document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183940>Question 1183940</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Question 5 <BR>\n" );
document.write( "For each of the following tobs values, determine the highest level of significance associated with the decision taken. \r<BR>\n" );
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document.write( "tobs  		df 	Decision <BR>\n" );
document.write( "(i) 	4.000 	17 	Reject Ha for a two-tailed test <BR>\n" );
document.write( "(ii) 	1.200 	120 	Reject Ha for a one-tailed test <BR>\n" );
document.write( "(iii) 	-2.660 	16 	Reject Ha for both one-tailed and two-tailed test <BR>\n" );
document.write( "(iv) 	-1.586 	60 	Reject Ha for a one-tailed test \r<BR>\n" );
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document.write( "NB. I have seen already answered answers of this same question but it came with only the answers. I want to see step by step process to how the answer was arrived at. And Excel formulas doesn't help too. Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849892 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849892\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to determine the highest level of significance for each t-value:\r<BR>\n" );
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document.write( "The level of significance (alpha, α) is the probability of rejecting the null hypothesis when it is actually true (Type I error).  We find it by looking up the *area in the tails* of the t-distribution that corresponds to our calculated t-statistic (tobs).\r<BR>\n" );
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document.write( "**Understanding the Process**\r<BR>\n" );
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document.write( "1. **Degrees of Freedom (df):** This value is crucial for looking up the correct probability in a t-table or using a statistical calculator.\r<BR>\n" );
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document.write( "2. **One-tailed vs. Two-tailed Test:** This determines how we interpret the t-value and where we look for the probability.\r<BR>\n" );
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document.write( "   * **Two-tailed:** We are interested in differences in *either* direction (greater than or less than).  We split the alpha in half and look at both tails.<BR>\n" );
document.write( "   * **One-tailed:** We are interested in a difference in *only one* direction (either greater than *or* less than).  We look at only one tail.\r<BR>\n" );
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document.write( "3. **Finding Alpha:** We are given a t-value and a decision (reject Hₐ). This means the t-value is *in the rejection region*. We want to find the smallest alpha where this would be true.\r<BR>\n" );
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document.write( "**Calculations**\r<BR>\n" );
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document.write( "Here's the breakdown for each case:\r<BR>\n" );
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document.write( "**(i) tobs = 4.000, df = 17, Two-tailed**\r<BR>\n" );
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document.write( "* Look up 4.000 in a t-table with 17 df.  Since it is two-tailed, we are looking for the area in *both* tails.<BR>\n" );
document.write( "* You'll find that 4.000 is beyond most t-table values, indicating a very small alpha.<BR>\n" );
document.write( "* Using a calculator, the p-value (area in both tails) is approximately 0.0008.<BR>\n" );
document.write( "* Therefore, the highest level of significance is approximately **0.0008** or **0.08%**.\r<BR>\n" );
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document.write( "**(ii) tobs = 1.200, df = 120, One-tailed**\r<BR>\n" );
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document.write( "* Look up 1.200 in a t-table with 120 df. Since it is one-tailed, we are looking at only one tail.<BR>\n" );
document.write( "* Because this is a one-tailed test and we are rejecting Hₐ, it means the test is for greater than.<BR>\n" );
document.write( "* Using a t-table or calculator, the p-value will be approximately 0.1151.<BR>\n" );
document.write( "* The highest level of significance is approximately **0.1151** or **11.51%**.\r<BR>\n" );
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document.write( "**(iii) tobs = -2.660, df = 16, Both One-tailed and Two-tailed**\r<BR>\n" );
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document.write( "* **Two-tailed:** Look up 2.660 (the absolute value) in a t-table with 16 df. Double the value you find because it is two-tailed. The p-value will be approximately 0.0166.<BR>\n" );
document.write( "* **One-tailed:** Look up 2.660 in a t-table with 16 df. The p-value will be approximately 0.0083.<BR>\n" );
document.write( "* Since the decision is to reject Hₐ for both, we take the smaller p-value.<BR>\n" );
document.write( "* The highest level of significance is approximately **0.0083** or **0.83%**.\r<BR>\n" );
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document.write( "**(iv) tobs = -1.586, df = 60, One-tailed**\r<BR>\n" );
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document.write( "* Look up 1.586 (the absolute value) in a t-table with 60 df. Because the test is one-tailed and tobs is negative, it means the test is for less than.<BR>\n" );
document.write( "* The p-value will be approximately 0.0588.<BR>\n" );
document.write( "* The highest level of significance is approximately **0.0588** or **5.88%**.<BR>\n" );
document.write( " </SPAN>\n" );
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