document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206173>Question 1206173</A>: <I><SPAN STYLE=\"word-break: break-all;\"> i have tried answering this problem, i can figure out the critcal values & Z score but my answers don't match up to the choices given...\r<BR>\n" );
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document.write( "It is desired to test H0: μ = 55 against H1: μ < 55 using α = 0.10. The population in question is normally distributed with a standard deviation of 20. A random sample of 64 will be drawn from this population. If μ is really equal to 50, what is the probability that the hypothesis test would lead the investigator to commit a Type II error? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #843447 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=843447\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> two tail 90% confidence interval generates a critical z-score of plus or minus 1.644885.\r<BR>\n" );
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document.write( "the population mean is assumed to be 55<BR>\n" );
document.write( "the population standard deviation is equal to 20.\r<BR>\n" );
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document.write( "the standard error is used.<BR>\n" );
document.write( "standard error = standard deviation / sqrt(sample size).\r<BR>\n" );
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document.write( "sample size is 64.<BR>\n" );
document.write( "standard error = 20 / sqrt(64) = 2.5.\r<BR>\n" );
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document.write( "z-score formula is z = (x-m)/s\r<BR>\n" );
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document.write( "when z = -1.644885, formula becomes -1.644885 = (x-55)/2.5<BR>\n" );
document.write( "solve for x to get x = 50.888\r<BR>\n" );
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document.write( "when z = 1.644885, formula becomes 1.644885 = (x-55)/2.5<BR>\n" );
document.write( "solve for x to get x = 59.112\r<BR>\n" );
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document.write( "90% confidence interval is 50.888 to 59.112\r<BR>\n" );
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document.write( "if we assume that the sample mean was 50, then z-score formula becomes z = (50 - 55)/2.5 = -2.\r<BR>\n" );
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document.write( "since the critical z-score is -1.64485, the results would be significant and the conclusion would be that 55 is not the real mean, and that the real mean was closer to 50.\r<BR>\n" );
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document.write( "a type 1 error is when you reject the null hypothesis but it is actually true.<BR>\n" );
document.write( "a type 2 error is when you reject the alternate hypothesis but it is actually true.\r<BR>\n" );
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document.write( "i'm not exactly sure how to calculate the probability of a type 2 error in this problem, so i declined to answer this part of your problem.\r<BR>\n" );
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document.write( "let me know if what i provided you helped and, if not, where the deficiency lies.\r<BR>\n" );
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document.write( "here's a reference on type 1 and type 2 errors.<BR>\n" );
document.write( "it may help, although it wasn't enough for me to understand fully how to calculate the probability of a type 2 error.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2996198/#:~:text=A%20type%20I%20error%20(false,actually%20false%20in%20the%20population.\" target = \"_blank\">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2996198/#:~:text=A%20type%20I%20error%20(false,actually%20false%20in%20the%20population.</a>\r<BR>\n" );
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document.write( "theo\r<BR>\n" );
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document.write( "P.S.\r<BR>\n" );
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document.write( "i found a calculator online that calculates power and beta of a test.<BR>\n" );
document.write( "it can be found at <a  rel=nofollow HREF= \"https://www.statology.org/type-ii-error-calculator/\" target = \"_blank\">https://www.statology.org/type-ii-error-calculator/</a>\r<BR>\n" );
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document.write( "it told me that beta = .23624.<BR>\n" );
document.write( "power of the test is 1 minus beta.\r<BR>\n" );
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document.write( "here's what the results of my using that calculator looks like.\r<BR>\n" );
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document.write( "i still don't understand how it was calculated, but at least i got an answer.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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