document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192384>Question 1192384</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The weight in pounds of a random sample of 7 babies were recorded before and after<BR>\n" );
document.write( "taking a certain vitamin syrup for a period of one month.<BR>\n" );
document.write( " a) Construct a 95% confidence interval of the difference of the mean weights.<BR>\n" );
document.write( " b) Is there a significant difference in their weights at 0.05 level? <BR>\n" );
document.write( "before 29 22 25 29 26 24 31<BR>\n" );
document.write( "after 39 26 25 35 33 36 32 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #824297 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=824297\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part a)\r<BR>\n" );
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document.write( "Hypothesis:<BR>\n" );
document.write( "H0: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \mu_d = 0\"><BR>\n" );
document.write( "H1: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \mu_d \not=  0\"><BR>\n" );
document.write( "This is a two-tailed test.\r<BR>\n" );
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document.write( "Given Data<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >before</TD><TD >after</TD></tr><tr><TD >29</TD><TD >39</TD></tr><tr><TD >22</TD><TD >26</TD></tr><tr><TD >25</TD><TD >25</TD></tr><tr><TD >29</TD><TD >35</TD></tr><tr><TD >26</TD><TD >33</TD></tr><tr><TD >24</TD><TD >36</TD></tr><tr><TD >31</TD><TD >32</TD></tr>\n" );
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document.write( "For each row separately, subtract the values in the order A-B to form a column of differences (d)<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >A</TD><TD >B</TD><TD >A-B</TD></tr><tr><TD >before</TD><TD >after</TD><TD >d = difference</TD></tr><tr><TD >29</TD><TD >39</TD><TD >-10</TD></tr><tr><TD >22</TD><TD >26</TD><TD >-4</TD></tr><tr><TD >25</TD><TD >25</TD><TD >0</TD></tr><tr><TD >29</TD><TD >35</TD><TD >-6</TD></tr><tr><TD >26</TD><TD >33</TD><TD >-7</TD></tr><tr><TD >24</TD><TD >36</TD><TD >-12</TD></tr><tr><TD >31</TD><TD >32</TD><TD >-1</TD></tr>\n" );
document.write( "</TABLE>Using a calculator, compute the sample mean and sample standard deviation of the list of difference values in that new column.<BR>\n" );
document.write( "You should get <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \overline{d} \approx -5.714286\"> for the sample mean and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large s_d \approx 4.423961\"> for the sample standard deviation.\r<BR>\n" );
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document.write( "The sample size is n = 7 which means the degrees of freedom (df) are<BR>\n" );
document.write( "df = n-1<BR>\n" );
document.write( "df = 7-1<BR>\n" );
document.write( "df = 6<BR>\n" );
document.write( "Refer to a T table like this one<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( "Locate the row that has df = 6<BR>\n" );
document.write( "Locate the column that has 0.05 as the two tails<BR>\n" );
document.write( "Intersecting the row and column yields the approximate value t = 2.447<BR>\n" );
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document.write( "Now let's compute the margin of error.<BR>\n" );
document.write( "E = margin of error\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large E = t*\frac{s_d}{\sqrt{n}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large E \approx 2.447*\frac{4.423961}{\sqrt{7}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large E \approx 4.091629\">\r<BR>\n" );
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document.write( "We can then find the lower bound L<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large L = \overline{d} - E\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large L \approx -5.714286 - 4.091629\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large L \approx -9.805915\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large L \approx -9.81\">\r<BR>\n" );
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document.write( "And the upper bound (U) as well<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large U = \overline{d} + E\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large U \approx -5.714286 + 4.091629\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large U \approx -1.622657\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large U \approx -1.62\">\r<BR>\n" );
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document.write( "The 95% confidence interval in the format (L, U) is roughly (-9.81, -1.62)<BR>\n" );
document.write( "Negative values are valid because it represents the idea that the \"after\" weights are larger than the \"before\" ones.\r<BR>\n" );
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document.write( "Part b)\r<BR>\n" );
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document.write( "The result of part a) is equivalent to saying we are 95% confident that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large -9.81 < \mu_d < -1.62\"> when using the format <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large L < \mu_d < U\"><BR>\n" );
document.write( "In other words, we are 95% confident that the pameter we're after (<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \mu_d\">) is somewhere between -9.81 and -1.62\r<BR>\n" );
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document.write( "Is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \mu_d = 0\"> in this confidence interval? No, it is not. <BR>\n" );
document.write( "So it's unlikely that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \mu_d = 0\"> is the case.<BR>\n" );
document.write( "There appears to be a significant difference in the weights.\r<BR>\n" );
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document.write( "This means we'd reject the null and side with the alternative hypothesis to conclude that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \mu_d \not= 0\"> <BR>\n" );
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document.write( " </SPAN>\n" );
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