document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206695>Question 1206695</A>: <I><SPAN STYLE=\"word-break: break-all;\"> HighTech Inc. randomly tests its employees about company policies. Last year in the 390 random tests conducted, 12 employees failed the test.\r<BR>\n" );
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document.write( "Develop a 95% confidence interval for the proportion of applicants that fail the test. (Round to three decimal places.)<BR>\n" );
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document.write( "p = population proportion<BR>\n" );
document.write( "phat = sample proportion<BR>\n" );
document.write( "The job of phat is to estimate p.\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( "This is something to memorize or have on a reference sheet.\r<BR>\n" );
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document.write( "You can use a table such as 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 determine the z critical values. Look at the bottom row and at the value just above the 95% confidence level.\r<BR>\n" );
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document.write( "What this means is that P(-1.96 < z < 1.96) = 0.95 approximately.\r<BR>\n" );
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document.write( "Another way to determine this z critical value is to use a stats calculator such as a TI84. \r<BR>\n" );
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document.write( "n = 390 = sample size<BR>\n" );
document.write( "x = 12 employees failed<BR>\n" );
document.write( "phat = sample proportion of those who failed<BR>\n" );
document.write( "phat = x/n<BR>\n" );
document.write( "phat = 12/390<BR>\n" );
document.write( "phat = 0.03076923 approximately<BR>\n" );
document.write( "Around 3.077% of the sample of employees failed the test.<BR>\n" );
document.write( "This is the center of the confidence interval.\r<BR>\n" );
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document.write( "E = margin of error<BR>\n" );
document.write( "E = z*sqrt(phat*(1-phat)/n)<BR>\n" );
document.write( "E = 1.96*sqrt(0.03076923*(1-0.03076923)/390)<BR>\n" );
document.write( "E = 0.01713940 approximately<BR>\n" );
document.write( "This helps determine how wide or spread out the confidence interval is.\r<BR>\n" );
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document.write( "L = lower bound of confidence interval<BR>\n" );
document.write( "L = phat - E<BR>\n" );
document.write( "L = 0.03076923 - 0.01713940<BR>\n" );
document.write( "L = 0.01362983<BR>\n" );
document.write( "L = 0.014\r<BR>\n" );
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document.write( "U = upper bound of confidence interval<BR>\n" );
document.write( "U = phat + E<BR>\n" );
document.write( "U = 0.03076923 + 0.01713940<BR>\n" );
document.write( "U = 0.04790863<BR>\n" );
document.write( "U = 0.048\r<BR>\n" );
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document.write( "Answer:<BR>\n" );
document.write( "The confidence interval in the format (L, U) would be approximately <font color=red>(0.014, 0.048)</font>\r<BR>\n" );
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document.write( "This is equivalent to writing <font color=red>0.014 < p < 0.048</font> which provides more context of which parameter we're trying to estimate. <BR>\n" );
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