR>\n" );
document.write( "    * E = (6.1165 - 5.8835) / 2<BR>\n" );
document.write( "    * E = 0.233 / 2<BR>\n" );
document.write( "    * E = 0.1165 kg\r<BR>\n" );
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document.write( "3.  **Find the critical z-value (z_c):**<BR>\n" );
document.write( "    * For a 98% confidence interval, z_c ≈ 2.33 (from a z-table or calculator).\r<BR>\n" );
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document.write( "4.  **Use the margin of error formula to solve for n:**<BR>\n" );
document.write( "    * E = z_c * (σ / √n)<BR>\n" );
document.write( "    * 0.1165 = 2.33 * (0.8 / √n)<BR>\n" );
document.write( "    * √n = (2.33 * 0.8) / 0.1165<BR>\n" );
document.write( "    * √n = 1.864 / 0.1165<BR>\n" );
document.write( "    * √n ≈ 16.00<BR>\n" );
document.write( "    * n = (16.00)²<BR>\n" );
document.write( "    * n = 256\r<BR>\n" );
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document.write( "5.  **Answer:**<BR>\n" );
document.write( "    * The sample size should be 256 apples.<BR>\n" );
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