document.write( "Question 588065: How do you get 1.645 when \r
\n" ); document.write( "\n" ); document.write( "If you use a 0.10 level of significance in a (two-tail) hypothesis test, what is your decision rule for rejecting a null hypothesis that the population mean is 500 if you use the Z test? \r
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Algebra.Com's Answer #374381 by Edwin McCravy(20059) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "The left-most green line is erected where z=-1.645. \r\n" );
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document.write( "The right-most green line is erected where z=+1.645. \r\n" );
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document.write( "The area bounded by the curve, the z-axis, and those two green lines is 90% of\r\n" );
document.write( "the total area between the curve and the z-axis.  The area to the left of the\r\n" );
document.write( "left green line is 5% of the total area, and the area to the right of the right\r\n" );
document.write( "green line is the other 5% of the total area between the curve and the z-axis.\r\n" );
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document.write( "Your are testing the hypothesis \r\n" );
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document.write( "H₀:  m = 500\r\n" );
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document.write( "against the alternative hypothess\r\n" );
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document.write( "H_a:  m ≠ 500\r\n" );
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document.write( "Suppose you took a sample of size n=100 and \r\n" );
document.write( "found that the sample mean was ⴳ = 493 and\r\n" );
document.write( "the standard deviation were s = 50\r\n" );
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document.write( "Then you would calculate this test statistic:\r\n" );
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document.write( "     ⴳ - m \r\n" );
document.write( "z = 覧覧覧覧\r\n" );
document.write( "       \r\n" );
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document.write( "      493 - 500 \r\n" );
document.write( "z = 覧覧覧覧覧覧 = -1.4\r\n" );
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document.write( "This would correcpond to the red line drawn\r\n" );
document.write( "at -1.4\r\n" );
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document.write( "and since the 1.4 value of z is between the\r\n" );
document.write( "two green bars, the mean of 493 is close\r\n" );
document.write( "enough to 500 with a sample that size with that\r\n" );
document.write( "standard deviation, and so we would fail to\r\n" );
document.write( "reject the hypothesis.\r\n" );
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document.write( "However, suppose you took a sample of size n=64 and \r\n" );
document.write( "found that the sample mean was ⴳ = 495 and\r\n" );
document.write( "the standard deviation were s = 20\r\n" );
document.write( "Then you would calculate this test statistic:\r\n" );
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document.write( "     ⴳ - m \r\n" );
document.write( "z = 覧覧覧覧\r\n" );
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document.write( "      495 - 500 \r\n" );
document.write( "z = 覧覧覧覧覧覧 = -2.0\r\n" );
document.write( "         \r\n" );
document.write( "This would correcpond to the red line drawn\r\n" );
document.write( "at -2.0\r\n" );
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document.write( "and since the 2.0 value of z is NOT between the two green bars, the mean of 495\r\n" );
document.write( "with a sample that size with that standard deviation, is not close enough to\r\n" );
document.write( "500, and so we would reject the null hypothesis.\r\n" );
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document.write( "Notice that a lot depends on the size and the standard deviation of the sample we take.\r\n" );
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document.write( "Now since this is a 2-tail test, the sample mean could be larger than 500\r\n" );
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document.write( "Suppose you took a sample of size n=81 and \r\n" );
document.write( "found that the sample mean was ⴳ = 510 and\r\n" );
document.write( "the standard deviation were s = 70\r\n" );
document.write( "Then you would calculate this test statistic:\r\n" );
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document.write( "     ⴳ - m \r\n" );
document.write( "z = 覧覧覧覧\r\n" );
document.write( "       \r\n" );
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document.write( "      510 - 500 \r\n" );
document.write( "z = 覧覧覧覧覧覧 = -1.29\r\n" );
document.write( "         \r\n" );
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document.write( "This would correcpond to the red line drawn\r\n" );
document.write( "at -1.29\r\n" );
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document.write( "and since the 1.29 value of z is between the\r\n" );
document.write( "two green bars, the mean of 510 is close\r\n" );
document.write( "enough to 500 with a sample that size with that\r\n" );
document.write( "standard deviation, and so we would fail to\r\n" );
document.write( "reject the hypothesis.\r\n" );
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document.write( "However, suppose you took a sample of size n=150 and \r\n" );
document.write( "found that the sample mean was ⴳ = 506 and\r\n" );
document.write( "the standard deviation were s = 40\r\n" );
document.write( "Then you would calculate this test statistic:\r\n" );
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document.write( "     ⴳ - m \r\n" );
document.write( "z = 覧覧覧覧\r\n" );
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document.write( "      506 - 500 \r\n" );
document.write( "z = 覧覧覧覧覧覧 = 1.84\r\n" );
document.write( "         \r\n" );
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document.write( "This would correcpond to the red line drawn\r\n" );
document.write( "at +1.84\r\n" );
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document.write( "and since the 1.84 value of z is NOT between the two green bars, the mean of 506\r\n" );
document.write( "with a sample that size with that standard deviation, is not close enough to\r\n" );
document.write( "500, and so we would reject the null hypothesis.\r\n" );
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document.write( "Edwin

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