document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193182>Question 1193182</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A sociologist found that in a sample of 50 retired men, the average number of jobs they had during their lifetimes was 7.2. The population standard deviation is 2.1.<BR>\n" );
document.write( "a) Find the best point estimate of the mean.<BR>\n" );
document.write( "b) Find the 95% confidence interval of the mean number of jobs. <BR>\n" );
document.write( "c) Find the 99% confidence interval of the mean number of jobs. </SPAN>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "The best point estimate of the population mean is the sample mean. It will be handy later to form the confidence intervals.<BR>\n" );
document.write( "We can use the notation xbar to indicate the sample mean.\r<BR>\n" );
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document.write( "Answer: 7.2\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "n = 50 = sample size<BR>\n" );
document.write( "sigma = 2.1 = population standard deviation\r<BR>\n" );
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document.write( "At 95% confidence, the z critical value is roughly z = 1.960<BR>\n" );
document.write( "You'll need to either memorize this value or have it on a reference table. A specialized calculator is also another option.\r<BR>\n" );
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document.write( "E = margin of error<BR>\n" );
document.write( "E = z*sigma/sqrt(n)<BR>\n" );
document.write( "E = 1.960*2.1/sqrt(50)<BR>\n" );
document.write( "E = 0.58209030227277<BR>\n" );
document.write( "E = 0.582090\r<BR>\n" );
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document.write( "L = lower bound of confidence interval<BR>\n" );
document.write( "L = xbar - E<BR>\n" );
document.write( "L = 7.2 - 0.582090<BR>\n" );
document.write( "L = 6.61791<BR>\n" );
document.write( "L = 6.62\r<BR>\n" );
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document.write( "U = upper bound of confidence interval<BR>\n" );
document.write( "U = xbar + E<BR>\n" );
document.write( "U = 7.2 + 0.582090<BR>\n" );
document.write( "U = 7.78209<BR>\n" );
document.write( "U = 7.78\r<BR>\n" );
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document.write( "Notice how the sample mean xbar = 7.2 is the center of the confidence interval.\r<BR>\n" );
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document.write( "Answer: (6.62, 7.78)\r<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "We use the same idea as part (b)<BR>\n" );
document.write( "This time we have z = 2.576 as the critical value for a 99% confidence interval.<BR>\n" );
document.write( "Every other variable stays the same.\r<BR>\n" );
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document.write( "E = margin of error<BR>\n" );
document.write( "E = z*sigma/sqrt(n)<BR>\n" );
document.write( "E = 2.576*2.1/sqrt(50)<BR>\n" );
document.write( "E = 0.76503296870134<BR>\n" );
document.write( "E = 0.765033\r<BR>\n" );
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document.write( "L = lower bound of confidence interval<BR>\n" );
document.write( "L = xbar - E<BR>\n" );
document.write( "L = 7.2 - 0.765033<BR>\n" );
document.write( "L = 6.434967<BR>\n" );
document.write( "L = 6.43\r<BR>\n" );
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document.write( "U = upper bound of confidence interval<BR>\n" );
document.write( "U = xbar + E<BR>\n" );
document.write( "U = 7.2 + 0.765033<BR>\n" );
document.write( "U = 7.965033<BR>\n" );
document.write( "U = 7.97\r<BR>\n" );
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document.write( "The point estimate is the same as last time. The only thing that changed is the margin of error. <BR>\n" );
document.write( "The higher the confidence level, the larger the margin of error will be. This in turn creates a wider confidence interval.\r<BR>\n" );
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document.write( "It's like trying to catch an elusive fish. The wider the net (i.e. the wider the interval), the more confident we are in catching the fish. \r<BR>\n" );
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document.write( "Answer: (6.43, 7.97)<BR>\n" );
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