document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201612>Question 1201612</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You are given the sample mean and the population standard deviation. Use this information to construct the​ 90% and​ 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. If​ convenient, use technology to construct the confidence intervals.\r<BR>\n" );
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document.write( "A random sample of 55 home theater systems has a mean price of ​$117.00. Assume the population standard deviation is ​$18.30. </SPAN>\n" );
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document.write( "<font color=red>Answers:</font><BR>\n" );
document.write( "90% confidence interval is (112.94, 121.06)<BR>\n" );
document.write( "95% confidence interval is (112.16, 121.84)<BR>\n" );
document.write( "The 95% confidence interval is wider.\r<BR>\n" );
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document.write( "Further explanation is down below.\r<BR>\n" );
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document.write( "At 90% confidence, the z critical value is roughly z = 1.645<BR>\n" );
document.write( "Use a table like this<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf\">https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf</a><BR>\n" );
document.write( "to get that value. Look at the bottom row labeled \"Z\" and above the 90% confidence level.<BR>\n" );
document.write( "A stats calculator can also compute this value.\r<BR>\n" );
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document.write( "xbar = 117.00 = sample mean<BR>\n" );
document.write( "sigma = 18.30 = population standard deviation<BR>\n" );
document.write( "n = 55 = sample size\r<BR>\n" );
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document.write( "Let's compute the margin of error<BR>\n" );
document.write( "E = margin of error<BR>\n" );
document.write( "E = z*sigma/sqrt(n)<BR>\n" );
document.write( "E = 1.645*18.30/sqrt(55)<BR>\n" );
document.write( "E = 4.05915511193244<BR>\n" );
document.write( "E = 4.059155\r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "L = lower boundary<BR>\n" );
document.write( "L = xbar - E<BR>\n" );
document.write( "L = 117 - 4.059155<BR>\n" );
document.write( "L = 112.940845<BR>\n" );
document.write( "L = 112.94<BR>\n" );
document.write( "and<BR>\n" );
document.write( "U = upper boundary<BR>\n" );
document.write( "U = xbar + E<BR>\n" );
document.write( "U = 117 + 4.059155<BR>\n" );
document.write( "U = 121.059155<BR>\n" );
document.write( "U = 121.06\r<BR>\n" );
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document.write( "The 90% confidence interval for the population mean is 112.94 < mu < 121.06<BR>\n" );
document.write( "That is in the format L < mu < U\r<BR>\n" );
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document.write( "It can be condensed to the format (L, U) and we get (112.94, 121.06)<BR>\n" );
document.write( "This second format is more common, but the drawback is that we don't know what population parameter we're estimating (unless further context is given).\r<BR>\n" );
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document.write( "Interpretation: We are 90% confident the population mean price of a home theater system is somewhere between $112.94 and $121.06\r<BR>\n" );
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document.write( "We'll repeat the process but this time for a 95% confidence interval.\r<BR>\n" );
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document.write( "At 95% confidence, the z critical value is roughly z = 1.96<BR>\n" );
document.write( "Refer to that table I mentioned earlier.\r<BR>\n" );
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document.write( "The other input values are the same as before.<BR>\n" );
document.write( "E = margin of error<BR>\n" );
document.write( "E = z*sigma/sqrt(n)<BR>\n" );
document.write( "E = 1.96*18.30/sqrt(55)<BR>\n" );
document.write( "E = 4.83644013336631<BR>\n" );
document.write( "E = 4.836440<BR>\n" );
document.write( "The margin of error is larger than previously calculated.<BR>\n" );
document.write( "This will widen the confidence interval.\r<BR>\n" );
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document.write( "L = lower boundary<BR>\n" );
document.write( "L = xbar - E<BR>\n" );
document.write( "L = 117 - 4.836440<BR>\n" );
document.write( "L = 112.16356<BR>\n" );
document.write( "L = 112.16<BR>\n" );
document.write( "and<BR>\n" );
document.write( "U = upper boundary<BR>\n" );
document.write( "U = xbar + E<BR>\n" );
document.write( "U = 117 + 4.836440<BR>\n" );
document.write( "U = 121.83644<BR>\n" );
document.write( "U = 121.84\r<BR>\n" );
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document.write( "The 95% confidence interval can be expressed as 112.16 < mu < 121.84<BR>\n" );
document.write( "Or can be expressed as (112.16, 121.84)\r<BR>\n" );
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document.write( "Interpretation: We are 95% confident the population mean price of a home theater system is somewhere between $112.16 and $121.84\r<BR>\n" );
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document.write( "Let's compare the confidence intervals.<BR>\n" );
document.write( "90% confidence interval: (112.94, 121.06)<BR>\n" );
document.write( "95% confidence interval: (112.16, 121.84)<BR>\n" );
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document.write( "The 95% confidence interval is wider because the margin of error is larger.<BR>\n" );
document.write( "The more confident we get, the wider the interval will become.\r<BR>\n" );
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document.write( "Imagine you are looking for a certain rare species of fish in the ocean.<BR>\n" );
document.write( "To be more confident you caught the fish you want, you could increase the size of the net. <BR>\n" );
document.write( "The size of the net is analogous to the width of the confidence interval.<BR>\n" );
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