document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1087484>Question 1087484</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A reporter bought hamburgers at randomly selected stores of two different restaurant chains, and had the number of Calories in each hamburger measured. Can the reporter conclude, at alpha = 0.05, that the hamburgers from the two chains have a different number of Calories??\r<BR>\n" );
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document.write( "Chain A sample size 7, sample mean 280 cal, sample standard deviation 21 cal<BR>\n" );
document.write( "Chain B sample size 8, sample mean 315 cal, sample standard deviation 27 cal\r<BR>\n" );
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document.write( "a. No, because the test value –0.23 is inside the noncritical region\r<BR>\n" );
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document.write( "b. Yes, because the test value –0.23 is inside the noncritical region\r<BR>\n" );
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document.write( "c. Yes, because the test value –2.82 is outside the noncritical region\r<BR>\n" );
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document.write( "d. No, because the test value –1.29 is inside the noncritical region </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #701797 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=701797\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">\r<BR>\n" );
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document.write( "Null Hypothesis Using Symbols\r<BR>\n" );
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document.write( "H0: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \mu_1 = \mu_2\">, which is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \mu_1 - \mu_2 = 0\">\r<BR>\n" );
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document.write( "Alternative Hypothesis Using Symbols\r<BR>\n" );
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document.write( "H1: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \mu_1 \neq \mu_2\">, which is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \mu_1 - \mu_2 \neq 0\">\r<BR>\n" );
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document.write( "The symbol <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \mu\"> is the greek letter mu.\r<BR>\n" );
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document.write( "Null Hypothesis in English: The two hamburger chains have burgers with the same number of Calories\r<BR>\n" );
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document.write( "Alternative Hypothesis in English: The two hamburger chains have burgers with a different number of Calories\r<BR>\n" );
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document.write( "This is a two tailed test. We're going to use a two sample unpaired T test to test the hypothesis. \r<BR>\n" );
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document.write( "Given Information:\r<BR>\n" );
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document.write( "For chain A we have <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n_1 = 7\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \overline{x}_1 = 280\">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s_1 = 21\"> (to represent the sample size, sample mean, and sample standard deviation in that order)\r<BR>\n" );
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document.write( "For chain B we are told <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n_2 = 8\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \overline{x}_2 = 315\">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s_2 = 27\"> (again representing the sample size, sample mean, and sample standard deviation in that order but this time we have '2's attached to the variable's instead of '1's to indicate a different label)\r<BR>\n" );
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document.write( "Using the info about the sample sizes and sample standard deviations, let's compute the Standard Error (SE)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE SE = \sqrt{\frac{\left(s_1\right)^2}{n_1}+\frac{\left(s_2\right)^2}{n_2}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE SE = \sqrt{\frac{\left(21\right)^2}{7}+\frac{\left(27\right)^2}{8}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE SE = \sqrt{\frac{441}{7}+\frac{729}{8}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE SE = \sqrt{63+91.125\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE SE = \sqrt{154.125\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE SE \approx 12.4147090179351\">\r<BR>\n" );
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document.write( "With the SE, we can find the T test statistic\r<BR>\n" );
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document.write( "I'll call this variable <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_s\"> (T sub s) to differentiate it from the T critical values later on.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_s = \frac{\overline{x}_1-\overline{x}_2}{SE}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_s \approx \frac{280-315}{12.4147090179351}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_s \approx \frac{-35}{12.4147090179351}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_s \approx -2.81923643554083\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_s \approx -2.82\">\r<BR>\n" );
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document.write( "The T test statistic is approximately -2.82\r<BR>\n" );
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document.write( "The T critical values, which I'll call <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_L\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_R\"> (for the left and right critical values respectively) are approximately\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_L \approx -2.160\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_R \approx 2.160\">\r<BR>\n" );
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document.write( "Use a table or calculator to find these critical values. In my case, I used <a  rel=nofollow HREF=\"http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf\">this table</a> to find the critical values. How? By looking in the df = 13 row (df = n1+n2-2 = 7+8-2 = 15-2 = 13; see note below). Then look in the column that has \"two tails = 0.05\" to find 2.160 in the table. This means that P(-2.160 < T < 2.160) is roughly equal to 0.95 and the area of 0.05 is in the tails (0.025 in each individual tail)\r<BR>\n" );
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document.write( "Recall that alpha = 0.05 is the significance level in this case.\r<BR>\n" );
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document.write( "Note: I'm assuming the population variances are equal. This makes the df value much easier to compute. If they were assumed to be unequal, then we'd have to use a nasty formula to compute the df. I checked both versions of the df and got roughly 12.855898757298 when variances were assumed to be unequal, which is close enough to 13 in my opinion. See <a  rel=nofollow HREF=\"http://www.statsdirect.com/help/parametric_methods/utt.htm\">this page</a> for further reading. \r<BR>\n" );
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document.write( "To recap so far, we found the following\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_s \approx -2.82\"> which is the T statistic\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_L \approx -2.160\"> which is the left critical T value\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_R \approx 2.160\"> which is the right critical T value\r<BR>\n" );
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document.write( "It's clear that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_s\"> is NOT between <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_L\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_R\">. So we are not in the noncritical region. The test statistic is in the critical region. \r<BR>\n" );
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document.write( "This visual shows <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_s\"> insinde the red rejection or critical region and outside of the blue noncritical region. The critical region is formed by the fenceposts <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_L\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T_R\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/44udOk0.png\"><BR>\n" );
document.write( "(Image generated by <a  rel=nofollow HREF= \"https://www.geogebra.org/home?ggbLang=en\">GeoGebra</a> which is free graphing software)\r<BR>\n" );
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document.write( "The decision is therefore to reject the null hypothesis (H0). So we accept the alternate hypothesis (H1).\r<BR>\n" );
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document.write( "The conclusion, translated to common english, is that the burgers do have different calorie counts between the two burger chains. \r<BR>\n" );
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document.write( "So the short answer is \"yes, the reporter can conclude that the two chains have different number of calories (at alpha = 0.05 significance level)\"\r<BR>\n" );
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document.write( "The final answer is choice C.</font> </SPAN>\n" );
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